Daily Papers

Detecting and localizing leaks in water distribution network systems is an important topic with direct environmental, economic, and social impact. Our paper is the first to explore the use of factor graph optimization techniques for leak localization in water distribution networks, enabling us to perform sensor fusion between pressure and demand sensor readings and to estimate the network's temporal and structural state evolution across all network nodes. The methodology introduces specific water network factors and proposes a new architecture composed of two factor graphs: a leak-free state estimation factor graph and a leak localization factor graph. When a new sensor reading is obtained, unlike Kalman and other interpolation-based methods, which estimate only the current network state, factor graphs update both current and past states. Results on Modena, L-TOWN and synthetic networks show that factor graphs are much faster than nonlinear Kalman-based alternatives such as the UKF, while also providing improvements in localization compared to state-of-the-art estimation-localization approaches. Implementation and benchmarks are available at https://github.com/pirofti/FGLL.

This paper presents the first photo-realistic LiDAR-Inertial-Camera Gaussian Splatting SLAM system that simultaneously addresses visual quality, geometric accuracy, and real-time performance. The proposed method performs robust and accurate pose estimation within a continuous-time trajectory optimization framework, while incrementally reconstructing a 3D Gaussian map using camera and LiDAR data, all in real time. The resulting map enables high-quality, real-time novel view rendering of both RGB images and depth maps. To effectively address under-reconstruction in regions not covered by the LiDAR, we employ a lightweight zero-shot depth model that synergistically combines RGB appearance cues with sparse LiDAR measurements to generate dense depth maps. The depth completion enables reliable Gaussian initialization in LiDAR-blind areas, significantly improving system applicability for sparse LiDAR sensors. To enhance geometric accuracy, we use sparse but precise LiDAR depths to supervise Gaussian map optimization and accelerate it with carefully designed CUDA-accelerated strategies. Furthermore, we explore how the incrementally reconstructed Gaussian map can improve the robustness of odometry. By tightly incorporating photometric constraints from the Gaussian map into the continuous-time factor graph optimization, we demonstrate improved pose estimation under LiDAR degradation scenarios. We also showcase downstream applications via extending our elaborate system, including video frame interpolation and fast 3D mesh extraction. To support rigorous evaluation, we construct a dedicated LiDAR-Inertial-Camera dataset featuring ground-truth poses, depth maps, and extrapolated trajectories for assessing out-of-sequence novel view synthesis. Both the dataset and code will be made publicly available on project page https://xingxingzuo.github.io/gaussian_lic2.

Fair graph clustering seeks partitions that respect network structure while maintaining proportional representation across sensitive groups, with applications spanning community detection, team formation, resource allocation, and social network analysis. Many existing approaches enforce rigid constraints or rely on multi-stage pipelines (e.g., spectral embedding followed by k-means), limiting trade-off control, interpretability, and scalability. We introduce DFNMF, an end-to-end deep nonnegative tri-factorization tailored to graphs that directly optimizes cluster assignments with a soft statistical-parity regularizer. A single parameter lambda tunes the fairness--utility balance, while nonnegativity yields parts-based factors and transparent soft memberships. The optimization uses sparse-friendly alternating updates and scales near-linearly with the number of edges. Across synthetic and real networks, DFNMF achieves substantially higher group balance at comparable modularity, often dominating state-of-the-art baselines on the Pareto front. The code is available at https://github.com/SiamakGhodsi/DFNMF.git.

Image-based surface reconstruction and characterization is crucial for missions to small celestial bodies, as it informs mission planning, navigation, and scientific analysis. However, current state-of-the-practice methods, such as stereophotoclinometry (SPC), rely heavily on human-in-the-loop verification and high-fidelity a priori information. This paper proposes Photoclinometry-from-Motion (PhoMo), a novel framework that incorporates photoclinometry techniques into a keypoint-based structure-from-motion (SfM) system to estimate the surface normal and albedo at detected landmarks to improve autonomous surface and shape characterization of small celestial bodies from in-situ imagery. In contrast to SPC, we forego the expensive maplet estimation step and instead use dense keypoint measurements and correspondences from an autonomous keypoint detection and matching method based on deep learning. Moreover, we develop a factor graph-based approach allowing for simultaneous optimization of the spacecraft's pose, landmark positions, Sun-relative direction, and surface normals and albedos via fusion of Sun vector measurements and image keypoint measurements. The proposed framework is validated on real imagery taken by the Dawn mission to the asteroid 4 Vesta and the minor planet 1 Ceres and compared against an SPC reconstruction, where we demonstrate superior rendering performance compared to an SPC solution and precise alignment to a stereophotogrammetry (SPG) solution without relying on any a priori camera pose and topography information or humans-in-the-loop.

Graph Neural Networks (GNNs) have displayed considerable promise in graph representation learning across various applications. The core learning process requires the initialization of model weight matrices within each GNN layer, which is typically accomplished via classic initialization methods such as Xavier initialization. However, these methods were originally motivated to stabilize the variance of hidden embeddings and gradients across layers of Feedforward Neural Networks (FNNs) and Convolutional Neural Networks (CNNs) to avoid vanishing gradients and maintain steady information flow. In contrast, within the GNN context classical initializations disregard the impact of the input graph structure and message passing on variance. In this paper, we analyze the variance of forward and backward propagation across GNN layers and show that the variance instability of GNN initializations comes from the combined effect of the activation function, hidden dimension, graph structure and message passing. To better account for these influence factors, we propose a new initialization method for Variance Instability Reduction within GNN Optimization (Virgo), which naturally tends to equate forward and backward variances across successive layers. We conduct comprehensive experiments on 15 datasets to show that Virgo can lead to superior model performance and more stable variance at initialization on node classification, link prediction and graph classification tasks. Codes are in https://github.com/LspongebobJH/virgo_icml2023.

Various human-designed prompt engineering techniques have been proposed to improve problem solvers based on Large Language Models (LLMs), yielding many disparate code bases. We unify these approaches by describing LLM-based agents as computational graphs. The nodes implement functions to process multimodal data or query LLMs, and the edges describe the information flow between operations. Graphs can be recursively combined into larger composite graphs representing hierarchies of inter-agent collaboration (where edges connect operations of different agents). Our novel automatic graph optimizers (1) refine node-level LLM prompts (node optimization) and (2) improve agent orchestration by changing graph connectivity (edge optimization). Experiments demonstrate that our framework can be used to efficiently develop, integrate, and automatically improve various LLM agents. The code can be found at https://github.com/metauto-ai/gptswarm.

Large language models (LLMs) have achieved impressive success across several fields, but their proficiency in understanding and resolving complex graph problems is less explored. To bridge this gap, we introduce GraphInstruct, a novel and comprehensive instruction-tuning dataset designed to equip language models with the ability to tackle a broad spectrum of graph problems using explicit reasoning paths. Utilizing GraphInstruct, we build GraphWiz, an open-source language model capable of resolving various graph problem types while generating clear reasoning processes. To enhance the model's capability and reliability, we incorporate the Direct Preference Optimization (DPO) framework into the graph problem-solving context. The enhanced model, GraphWiz-DPO, achieves an average accuracy of 65% across nine tasks with different complexity levels, surpassing GPT-4 which has an average accuracy of 43.8%. Moreover, our research delves into the delicate balance between training data volume and model performance, highlighting the potential for overfitting with increased data. We also explore the transferability of the model's reasoning ability across different graph tasks, indicating the model's adaptability and practical application potential. Our investigation offers a new blueprint and valuable insights for developing LLMs specialized in graph reasoning and problem-solving.

Graphs can be used to represent and reason about systems and a variety of metrics have been devised to quantify their global characteristics. However, little is currently known about how to construct a graph or improve an existing one given a target objective. In this work, we formulate the construction of a graph as a decision-making process in which a central agent creates topologies by trial and error and receives rewards proportional to the value of the target objective. By means of this conceptual framework, we propose an algorithm based on reinforcement learning and graph neural networks to learn graph construction and improvement strategies. Our core case study focuses on robustness to failures and attacks, a property relevant for the infrastructure and communication networks that power modern society. Experiments on synthetic and real-world graphs show that this approach can outperform existing methods while being cheaper to evaluate. It also allows generalization to out-of-sample graphs, as well as to larger out-of-distribution graphs in some cases. The approach is applicable to the optimization of other global structural properties of graphs.

Graph In-Context Learning, with the ability to adapt pre-trained graph models to novel and diverse downstream graphs without updating any parameters, has gained much attention in the community. The key to graph in-context learning is to perform downstream graphs conditioned on chosen prompt examples. Existing methods randomly select subgraphs or edges as prompts, leading to noisy graph prompts and inferior model performance. Additionally, due to the gap between pre-training and testing graphs, when the number of classes in the testing graphs is much greater than that in the training, the in-context learning ability will also significantly deteriorate. To tackle the aforementioned challenges, we develop a multi-stage adaptive prompt optimization method GraphPrompter, which optimizes the entire process of generating, selecting, and using graph prompts for better in-context learning capabilities. Firstly, Prompt Generator introduces a reconstruction layer to highlight the most informative edges and reduce irrelevant noise for graph prompt construction. Furthermore, in the selection stage, Prompt Selector employs the k-nearest neighbors algorithm and pre-trained selection layers to dynamically choose appropriate samples and minimize the influence of irrelevant prompts. Finally, we leverage a Prompt Augmenter with a cache replacement strategy to enhance the generalization capability of the pre-trained model on new datasets. Extensive experiments show that GraphPrompter effectively enhances the in-context learning ability of graph models. On average across all the settings, our approach surpasses the state-of-the-art baselines by over 8%. Our code is released at https://github.com/karin0018/GraphPrompter.

Optimization problems over dynamic networks have been extensively studied and widely used in the past decades to formulate numerous real-world problems. However, (1) traditional optimization-based approaches do not scale to large networks, and (2) the design of good heuristics or approximation algorithms often requires significant manual trial-and-error. In this work, we argue that data-driven strategies can automate this process and learn efficient algorithms without compromising optimality. To do so, we present network control problems through the lens of reinforcement learning and propose a graph network-based framework to handle a broad class of problems. Instead of naively computing actions over high-dimensional graph elements, e.g., edges, we propose a bi-level formulation where we (1) specify a desired next state via RL, and (2) solve a convex program to best achieve it, leading to drastically improved scalability and performance. We further highlight a collection of desirable features to system designers, investigate design decisions, and present experiments on real-world control problems showing the utility, scalability, and flexibility of our framework.

Foundation models have emerged as critical components in a variety of artificial intelligence applications, and showcase significant success in natural language processing and several other domains. Meanwhile, the field of graph machine learning is witnessing a paradigm transition from shallow methods to more sophisticated deep learning approaches. The capabilities of foundation models to generalize and adapt motivate graph machine learning researchers to discuss the potential of developing a new graph learning paradigm. This paradigm envisions models that are pre-trained on extensive graph data and can be adapted for various graph tasks. Despite this burgeoning interest, there is a noticeable lack of clear definitions and systematic analyses pertaining to this new domain. To this end, this article introduces the concept of Graph Foundation Models (GFMs), and offers an exhaustive explanation of their key characteristics and underlying technologies. We proceed to classify the existing work related to GFMs into three distinct categories, based on their dependence on graph neural networks and large language models. In addition to providing a thorough review of the current state of GFMs, this article also outlooks potential avenues for future research in this rapidly evolving domain.

Integer Linear Programs (ILPs) are powerful tools for modeling and solving a large number of combinatorial optimization problems. Recently, it has been shown that Large Neighborhood Search (LNS), as a heuristic algorithm, can find high quality solutions to ILPs faster than Branch and Bound. However, how to find the right heuristics to maximize the performance of LNS remains an open problem. In this paper, we propose a novel approach, CL-LNS, that delivers state-of-the-art anytime performance on several ILP benchmarks measured by metrics including the primal gap, the primal integral, survival rates and the best performing rate. Specifically, CL-LNS collects positive and negative solution samples from an expert heuristic that is slow to compute and learns a new one with a contrastive loss. We use graph attention networks and a richer set of features to further improve its performance.

We introduce fast algorithms for correlation clustering with respect to the Min Max objective that provide constant factor approximations on complete graphs. Our algorithms are the first purely combinatorial approximation algorithms for this problem. We construct a novel semi-metric on the set of vertices, which we call the correlation metric, that indicates to our clustering algorithms whether pairs of nodes should be in the same cluster. The paper demonstrates empirically that, compared to prior work, our algorithms sacrifice little in the objective quality to obtain significantly better run-time. Moreover, our algorithms scale to larger networks that are effectively intractable for known algorithms.

The higher-order correlation clustering problem is an expressive model, and recently, local search heuristics have been proposed for several applications. Certifying optimality, however, is NP-hard and practically hampered already by the complexity of the problem statement. Here, we focus on establishing partial optimality conditions for the special case of complete graphs and cubic objective functions. In addition, we define and implement algorithms for testing these conditions and examine their effect numerically, on two datasets.

Graph Neural Networks (GNNs) have greatly advanced the semi-supervised node classification task on graphs. The majority of existing GNNs are trained in an end-to-end manner that can be viewed as tackling a bi-level optimization problem. This process is often inefficient in computation and memory usage. In this work, we propose a new optimization framework for semi-supervised learning on graphs. The proposed framework can be conveniently solved by the alternating optimization algorithms, resulting in significantly improved efficiency. Extensive experiments demonstrate that the proposed method can achieve comparable or better performance with state-of-the-art baselines while it has significantly better computation and memory efficiency.

The deployment and training of neural networks on edge computing devices pose many challenges. The low memory nature of edge devices is often one of the biggest limiting factors encountered in the deployment of large neural network models. Tensor rematerialization or recompute is a way to address high memory requirements for neural network training and inference. In this paper we consider the problem of execution time minimization of compute graphs subject to a memory budget. In particular, we develop a new constraint programming formulation called Moccasin with only O(n) integer variables, where n is the number of nodes in the compute graph. This is a significant improvement over the works in the recent literature that propose formulations with O(n^2) Boolean variables. We present numerical studies that show that our approach is up to an order of magnitude faster than recent work especially for large-scale graphs.

We propose algorithms for approximate filtering and smoothing in high-dimensional Factorial hidden Markov models. The approximation involves discarding, in a principled way, likelihood factors according to a notion of locality in a factor graph associated with the emission distribution. This allows the exponential-in-dimension cost of exact filtering and smoothing to be avoided. We prove that the approximation accuracy, measured in a local total variation norm, is "dimension-free" in the sense that as the overall dimension of the model increases the error bounds we derive do not necessarily degrade. A key step in the analysis is to quantify the error introduced by localizing the likelihood function in a Bayes' rule update. The factorial structure of the likelihood function which we exploit arises naturally when data have known spatial or network structure. We demonstrate the new algorithms on synthetic examples and a London Underground passenger flow problem, where the factor graph is effectively given by the train network.

Graph neural networks (GNNs) have been demonstrated to be powerful in modeling graph-structured data. However, training GNNs usually requires abundant task-specific labeled data, which is often arduously expensive to obtain. One effective way to reduce the labeling effort is to pre-train an expressive GNN model on unlabeled data with self-supervision and then transfer the learned model to downstream tasks with only a few labels. In this paper, we present the GPT-GNN framework to initialize GNNs by generative pre-training. GPT-GNN introduces a self-supervised attributed graph generation task to pre-train a GNN so that it can capture the structural and semantic properties of the graph. We factorize the likelihood of the graph generation into two components: 1) Attribute Generation and 2) Edge Generation. By modeling both components, GPT-GNN captures the inherent dependency between node attributes and graph structure during the generative process. Comprehensive experiments on the billion-scale Open Academic Graph and Amazon recommendation data demonstrate that GPT-GNN significantly outperforms state-of-the-art GNN models without pre-training by up to 9.1% across various downstream tasks.

Estimating the structure of directed acyclic graphs (DAGs, also known as Bayesian networks) is a challenging problem since the search space of DAGs is combinatorial and scales superexponentially with the number of nodes. Existing approaches rely on various local heuristics for enforcing the acyclicity constraint. In this paper, we introduce a fundamentally different strategy: We formulate the structure learning problem as a purely continuous optimization problem over real matrices that avoids this combinatorial constraint entirely. This is achieved by a novel characterization of acyclicity that is not only smooth but also exact. The resulting problem can be efficiently solved by standard numerical algorithms, which also makes implementation effortless. The proposed method outperforms existing ones, without imposing any structural assumptions on the graph such as bounded treewidth or in-degree. Code implementing the proposed algorithm is open-source and publicly available at https://github.com/xunzheng/notears.

The structure learning problem consists of fitting data generated by a Directed Acyclic Graph (DAG) to correctly reconstruct its arcs. In this context, differentiable approaches constrain or regularize the optimization problem using a continuous relaxation of the acyclicity property. The computational cost of evaluating graph acyclicity is cubic on the number of nodes and significantly affects scalability. In this paper we introduce COSMO, a constraint-free continuous optimization scheme for acyclic structure learning. At the core of our method, we define a differentiable approximation of an orientation matrix parameterized by a single priority vector. Differently from previous work, our parameterization fits a smooth orientation matrix and the resulting acyclic adjacency matrix without evaluating acyclicity at any step. Despite the absence of explicit constraints, we prove that COSMO always converges to an acyclic solution. In addition to being asymptotically faster, our empirical analysis highlights how COSMO performance on graph reconstruction compares favorably with competing structure learning methods.

In recent years, Graph Neural Networks (GNNs) have made significant advances in processing structured data. However, most of them primarily adopted a model-centric approach, which simplifies graphs by converting them into undirected formats and emphasizes model designs. This approach is inherently limited in real-world applications due to the unavoidable information loss in simple undirected graphs and the model optimization challenges that arise when exceeding the upper bounds of this sub-optimal data representational capacity. As a result, there has been a shift toward data-centric methods that prioritize improving graph quality and representation. Specifically, various types of graphs can be derived from naturally structured data, including heterogeneous graphs, hypergraphs, and directed graphs. Among these, directed graphs offer distinct advantages in topological systems by modeling causal relationships, and directed GNNs have been extensively studied in recent years. However, a comprehensive survey of this emerging topic is still lacking. Therefore, we aim to provide a comprehensive review of directed graph learning, with a particular focus on a data-centric perspective. Specifically, we first introduce a novel taxonomy for existing studies. Subsequently, we re-examine these methods from the data-centric perspective, with an emphasis on understanding and improving data representation. It demonstrates that a deep understanding of directed graphs and their quality plays a crucial role in model performance. Additionally, we explore the diverse applications of directed GNNs across 10+ domains, highlighting their broad applicability. Finally, we identify key opportunities and challenges within the field, offering insights that can guide future research and development in directed graph learning.

We propose constrained causal Bayesian optimization (cCBO), an approach for finding interventions in a known causal graph that optimize a target variable under some constraints. cCBO first reduces the search space by exploiting the graph structure and, if available, an observational dataset; and then solves the restricted optimization problem by modelling target and constraint quantities using Gaussian processes and by sequentially selecting interventions via a constrained expected improvement acquisition function. We propose different surrogate models that enable to integrate observational and interventional data while capturing correlation among effects with increasing levels of sophistication. We evaluate cCBO on artificial and real-world causal graphs showing successful trade off between fast convergence and percentage of feasible interventions.

This work studies a central extremal graph theory problem inspired by a 1975 conjecture of Erdos, which aims to find graphs with a given size (number of nodes) that maximize the number of edges without having 3- or 4-cycles. We formulate this problem as a sequential decision-making problem and compare AlphaZero, a neural network-guided tree search, with tabu search, a heuristic local search method. Using either method, by introducing a curriculum -- jump-starting the search for larger graphs using good graphs found at smaller sizes -- we improve the state-of-the-art lower bounds for several sizes. We also propose a flexible graph-generation environment and a permutation-invariant network architecture for learning to search in the space of graphs.

Mixed-Integer Linear Programming (MILP) is a cornerstone of combinatorial optimization, yet solving large-scale instances remains a significant computational challenge. Recently, Graph Neural Networks (GNNs) have shown promise in accelerating MILP solvers by predicting high-quality solutions. However, we identify that existing methods misalign with the intrinsic structure of MILP problems at two levels. At the leaning objective level, the Binary Cross-Entropy (BCE) loss treats variables independently, neglecting their relative priority and yielding plausible logits. At the model architecture level, standard GNN message passing inherently smooths the representations across variables, missing the natural competitive relationships within constraints. To address these challenges, we propose CoCo-MILP, which explicitly models inter-variable Contrast and intra-constraint Competition for advanced MILP solution prediction. At the objective level, CoCo-MILP introduces the Inter-Variable Contrastive Loss (VCL), which explicitly maximizes the embedding margin between variables assigned one versus zero. At the architectural level, we design an Intra-Constraint Competitive GNN layer that, instead of homogenizing features, learns to differentiate representations of competing variables within a constraint, capturing their exclusionary nature. Experimental results on standard benchmarks demonstrate that CoCo-MILP significantly outperforms existing learning-based approaches, reducing the solution gap by up to 68.12% compared to traditional solvers. Our code is available at https://github.com/happypu326/CoCo-MILP.

Combinatorial optimization finds an optimal solution within a discrete set of variables and constraints. The field has seen tremendous progress both in research and industry. With the success of deep learning in the past decade, a recent trend in combinatorial optimization has been to improve state-of-the-art combinatorial optimization solvers by replacing key heuristic components with machine learning (ML) models. In this paper, we investigate two essential aspects of machine learning algorithms for combinatorial optimization: temporal characteristics and attention. We argue that for the task of variable selection in the branch-and-bound (B&B) algorithm, incorporating the temporal information as well as the bipartite graph attention improves the solver's performance. We support our claims with intuitions and numerical results over several standard datasets used in the literature and competitions. Code is available at: https://developer.huaweicloud.com/develop/aigallery/notebook/detail?id=047c6cf2-8463-40d7-b92f-7b2ca998e935

Foundation Models (FMs) serve as a general class for the development of artificial intelligence systems, offering broad potential for generalization across a spectrum of downstream tasks. Despite extensive research into self-supervised learning as the cornerstone of FMs, several outstanding issues persist in Graph Foundation Models that rely on graph self-supervised learning, namely: 1) Homogenization. The extent of generalization capability on downstream tasks remains unclear. 2) Scalability. It is unknown how effectively these models can scale to large datasets. 3) Efficiency. The training time and memory usage of these models require evaluation. 4) Training Stop Criteria. Determining the optimal stopping strategy for pre-training across multiple tasks to maximize performance on downstream tasks. To address these questions, we have constructed a rigorous benchmark that thoroughly analyzes and studies the generalization and scalability of self-supervised Graph Neural Network (GNN) models. Regarding generalization, we have implemented and compared the performance of various self-supervised GNN models, trained to generate node representations, across tasks such as node classification, link prediction, and node clustering. For scalability, we have compared the performance of various models after training using full-batch and mini-batch strategies. Additionally, we have assessed the training efficiency of these models by conducting experiments to test their GPU memory usage and throughput. Through these experiments, we aim to provide insights to motivate future research. The code for this benchmark is publicly available at https://github.com/NYUSHCS/GraphFM.

Personalized PageRank (PPR) has enormous applications, such as link prediction and recommendation systems for social networks, which often require the fully PPR to be known. Besides, most of real-life graphs are edge-weighted, e.g., the interaction between users on the Facebook network. However, it is computationally difficult to compute the fully PPR, especially on large graphs, not to mention that most existing approaches do not consider the weights of edges. In particular, the existing approach cannot handle graphs with billion edges on a moderate-size cluster. To address this problem, this paper presents a novel study on the computation of fully edge-weighted PPR on large graphs using the distributed computing framework. Specifically, we employ the Monte Carlo approximation that performs a large number of random walks from each node of the graph, and exploits the parallel pipeline framework to reduce the overall running time of the fully PPR. Based on that, we develop several optimization techniques which (i) alleviate the issue of large nodes that could explode the memory space, (ii) pre-compute short walks for small nodes that largely speedup the computation of random walks, and (iii) optimize the amount of random walks to compute in each pipeline that significantly reduces the overhead. With extensive experiments on a variety of real-life graph datasets, we demonstrate that our solution is several orders of magnitude faster than the state-of-the-arts, and meanwhile, largely outperforms the baseline algorithms in terms of accuracy.

Recently, an intriguing class of non-convex optimization problems has emerged in the context of learning directed acyclic graphs (DAGs). These problems involve minimizing a given loss or score function, subject to a non-convex continuous constraint that penalizes the presence of cycles in a graph. In this work, we delve into the optimization challenges associated with this class of non-convex programs. To address these challenges, we propose a bi-level algorithm that leverages the non-convex constraint in a novel way. The outer level of the algorithm optimizes over topological orders by iteratively swapping pairs of nodes within the topological order of a DAG. A key innovation of our approach is the development of an effective method for generating a set of candidate swapping pairs for each iteration. At the inner level, given a topological order, we utilize off-the-shelf solvers that can handle linear constraints. The key advantage of our proposed algorithm is that it is guaranteed to find a local minimum or a KKT point under weaker conditions compared to previous work and finds solutions with lower scores. Extensive experiments demonstrate that our method outperforms state-of-the-art approaches in terms of achieving a better score. Additionally, our method can also be used as a post-processing algorithm to significantly improve the score of other algorithms. Code implementing the proposed method is available at https://github.com/duntrain/topo.

Graph-based collaborative filtering methods have prevailing performance for recommender systems since they can capture high-order information between users and items, in which the graphs are constructed from the observed user-item interactions that might miss links or contain spurious positive interactions in industrial scenarios. The Bayesian Graph Neural Network framework approaches this issue with generative models for the interaction graphs. The critical problem is to devise a proper family of graph generative models tailored to recommender systems. We propose an efficient generative model that jointly considers the preferences of users, the concurrence of items and some important graph structure information. Experiments on four popular benchmark datasets demonstrate the effectiveness of our proposed graph generative methods for recommender systems.

Graph generation is a critical task in numerous domains, including molecular design and social network analysis, due to its ability to model complex relationships and structured data. While most modern graph generative models utilize adjacency matrix representations, this work revisits an alternative approach that represents graphs as sequences of node set and edge set. We advocate for this approach due to its efficient encoding of graphs and propose a novel representation. Based on this representation, we introduce the Graph Generative Pre-trained Transformer (G2PT), an auto-regressive model that learns graph structures via next-token prediction. To further exploit G2PT's capabilities as a general-purpose foundation model, we explore fine-tuning strategies for two downstream applications: goal-oriented generation and graph property prediction. We conduct extensive experiments across multiple datasets. Results indicate that G2PT achieves superior generative performance on both generic graph and molecule datasets. Furthermore, G2PT exhibits strong adaptability and versatility in downstream tasks from molecular design to property prediction.

Many graph algorithms can be viewed as sets of rules that are iteratively applied, with the number of iterations dependent on the size and complexity of the input graph. Existing machine learning architectures often struggle to represent these algorithmic decisions as discrete state transitions. Therefore, we propose a novel framework: GraphFSA (Graph Finite State Automaton). GraphFSA is designed to learn a finite state automaton that runs on each node of a given graph. We test GraphFSA on cellular automata problems, showcasing its abilities in a straightforward algorithmic setting. For a comprehensive empirical evaluation of our framework, we create a diverse range of synthetic problems. As our main application, we then focus on learning more elaborate graph algorithms. Our findings suggest that GraphFSA exhibits strong generalization and extrapolation abilities, presenting an alternative approach to represent these algorithms.

This paper studies the problem of embedding very large information networks into low-dimensional vector spaces, which is useful in many tasks such as visualization, node classification, and link prediction. Most existing graph embedding methods do not scale for real world information networks which usually contain millions of nodes. In this paper, we propose a novel network embedding method called the "LINE," which is suitable for arbitrary types of information networks: undirected, directed, and/or weighted. The method optimizes a carefully designed objective function that preserves both the local and global network structures. An edge-sampling algorithm is proposed that addresses the limitation of the classical stochastic gradient descent and improves both the effectiveness and the efficiency of the inference. Empirical experiments prove the effectiveness of the LINE on a variety of real-world information networks, including language networks, social networks, and citation networks. The algorithm is very efficient, which is able to learn the embedding of a network with millions of vertices and billions of edges in a few hours on a typical single machine. The source code of the LINE is available online.

Bayesian Optimization (BO) is a foundational strategy in the field of engineering design optimization for efficiently handling black-box functions with many constraints and expensive evaluations. This paper introduces a fast and accurate BO framework that leverages Pre-trained Transformers for Bayesian Optimization (PFN4sBO) to address constrained optimization problems in engineering. Unlike traditional BO methods that rely heavily on Gaussian Processes (GPs), our approach utilizes Prior-data Fitted Networks (PFNs), a type of pre-trained transformer, to infer constraints and optimal solutions without requiring any iterative retraining. We demonstrate the effectiveness of PFN-based BO through a comprehensive benchmark consisting of fifteen test problems, encompassing synthetic, structural, and engineering design challenges. Our findings reveal that PFN-based BO significantly outperforms Constrained Expected Improvement and Penalty-based GP methods by an order of magnitude in speed while also outperforming them in accuracy in identifying feasible, optimal solutions. This work showcases the potential of integrating machine learning with optimization techniques in solving complex engineering challenges, heralding a significant leap forward for optimization methodologies, opening up the path to using PFN-based BO to solve other challenging problems, such as enabling user-guided interactive BO, adaptive experiment design, or multi-objective design optimization. Additionally, we establish a benchmark for evaluating BO algorithms in engineering design, offering a robust platform for future research and development in the field. This benchmark framework for evaluating new BO algorithms in engineering design will be published at https://github.com/rosenyu304/BOEngineeringBenchmark.

Despite the widespread application of latent factor analysis, existing methods suffer from the following weaknesses: requiring the number of factors to be known, lack of theoretical guarantees for learning the model structure, and nonidentifiability of the parameters due to rotation invariance properties of the likelihood. We address these concerns by proposing a fast correlation thresholding (CT) algorithm that simultaneously learns the number of latent factors and a rotationally identifiable model structure. Our novel approach translates this structure learning problem into the search for so-called independent maximal cliques in a thresholded correlation graph that can be easily constructed from the observed data. Our clique analysis technique scales well up to thousands of variables, while competing methods are not applicable in a reasonable amount of running time. We establish a finite-sample error bound and high-dimensional consistency for the structure learning of our method. Through a series of simulation studies and a real data example, we show that the CT algorithm is an accurate method for learning the structure of factor analysis models and is robust to violations of its assumptions.

Aiming at better representing multivariate relationships, this paper investigates a motif dimensional framework for higher-order graph learning. The graph learning effectiveness can be improved through OFFER. The proposed framework mainly aims at accelerating and improving higher-order graph learning results. We apply the acceleration procedure from the dimensional of network motifs. Specifically, the refined degree for nodes and edges are conducted in two stages: (1) employ motif degree of nodes to refine the adjacency matrix of the network; and (2) employ motif degree of edges to refine the transition probability matrix in the learning process. In order to assess the efficiency of the proposed framework, four popular network representation algorithms are modified and examined. By evaluating the performance of OFFER, both link prediction results and clustering results demonstrate that the graph representation learning algorithms enhanced with OFFER consistently outperform the original algorithms with higher efficiency.

This monograph presents the main complexity theorems in convex optimization and their corresponding algorithms. Starting from the fundamental theory of black-box optimization, the material progresses towards recent advances in structural optimization and stochastic optimization. Our presentation of black-box optimization, strongly influenced by Nesterov's seminal book and Nemirovski's lecture notes, includes the analysis of cutting plane methods, as well as (accelerated) gradient descent schemes. We also pay special attention to non-Euclidean settings (relevant algorithms include Frank-Wolfe, mirror descent, and dual averaging) and discuss their relevance in machine learning. We provide a gentle introduction to structural optimization with FISTA (to optimize a sum of a smooth and a simple non-smooth term), saddle-point mirror prox (Nemirovski's alternative to Nesterov's smoothing), and a concise description of interior point methods. In stochastic optimization we discuss stochastic gradient descent, mini-batches, random coordinate descent, and sublinear algorithms. We also briefly touch upon convex relaxation of combinatorial problems and the use of randomness to round solutions, as well as random walks based methods.

Graph Neural Networks (GNNs) have demonstrated impressive performance on task-specific benchmarks, yet their ability to generalize across diverse domains and tasks remains limited. Existing approaches often struggle with negative transfer, scalability issues, and high adaptation costs. To address these challenges, we propose GMoPE (Graph Mixture of Prompt-Experts), a novel framework that seamlessly integrates the Mixture-of-Experts (MoE) architecture with prompt-based learning for graphs. GMoPE leverages expert-specific prompt vectors and structure-aware MoE routing to enable each expert to specialize in distinct subdomains and dynamically contribute to predictions. To promote diversity and prevent expert collapse, we introduce a soft orthogonality constraint across prompt vectors, encouraging expert specialization and facilitating a more balanced expert utilization. Additionally, we adopt a prompt-only fine-tuning strategy that significantly reduces spatiotemporal complexity during transfer. We validate GMoPE through extensive experiments under various pretraining strategies and multiple downstream tasks. Results show that GMoPE consistently outperforms state-of-the-art baselines and achieves performance comparable to full parameter fine-tuning-while requiring only a fraction of the adaptation overhead. Our work provides a principled and scalable framework for advancing generalizable and efficient graph foundation models.

Generating graph-structured data requires learning the underlying distribution of graphs. Yet, this is a challenging problem, and the previous graph generative methods either fail to capture the permutation-invariance property of graphs or cannot sufficiently model the complex dependency between nodes and edges, which is crucial for generating real-world graphs such as molecules. To overcome such limitations, we propose a novel score-based generative model for graphs with a continuous-time framework. Specifically, we propose a new graph diffusion process that models the joint distribution of the nodes and edges through a system of stochastic differential equations (SDEs). Then, we derive novel score matching objectives tailored for the proposed diffusion process to estimate the gradient of the joint log-density with respect to each component, and introduce a new solver for the system of SDEs to efficiently sample from the reverse diffusion process. We validate our graph generation method on diverse datasets, on which it either achieves significantly superior or competitive performance to the baselines. Further analysis shows that our method is able to generate molecules that lie close to the training distribution yet do not violate the chemical valency rule, demonstrating the effectiveness of the system of SDEs in modeling the node-edge relationships. Our code is available at https://github.com/harryjo97/GDSS.

We introduce Mirage, the first multi-level superoptimizer for tensor programs. A key idea in Mirage is μGraphs, a uniform representation of tensor programs at the kernel, thread block, and thread levels of the GPU compute hierarchy. μGraphs enable Mirage to discover novel optimizations that combine algebraic transformations, schedule transformations, and generation of new custom kernels. To navigate the large search space, Mirage introduces a pruning technique based on abstraction that significantly reduces the search space and provides a certain optimality guarantee. To ensure that the optimized μGraph is equivalent to the input program, Mirage introduces a probabilistic equivalence verification procedure with strong theoretical guarantees. Our evaluation shows that Mirage outperforms existing approaches by up to 3.3times even for DNNs that are widely used and heavily optimized. Mirage is publicly available at https://github.com/mirage-project/mirage.

In this paper, we propose the first framework that enables solving graph learning tasks of all levels (node, edge and graph) and all types (generation, regression and classification) with one model. We first propose Latent Graph Diffusion (LGD), a generative model that can generate node, edge, and graph-level features of all categories simultaneously. We achieve this goal by embedding the graph structures and features into a latent space leveraging a powerful encoder which can also be decoded, then training a diffusion model in the latent space. LGD is also capable of conditional generation through a specifically designed cross-attention mechanism. Then we formulate prediction tasks including regression and classification as (conditional) generation, which enables our LGD to solve tasks of all levels and all types with provable guarantees. We verify the effectiveness of our framework with extensive experiments, where our models achieve state-of-the-art or highly competitive results across generation and regression tasks.

Node features and structural information of a graph are both crucial for semi-supervised node classification problems. A variety of graph neural network (GNN) based approaches have been proposed to tackle these problems, which typically determine output labels through feature aggregation. This can be problematic, as it implies conditional independence of output nodes given hidden representations, despite their direct connections in the graph. To learn the direct influence among output nodes in a graph, we propose the Explicit Pairwise Factorized Graph Neural Network (EPFGNN), which models the whole graph as a partially observed Markov Random Field. It contains explicit pairwise factors to model output-output relations and uses a GNN backbone to model input-output relations. To balance model complexity and expressivity, the pairwise factors have a shared component and a separate scaling coefficient for each edge. We apply the EM algorithm to train our model, and utilize a star-shaped piecewise likelihood for the tractable surrogate objective. We conduct experiments on various datasets, which shows that our model can effectively improve the performance for semi-supervised node classification on graphs.

We propose the convergent graph solver (CGS), a deep learning method that learns iterative mappings to predict the properties of a graph system at its stationary state (fixed point) with guaranteed convergence. CGS systematically computes the fixed points of a target graph system and decodes them to estimate the stationary properties of the system without the prior knowledge of existing solvers or intermediate solutions. The forward propagation of CGS proceeds in three steps: (1) constructing the input dependent linear contracting iterative maps, (2) computing the fixed-points of the linear maps, and (3) decoding the fixed-points to estimate the properties. The contractivity of the constructed linear maps guarantees the existence and uniqueness of the fixed points following the Banach fixed point theorem. To train CGS efficiently, we also derive a tractable analytical expression for its gradient by leveraging the implicit function theorem. We evaluate the performance of CGS by applying it to various network-analytic and graph benchmark problems. The results indicate that CGS has competitive capabilities for predicting the stationary properties of graph systems, irrespective of whether the target systems are linear or non-linear. CGS also shows high performance for graph classification problems where the existence or the meaning of a fixed point is hard to be clearly defined, which highlights the potential of CGS as a general graph neural network architecture.

The prevalence of multi-modal content on social media complicates automated moderation strategies. This calls for an enhancement in multi-modal classification and a deeper understanding of understated meanings in images and memes. Although previous efforts have aimed at improving model performance through fine-tuning, few have explored an end-to-end optimization pipeline that accounts for modalities, prompting, labeling, and fine-tuning. In this study, we propose an end-to-end conceptual framework for model optimization in complex tasks. Experiments support the efficacy of this traditional yet novel framework, achieving the highest accuracy and AUROC. Ablation experiments demonstrate that isolated optimizations are not ineffective on their own.

Recently, there has been a surge of interest in employing neural networks for graph generation, a fundamental statistical learning problem with critical applications like molecule design and community analysis. However, most approaches encounter significant limitations when generating large-scale graphs. This is due to their requirement to output the full adjacency matrices whose size grows quadratically with the number of nodes. In response to this challenge, we introduce a new, simple, and scalable graph representation named gap encoded edge list (GEEL) that has a small representation size that aligns with the number of edges. In addition, GEEL significantly reduces the vocabulary size by incorporating the gap encoding and bandwidth restriction schemes. GEEL can be autoregressively generated with the incorporation of node positional encoding, and we further extend GEEL to deal with attributed graphs by designing a new grammar. Our findings reveal that the adoption of this compact representation not only enhances scalability but also bolsters performance by simplifying the graph generation process. We conduct a comprehensive evaluation across ten non-attributed and two molecular graph generation tasks, demonstrating the effectiveness of GEEL.

Message-passing graph neural networks (MPNNs) emerged as powerful tools for processing graph-structured input. However, they operate on a fixed input graph structure, ignoring potential noise and missing information. Furthermore, their local aggregation mechanism can lead to problems such as over-squashing and limited expressive power in capturing relevant graph structures. Existing solutions to these challenges have primarily relied on heuristic methods, often disregarding the underlying data distribution. Hence, devising principled approaches for learning to infer graph structures relevant to the given prediction task remains an open challenge. In this work, leveraging recent progress in exact and differentiable k-subset sampling, we devise probabilistically rewired MPNNs (PR-MPNNs), which learn to add relevant edges while omitting less beneficial ones. For the first time, our theoretical analysis explores how PR-MPNNs enhance expressive power, and we identify precise conditions under which they outperform purely randomized approaches. Empirically, we demonstrate that our approach effectively mitigates issues like over-squashing and under-reaching. In addition, on established real-world datasets, our method exhibits competitive or superior predictive performance compared to traditional MPNN models and recent graph transformer architectures.

The advent of the "pre-train, prompt" paradigm has recently extended its generalization ability and data efficiency to graph representation learning, following its achievements in Natural Language Processing (NLP). Initial graph prompt tuning approaches tailored specialized prompting functions for Graph Neural Network (GNN) models pre-trained with specific strategies, such as edge prediction, thus limiting their applicability. In contrast, another pioneering line of research has explored universal prompting via adding prompts to the input graph's feature space, thereby removing the reliance on specific pre-training strategies. However, the necessity to add feature prompts to all nodes remains an open question. Motivated by findings from prompt tuning research in the NLP domain, which suggest that highly capable pre-trained models need less conditioning signal to achieve desired behaviors, we advocate for strategically incorporating necessary and lightweight feature prompts to certain graph nodes to enhance downstream task performance. This introduces a combinatorial optimization problem, requiring a policy to decide 1) which nodes to prompt and 2) what specific feature prompts to attach. We then address the problem by framing the prompt incorporation process as a sequential decision-making problem and propose our method, RELIEF, which employs Reinforcement Learning (RL) to optimize it. At each step, the RL agent selects a node (discrete action) and determines the prompt content (continuous action), aiming to maximize cumulative performance gain. Extensive experiments on graph and node-level tasks with various pre-training strategies in few-shot scenarios demonstrate that our RELIEF outperforms fine-tuning and other prompt-based approaches in classification performance and data efficiency.

Learning the behavior of large agent populations is an important task for numerous research areas. Although the field of multi-agent reinforcement learning (MARL) has made significant progress towards solving these systems, solutions for many agents often remain computationally infeasible and lack theoretical guarantees. Mean Field Games (MFGs) address both of these issues and can be extended to Graphon MFGs (GMFGs) to include network structures between agents. Despite their merits, the real world applicability of GMFGs is limited by the fact that graphons only capture dense graphs. Since most empirically observed networks show some degree of sparsity, such as power law graphs, the GMFG framework is insufficient for capturing these network topologies. Thus, we introduce the novel concept of Graphex MFGs (GXMFGs) which builds on the graph theoretical concept of graphexes. Graphexes are the limiting objects to sparse graph sequences that also have other desirable features such as the small world property. Learning equilibria in these games is challenging due to the rich and sparse structure of the underlying graphs. To tackle these challenges, we design a new learning algorithm tailored to the GXMFG setup. This hybrid graphex learning approach leverages that the system mainly consists of a highly connected core and a sparse periphery. After defining the system and providing a theoretical analysis, we state our learning approach and demonstrate its learning capabilities on both synthetic graphs and real-world networks. This comparison shows that our GXMFG learning algorithm successfully extends MFGs to a highly relevant class of hard, realistic learning problems that are not accurately addressed by current MARL and MFG methods.

We study a new connection between a technical measure called mu-conductance that arises in the study of Markov chains for sampling convex bodies and the network community profile that characterizes size-resolved properties of clusters and communities in social and information networks. The idea of mu-conductance is similar to the traditional graph conductance, but disregards sets with small volume. We derive a sequence of optimization problems including a low-rank semi-definite program from which we can derive a lower bound on the optimal mu-conductance value. These ideas give the first theoretically sound bound on the behavior of the network community profile for a wide range of cluster sizes. The algorithm scales up to graphs with hundreds of thousands of nodes and we demonstrate how our framework validates the predicted structures of real-world graphs.

Single-stage neural combinatorial optimization solvers have achieved near-optimal results on various small-scale combinatorial optimization (CO) problems without requiring expert knowledge. However, these solvers exhibit significant performance degradation when applied to large-scale CO problems. Recently, two-stage neural methods motivated by divide-and-conquer strategies have shown efficiency in addressing large-scale CO problems. Nevertheless, the performance of these methods highly relies on problem-specific heuristics in either the dividing or the conquering procedure, which limits their applicability to general CO problems. Moreover, these methods employ separate training schemes and ignore the interdependencies between the dividing and conquering strategies, often leading to sub-optimal solutions. To tackle these drawbacks, this article develops a unified neural divide-and-conquer framework (i.e., UDC) for solving general large-scale CO problems. UDC offers a Divide-Conquer-Reunion (DCR) training method to eliminate the negative impact of a sub-optimal dividing policy. Employing a high-efficiency Graph Neural Network (GNN) for global instance dividing and a fixed-length sub-path solver for conquering divided sub-problems, the proposed UDC framework demonstrates extensive applicability, achieving superior performance in 10 representative large-scale CO problems. The code is available at https://github.com/CIAM-Group/NCO_code/tree/main/single_objective/UDC-Large-scale-CO-master.

Graph-structured data are the commonly used and have wide application scenarios in the real world. For these diverse applications, the vast variety of learning tasks, graph domains, and complex graph learning procedures present challenges for human experts when designing versatile graph learning approaches. Facing these challenges, large language models (LLMs) offer a potential solution due to the extensive knowledge and the human-like intelligence. This paper proposes a novel conceptual prototype for designing versatile graph learning methods with LLMs, with a particular focus on the "where" and "how" perspectives. From the "where" perspective, we summarize four key graph learning procedures, including task definition, graph data feature engineering, model selection and optimization, deployment and serving. We then explore the application scenarios of LLMs in these procedures across a wider spectrum. In the "how" perspective, we align the abilities of LLMs with the requirements of each procedure. Finally, we point out the promising directions that could better leverage the strength of LLMs towards versatile graph learning methods.

Most real-world graphs exhibit a hierarchical structure, which is often overlooked by existing graph generation methods. To address this limitation, we propose a novel graph generative network that captures the hierarchical nature of graphs and successively generates the graph sub-structures in a coarse-to-fine fashion. At each level of hierarchy, this model generates communities in parallel, followed by the prediction of cross-edges between communities using separate neural networks. This modular approach enables scalable graph generation for large and complex graphs. Moreover, we model the output distribution of edges in the hierarchical graph with a multinomial distribution and derive a recursive factorization for this distribution. This enables us to generate community graphs with integer-valued edge weights in an autoregressive manner. Empirical studies demonstrate the effectiveness and scalability of our proposed generative model, achieving state-of-the-art performance in terms of graph quality across various benchmark datasets. The code is available at https://github.com/Karami-m/HiGen_main.

Machine learning on graphs has been extensively studied in both academic and industry. However, as the literature on graph learning booms with a vast number of emerging methods and techniques, it becomes increasingly difficult to manually design the optimal machine learning algorithm for different graph-related tasks. To solve this critical challenge, automated machine learning (AutoML) on graphs which combines the strength of graph machine learning and AutoML together, is gaining attention from the research community. Therefore, we comprehensively survey AutoML on graphs in this paper, primarily focusing on hyper-parameter optimization (HPO) and neural architecture search (NAS) for graph machine learning. We further overview libraries related to automated graph machine learning and in-depth discuss AutoGL, the first dedicated open-source library for AutoML on graphs. In the end, we share our insights on future research directions for automated graph machine learning. This paper is the first systematic and comprehensive review of automated machine learning on graphs to the best of our knowledge.

Diffusion-based generative graph models have been proven effective in generating high-quality small graphs. However, they need to be more scalable for generating large graphs containing thousands of nodes desiring graph statistics. In this work, we propose EDGE, a new diffusion-based generative graph model that addresses generative tasks with large graphs. To improve computation efficiency, we encourage graph sparsity by using a discrete diffusion process that randomly removes edges at each time step and finally obtains an empty graph. EDGE only focuses on a portion of nodes in the graph at each denoising step. It makes much fewer edge predictions than previous diffusion-based models. Moreover, EDGE admits explicitly modeling the node degrees of the graphs, further improving the model performance. The empirical study shows that EDGE is much more efficient than competing methods and can generate large graphs with thousands of nodes. It also outperforms baseline models in generation quality: graphs generated by our approach have more similar graph statistics to those of the training graphs.

The 2021 NeurIPS Machine Learning for Combinatorial Optimization (ML4CO) competition was designed with the goal of improving state-of-the-art combinatorial optimization solvers by replacing key heuristic components with machine learning models. The competition's main scientific question was the following: is machine learning a viable option for improving traditional combinatorial optimization solvers on specific problem distributions, when historical data is available? This was motivated by the fact that in many practical scenarios, the data changes only slightly between the repetitions of a combinatorial optimization problem, and this is an area where machine learning models are particularly powerful at. This paper summarizes the solution and lessons learned by the Huawei EI-OROAS team in the dual task of the competition. The submission of our team achieved the second place in the final ranking, with a very close distance to the first spot. In addition, our solution was ranked first consistently for several weekly leaderboard updates before the final evaluation. We provide insights gained from a large number of experiments, and argue that a simple Graph Convolutional Neural Network (GCNNs) can achieve state-of-the-art results if trained and tuned properly.

We introduce a framework for automatically defining and learning deep generative models with problem-specific structure. We tackle problem domains that are more traditionally solved by algorithms such as sorting, constraint satisfaction for Sudoku, and matrix factorization. Concretely, we train diffusion models with an architecture tailored to the problem specification. This problem specification should contain a graphical model describing relationships between variables, and often benefits from explicit representation of subcomputations. Permutation invariances can also be exploited. Across a diverse set of experiments we improve the scaling relationship between problem dimension and our model's performance, in terms of both training time and final accuracy. Our code can be found at https://github.com/plai-group/gsdm.

Combinatorial optimization (CO) is naturally discrete, making machine learning based on differentiable optimization inapplicable. Karalias & Loukas (2020) adapted the probabilistic method to incorporate CO into differentiable optimization. Their work ignited the research on unsupervised learning for CO, composed of two main components: probabilistic objectives and derandomization. However, each component confronts unique challenges. First, deriving objectives under various conditions (e.g., cardinality constraints and minimum) is nontrivial. Second, the derandomization process is underexplored, and the existing derandomization methods are either random sampling or naive rounding. In this work, we aim to tackle prevalent (i.e., commonly involved) conditions in unsupervised CO. First, we concretize the targets for objective construction and derandomization with theoretical justification. Then, for various conditions commonly involved in different CO problems, we derive nontrivial objectives and derandomization to meet the targets. Finally, we apply the derivations to various CO problems. Via extensive experiments on synthetic and real-world graphs, we validate the correctness of our derivations and show our empirical superiority w.r.t. both optimization quality and speed.

Understanding causal relationships between machines is crucial for fault diagnosis and optimization in manufacturing processes. Real-world datasets frequently exhibit up to 90% missing data and high dimensionality from hundreds of sensors. These datasets also include domain-specific expert knowledge and chronological order information, reflecting the recording order across different machines, which is pivotal for discerning causal relationships within the manufacturing data. However, previous methods for handling missing data in scenarios akin to real-world conditions have not been able to effectively utilize expert knowledge. Conversely, prior methods that can incorporate expert knowledge struggle with datasets that exhibit missing values. Therefore, we propose COKE to construct causal graphs in manufacturing datasets by leveraging expert knowledge and chronological order among sensors without imputing missing data. Utilizing the characteristics of the recipe, we maximize the use of samples with missing values, derive embeddings from intersections with an initial graph that incorporates expert knowledge and chronological order, and create a sensor ordering graph. The graph-generating process has been optimized by an actor-critic architecture to obtain a final graph that has a maximum reward. Experimental evaluations in diverse settings of sensor quantities and missing proportions demonstrate that our approach compared with the benchmark methods shows an average improvement of 39.9% in the F1-score. Moreover, the F1-score improvement can reach 62.6% when considering the configuration similar to real-world datasets, and 85.0% in real-world semiconductor datasets. The source code is available at https://github.com/OuTingYun/COKE.

Recent studies in using deep reinforcement learning (DRL) to solve Job-shop scheduling problems (JSSP) focus on construction heuristics. However, their performance is still far from optimality, mainly because the underlying graph representation scheme is unsuitable for modelling partial solutions at each construction step. This paper proposes a novel DRL-guided improvement heuristic for solving JSSP, where graph representation is employed to encode complete solutions. We design a Graph Neural-Network-based representation scheme, consisting of two modules to effectively capture the information of dynamic topology and different types of nodes in graphs encountered during the improvement process. To speed up solution evaluation during improvement, we present a novel message-passing mechanism that can evaluate multiple solutions simultaneously. We prove that the computational complexity of our method scales linearly with problem size. Experiments on classic benchmarks show that the improvement policy learned by our method outperforms state-of-the-art DRL-based methods by a large margin.

We propose a novel framework that leverages LLMs for full causal graph discovery. While previous LLM-based methods have used a pairwise query approach, this requires a quadratic number of queries which quickly becomes impractical for larger causal graphs. In contrast, the proposed framework uses a breadth-first search (BFS) approach which allows it to use only a linear number of queries. We also show that the proposed method can easily incorporate observational data when available, to improve performance. In addition to being more time and data-efficient, the proposed framework achieves state-of-the-art results on real-world causal graphs of varying sizes. The results demonstrate the effectiveness and efficiency of the proposed method in discovering causal relationships, showcasing its potential for broad applicability in causal graph discovery tasks across different domains.

Recent advancements in graph diffusion models (GDMs) have enabled the synthesis of realistic network structures, yet ensuring fairness in the generated data remains a critical challenge. Existing solutions attempt to mitigate bias by re-training the GDMs with ad-hoc fairness constraints. Conversely, with this work, we propose FAROS, a novel FAir graph geneRatiOn framework leveraging attribute Switching mechanisms and directly running in the generation process of the pre-trained GDM. Technically, our approach works by altering nodes' sensitive attributes during the generation. To this end, FAROS calculates the optimal fraction of switching nodes, and selects the diffusion step to perform the switch by setting tailored multi-criteria constraints to preserve the node-topology profile from the original distribution (a proxy for accuracy) while ensuring the edge independence on the sensitive attributes for the generated graph (a proxy for fairness). Our experiments on benchmark datasets for link prediction demonstrate that the proposed approach effectively reduces fairness discrepancies while maintaining comparable (or even higher) accuracy performance to other similar baselines. Noteworthy, FAROS is also able to strike a better accuracy-fairness trade-off than other competitors in some of the tested settings under the Pareto optimality concept, demonstrating the effectiveness of the imposed multi-criteria constraints.

Dynamic graph clustering aims to detect and track time-varying clusters in dynamic graphs, revealing the evolutionary mechanisms of complex real-world dynamic systems. Matrix factorization-based methods are promising approaches for this task; however, these methods often struggle with scalability and can be time-consuming when applied to large-scale dynamic graphs. Moreover, they tend to lack robustness and are vulnerable to real-world noisy data. To address these issues, we make three key contributions. First, to improve scalability, we propose temporal separated matrix factorization, where a single matrix is divided into multiple smaller matrices for independent factorization, resulting in faster computation. Second, to improve robustness, we introduce bi-clustering regularization, which jointly optimizes graph embedding and clustering, thereby filtering out noisy features from the graph embeddings. Third, to further enhance effectiveness and efficiency, we propose selective embedding updating, where we update only the embeddings of dynamic nodes while the embeddings of static nodes are fixed among different timestamps. Experimental results on six synthetic and five real-world benchmarks demonstrate the scalability, robustness and effectiveness of our proposed method. Source code is available at https://github.com/Clearloveyuan/DyG-MF.

In designing and applying graph neural networks, we often fall into some optimization pitfalls, the most deceptive of which is that we can only build a deep model by solving over-smoothing. The fundamental reason is that we do not understand how graph neural networks work. Stress graph drawing can offer a unique viewpoint to message iteration in the graph, such as the root of the over-smoothing problem lies in the inability of graph models to maintain an ideal distance between nodes. We further elucidate the trigger conditions of over-smoothing and propose Stress Graph Neural Networks. By introducing the attractive and repulsive message passing from stress iteration, we show how to build a deep model without preventing over-smoothing, how to use repulsive information, and how to optimize the current message-passing scheme to approximate the full stress message propagation. By performing different tasks on 23 datasets, we verified the effectiveness of our attractive and repulsive models and the derived relationship between stress iteration and graph neural networks. We believe that stress graph drawing will be a popular resource for understanding and designing graph neural networks.

Rationale discovery is defined as finding a subset of the input data that maximally supports the prediction of downstream tasks. In graph machine learning context, graph rationale is defined to locate the critical subgraph in the given graph topology, which fundamentally determines the prediction results. In contrast to the rationale subgraph, the remaining subgraph is named the environment subgraph. Graph rationalization can enhance the model performance as the mapping between the graph rationale and prediction label is viewed as invariant, by assumption. To ensure the discriminative power of the extracted rationale subgraphs, a key technique named "intervention" is applied. The core idea of intervention is that given any changing environment subgraphs, the semantics from the rationale subgraph is invariant, which guarantees the correct prediction result. However, most, if not all, of the existing rationalization works on graph data develop their intervention strategies on the graph level, which is coarse-grained. In this paper, we propose well-tailored intervention strategies on graph data. Our idea is driven by the development of Transformer models, whose self-attention module provides rich interactions between input nodes. Based on the self-attention module, our proposed invariant graph Transformer (IGT) can achieve fine-grained, more specifically, node-level and virtual node-level intervention. Our comprehensive experiments involve 7 real-world datasets, and the proposed IGT shows significant performance advantages compared to 13 baseline methods.

In the realm of generative models for graphs, extensive research has been conducted. However, most existing methods struggle with large graphs due to the complexity of representing the entire joint distribution across all node pairs and capturing both global and local graph structures simultaneously. To overcome these issues, we introduce a method that generates a graph by progressively expanding a single node to a target graph. In each step, nodes and edges are added in a localized manner through denoising diffusion, building first the global structure, and then refining the local details. The local generation avoids modeling the entire joint distribution over all node pairs, achieving substantial computational savings with subquadratic runtime relative to node count while maintaining high expressivity through multiscale generation. Our experiments show that our model achieves state-of-the-art performance on well-established benchmark datasets while successfully scaling to graphs with at least 5000 nodes. Our method is also the first to successfully extrapolate to graphs outside of the training distribution, showcasing a much better generalization capability over existing methods.

Deep generative models (DGMs) have been widely developed for graph data. However, much less investigation has been carried out on understanding the latent space of such pretrained graph DGMs. These understandings possess the potential to provide constructive guidelines for crucial tasks, such as graph controllable generation. Thus in this work, we are interested in studying this problem and propose GraphCG, a method for the unsupervised discovery of steerable factors in the latent space of pretrained graph DGMs. We first examine the representation space of three pretrained graph DGMs with six disentanglement metrics, and we observe that the pretrained representation space is entangled. Motivated by this observation, GraphCG learns the steerable factors via maximizing the mutual information between semantic-rich directions, where the controlled graph moving along the same direction will share the same steerable factors. We quantitatively verify that GraphCG outperforms four competitive baselines on two graph DGMs pretrained on two molecule datasets. Additionally, we qualitatively illustrate seven steerable factors learned by GraphCG on five pretrained DGMs over five graph datasets, including two for molecules and three for point clouds.

GraphRAG addresses significant challenges in Retrieval-Augmented Generation (RAG) by leveraging graphs with embedded knowledge to enhance the reasoning capabilities of Large Language Models (LLMs). Despite its promising potential, the GraphRAG community currently lacks a unified framework for fine-grained decomposition of the graph-based knowledge retrieval process. Furthermore, there is no systematic categorization or evaluation of existing solutions within the retrieval process. In this paper, we present LEGO-GraphRAG, a modular framework that decomposes the retrieval process of GraphRAG into three interconnected modules: subgraph-extraction, path-filtering, and path-refinement. We systematically summarize and classify the algorithms and neural network (NN) models relevant to each module, providing a clearer understanding of the design space for GraphRAG instances. Additionally, we identify key design factors, such as Graph Coupling and Computational Cost, that influence the effectiveness of GraphRAG implementations. Through extensive empirical studies, we construct high-quality GraphRAG instances using a representative selection of solutions and analyze their impact on retrieval and reasoning performance. Our findings offer critical insights into optimizing GraphRAG instance design, ultimately contributing to the advancement of more accurate and contextually relevant LLM applications.

We present FastRP, a scalable and performant algorithm for learning distributed node representations in a graph. FastRP is over 4,000 times faster than state-of-the-art methods such as DeepWalk and node2vec, while achieving comparable or even better performance as evaluated on several real-world networks on various downstream tasks. We observe that most network embedding methods consist of two components: construct a node similarity matrix and then apply dimension reduction techniques to this matrix. We show that the success of these methods should be attributed to the proper construction of this similarity matrix, rather than the dimension reduction method employed. FastRP is proposed as a scalable algorithm for network embeddings. Two key features of FastRP are: 1) it explicitly constructs a node similarity matrix that captures transitive relationships in a graph and normalizes matrix entries based on node degrees; 2) it utilizes very sparse random projection, which is a scalable optimization-free method for dimension reduction. An extra benefit from combining these two design choices is that it allows the iterative computation of node embeddings so that the similarity matrix need not be explicitly constructed, which further speeds up FastRP. FastRP is also advantageous for its ease of implementation, parallelization and hyperparameter tuning. The source code is available at https://github.com/GTmac/FastRP.

Combinatorial optimization (CO) problems are often NP-hard and thus out of reach for exact algorithms, making them a tempting domain to apply machine learning methods. The highly structured constraints in these problems can hinder either optimization or sampling directly in the solution space. On the other hand, GFlowNets have recently emerged as a powerful machinery to efficiently sample from composite unnormalized densities sequentially and have the potential to amortize such solution-searching processes in CO, as well as generate diverse solution candidates. In this paper, we design Markov decision processes (MDPs) for different combinatorial problems and propose to train conditional GFlowNets to sample from the solution space. Efficient training techniques are also developed to benefit long-range credit assignment. Through extensive experiments on a variety of different CO tasks with synthetic and realistic data, we demonstrate that GFlowNet policies can efficiently find high-quality solutions.

In recent years, graph pre-training has gained significant attention, focusing on acquiring transferable knowledge from unlabeled graph data to improve downstream performance. Despite these recent endeavors, the problem of negative transfer remains a major concern when utilizing graph pre-trained models to downstream tasks. Previous studies made great efforts on the issue of what to pre-train and how to pre-train by designing a variety of graph pre-training and fine-tuning strategies. However, there are cases where even the most advanced "pre-train and fine-tune" paradigms fail to yield distinct benefits. This paper introduces a generic framework W2PGNN to answer the crucial question of when to pre-train (i.e., in what situations could we take advantage of graph pre-training) before performing effortful pre-training or fine-tuning. We start from a new perspective to explore the complex generative mechanisms from the pre-training data to downstream data. In particular, W2PGNN first fits the pre-training data into graphon bases, each element of graphon basis (i.e., a graphon) identifies a fundamental transferable pattern shared by a collection of pre-training graphs. All convex combinations of graphon bases give rise to a generator space, from which graphs generated form the solution space for those downstream data that can benefit from pre-training. In this manner, the feasibility of pre-training can be quantified as the generation probability of the downstream data from any generator in the generator space. W2PGNN offers three broad applications: providing the application scope of graph pre-trained models, quantifying the feasibility of pre-training, and assistance in selecting pre-training data to enhance downstream performance. We provide a theoretically sound solution for the first application and extensive empirical justifications for the latter two applications.

This work studies the combinatorial optimization problem of finding an optimal core tensor shape, also called multilinear rank, for a size-constrained Tucker decomposition. We give an algorithm with provable approximation guarantees for its reconstruction error via connections to higher-order singular values. Specifically, we introduce a novel Tucker packing problem, which we prove is NP-hard, and give a polynomial-time approximation scheme based on a reduction to the 2-dimensional knapsack problem with a matroid constraint. We also generalize our techniques to tree tensor network decompositions. We implement our algorithm using an integer programming solver, and show that its solution quality is competitive with (and sometimes better than) the greedy algorithm that uses the true Tucker decomposition loss at each step, while also running up to 1000x faster.

Graph generation is integral to various engineering and scientific disciplines. Nevertheless, existing methodologies tend to overlook the generation of edge attributes. However, we identify critical applications where edge attributes are essential, making prior methods potentially unsuitable in such contexts. Moreover, while trivial adaptations are available, empirical investigations reveal their limited efficacy as they do not properly model the interplay among graph components. To address this, we propose a joint score-based model of nodes and edges for graph generation that considers all graph components. Our approach offers two key novelties: (i) node and edge attributes are combined in an attention module that generates samples based on the two ingredients; and (ii) node, edge and adjacency information are mutually dependent during the graph diffusion process. We evaluate our method on challenging benchmarks involving real-world and synthetic datasets in which edge features are crucial. Additionally, we introduce a new synthetic dataset that incorporates edge values. Furthermore, we propose a novel application that greatly benefits from the method due to its nature: the generation of traffic scenes represented as graphs. Our method outperforms other graph generation methods, demonstrating a significant advantage in edge-related measures.

Recently similarity graphs became the leading paradigm for efficient nearest neighbor search, outperforming traditional tree-based and LSH-based methods. Similarity graphs perform the search via greedy routing: a query traverses the graph and in each vertex moves to the adjacent vertex that is the closest to this query. In practice, similarity graphs are often susceptible to local minima, when queries do not reach its nearest neighbors, getting stuck in suboptimal vertices. In this paper we propose to learn the routing function that overcomes local minima via incorporating information about the graph global structure. In particular, we augment the vertices of a given graph with additional representations that are learned to provide the optimal routing from the start vertex to the query nearest neighbor. By thorough experiments, we demonstrate that the proposed learnable routing successfully diminishes the local minima problem and significantly improves the overall search performance.

Although Large Language Models (LLMs) have demonstrated remarkable progress, their proficiency in graph-related tasks remains notably limited, hindering the development of truly general-purpose models. Previous attempts, including pretraining graph foundation models or employing supervised fine-tuning, often face challenges such as the scarcity of large-scale, universally represented graph data. We introduce G1, a simple yet effective approach demonstrating that Reinforcement Learning (RL) on synthetic graph-theoretic tasks can significantly scale LLMs' graph reasoning abilities. To enable RL training, we curate Erd\~os, the largest graph reasoning dataset to date comprising 50 diverse graph-theoretic tasks of varying difficulty levels, 100k training data and 5k test data, all drived from real-world graphs. With RL on Erd\~os, G1 obtains substantial improvements in graph reasoning, where our finetuned 3B model even outperforms Qwen2.5-72B-Instruct (24x size). RL-trained models also show strong zero-shot generalization to unseen tasks, domains, and graph encoding schemes, including other graph-theoretic benchmarks as well as real-world node classification and link prediction tasks, without compromising general reasoning abilities. Our findings offer an efficient, scalable path for building strong graph reasoners by finetuning LLMs with RL on graph-theoretic tasks, which combines the strengths of pretrained LLM capabilities with abundant, automatically generated synthetic data, suggesting that LLMs possess graph understanding abilities that RL can elicit successfully.

Graph generation with Machine Learning is an open problem with applications in various research fields. In this work, we propose to cast the generative process of a graph into a sequential one, relying on a node ordering procedure. We use this sequential process to design a novel generative model composed of two recurrent neural networks that learn to predict the edges of graphs: the first network generates one endpoint of each edge, while the second network generates the other endpoint conditioned on the state of the first. We test our approach extensively on five different datasets, comparing with two well-known baselines coming from graph literature, and two recurrent approaches, one of which holds state of the art performances. Evaluation is conducted considering quantitative and qualitative characteristics of the generated samples. Results show that our approach is able to yield novel, and unique graphs originating from very different distributions, while retaining structural properties very similar to those in the training sample. Under the proposed evaluation framework, our approach is able to reach performances comparable to the current state of the art on the graph generation task.

Existing graph matching methods typically assume that there are similar structures between graphs and they are matchable. However, these assumptions do not align with real-world applications. This work addresses a more realistic scenario where graphs exhibit diverse modes, requiring graph grouping before or along with matching, a task termed mixture graph matching and clustering. We introduce Minorize-Maximization Matching and Clustering (M3C), a learning-free algorithm that guarantees theoretical convergence through the Minorize-Maximization framework and offers enhanced flexibility via relaxed clustering. Building on M3C, we develop UM3C, an unsupervised model that incorporates novel edge-wise affinity learning and pseudo label selection. Extensive experimental results on public benchmarks demonstrate that our method outperforms state-of-the-art graph matching and mixture graph matching and clustering approaches in both accuracy and efficiency. Source code will be made publicly available.

We study experiment design for unique identification of the causal graph of a simple SCM, where the graph may contain cycles. The presence of cycles in the structure introduces major challenges for experiment design as, unlike acyclic graphs, learning the skeleton of causal graphs with cycles may not be possible from merely the observational distribution. Furthermore, intervening on a variable in such graphs does not necessarily lead to orienting all the edges incident to it. In this paper, we propose an experiment design approach that can learn both cyclic and acyclic graphs and hence, unifies the task of experiment design for both types of graphs. We provide a lower bound on the number of experiments required to guarantee the unique identification of the causal graph in the worst case, showing that the proposed approach is order-optimal in terms of the number of experiments up to an additive logarithmic term. Moreover, we extend our result to the setting where the size of each experiment is bounded by a constant. For this case, we show that our approach is optimal in terms of the size of the largest experiment required for uniquely identifying the causal graph in the worst case.

To deduce new facts on a knowledge graph (KG), a link predictor learns from the graph structure and collects local evidence to find the answer to a given query. However, existing methods suffer from a severe scalability problem due to the utilization of the whole KG for prediction, which hinders their promise on large scale KGs and cannot be directly addressed by vanilla sampling methods. In this work, we propose the one-shot-subgraph link prediction to achieve efficient and adaptive prediction. The design principle is that, instead of directly acting on the whole KG, the prediction procedure is decoupled into two steps, i.e., (i) extracting only one subgraph according to the query and (ii) predicting on this single, query dependent subgraph. We reveal that the non-parametric and computation-efficient heuristics Personalized PageRank (PPR) can effectively identify the potential answers and supporting evidence. With efficient subgraph-based prediction, we further introduce the automated searching of the optimal configurations in both data and model spaces. Empirically, we achieve promoted efficiency and leading performances on five large-scale benchmarks. The code is publicly available at: https://github.com/tmlr-group/one-shot-subgraph.

The paper proposes a novel machine learning-based approach to the pathfinding problem on extremely large graphs. This method leverages diffusion distance estimation via a neural network and uses beam search for pathfinding. We demonstrate its efficiency by finding solutions for 4x4x4 and 5x5x5 Rubik's cubes with unprecedentedly short solution lengths, outperforming all available solvers and introducing the first machine learning solver beyond the 3x3x3 case. In particular, it surpasses every single case of the combined best results in the Kaggle Santa 2023 challenge, which involved over 1,000 teams. For the 3x3x3 Rubik's cube, our approach achieves an optimality rate exceeding 98%, matching the performance of task-specific solvers and significantly outperforming prior solutions such as DeepCubeA (60.3%) and EfficientCube (69.6%). Additionally, our solution is more than 26 times faster in solving 3x3x3 Rubik's cubes while requiring up to 18.5 times less model training time than the most efficient state-of-the-art competitor.

The era of foundation models has revolutionized AI research, yet Graph Foundation Models (GFMs) remain constrained by the scarcity of large-scale graph corpora. Traditional graph data synthesis techniques primarily focus on simplistic structural operations, lacking the capacity to generate semantically rich nodes with meaningful textual attributes: a critical limitation for real-world applications. While large language models (LLMs) demonstrate exceptional text generation capabilities, their direct application to graph synthesis is impeded by context window limitations, hallucination phenomena, and structural consistency challenges. To address these issues, we introduce GraphMaster, the first multi-agent framework specifically designed for graph data synthesis in data-limited environments. GraphMaster orchestrates four specialized LLM agents (Manager, Perception, Enhancement, and Evaluation) that collaboratively optimize the synthesis process through iterative refinement, ensuring both semantic coherence and structural integrity. To rigorously evaluate our approach, we create new data-limited "Sub" variants of six standard graph benchmarks, specifically designed to test synthesis capabilities under realistic constraints. Additionally, we develop a novel interpretability assessment framework that combines human evaluation with a principled Grassmannian manifold-based analysis, providing both qualitative and quantitative measures of semantic coherence. Experimental results demonstrate that GraphMaster significantly outperforms traditional synthesis methods across multiple datasets, establishing a strong foundation for advancing GFMs in data-scarce environments.

This systematic review explores the application of Large Language Models (LLMs) in Combinatorial Optimization (CO). We report our findings using the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) guidelines. We conduct a literature search via Scopus and Google Scholar, examining over 2,000 publications. We assess publications against four inclusion and four exclusion criteria related to their language, research focus, publication year, and type. Eventually, we select 103 studies. We classify these studies into semantic categories and topics to provide a comprehensive overview of the field, including the tasks performed by LLMs, the architectures of LLMs, the existing datasets specifically designed for evaluating LLMs in CO, and the field of application. Finally, we identify future directions for leveraging LLMs in this field.

Relational databases (RDBs) are composed of interconnected tables, where relationships between them are defined through foreign keys. Recent research on applying machine learning to RDBs has explored graph-based representations of RDBs, where rows of tables are modeled as nodes, and foreign key relationships are modeled as edges. RDB-to-graph modeling helps capture cross-table dependencies, ultimately leading to enhanced performance across diverse tasks. However, there are numerous ways to model RDBs as graphs, and performance varies significantly depending on the chosen graph model. In our analysis, applying a common heuristic rule for graph modeling leads to up to a 10% drop in performance compared to the best-performing graph model, which remains non-trivial to identify. To foster research on intelligent RDB-to-graph modeling, we introduce RDB2G-Bench, the first benchmark framework for evaluating such methods. We construct extensive datasets covering 5 real-world RDBs and 12 predictive tasks, resulting in around 50k graph-performance pairs for efficient and reproducible evaluations. Thanks to our precomputed datasets, we were able to benchmark 9 automatic RDB-to-graph modeling methods on the 12 tasks over 600x faster than on-the-fly evaluation, which requires repeated model training. Our analysis of the datasets and benchmark results reveals key structural patterns affecting graph model effectiveness, along with practical implications for effective graph modeling.

Existing methods for evaluating graph generative models primarily rely on Maximum Mean Discrepancy (MMD) metrics based on graph descriptors. While these metrics can rank generative models, they do not provide an absolute measure of performance. Their values are also highly sensitive to extrinsic parameters, namely kernel and descriptor parametrization, making them incomparable across different graph descriptors. We introduce PolyGraph Discrepancy (PGD), a new evaluation framework that addresses these limitations. It approximates the Jensen-Shannon distance of graph distributions by fitting binary classifiers to distinguish between real and generated graphs, featurized by these descriptors. The data log-likelihood of these classifiers approximates a variational lower bound on the JS distance between the two distributions. Resulting metrics are constrained to the unit interval [0,1] and are comparable across different graph descriptors. We further derive a theoretically grounded summary metric that combines these individual metrics to provide a maximally tight lower bound on the distance for the given descriptors. Thorough experiments demonstrate that PGD provides a more robust and insightful evaluation compared to MMD metrics. The PolyGraph framework for benchmarking graph generative models is made publicly available at https://github.com/BorgwardtLab/polygraph-benchmark.

Generation of graphs is a major challenge for real-world tasks that require understanding the complex nature of their non-Euclidean structures. Although diffusion models have achieved notable success in graph generation recently, they are ill-suited for modeling the topological properties of graphs since learning to denoise the noisy samples does not explicitly learn the graph structures to be generated. To tackle this limitation, we propose a generative framework that models the topology of graphs by explicitly learning the final graph structures of the diffusion process. Specifically, we design the generative process as a mixture of endpoint-conditioned diffusion processes which is driven toward the predicted graph that results in rapid convergence. We further introduce a simple parameterization of the mixture process and develop an objective for learning the final graph structure, which enables maximum likelihood training. Through extensive experimental validation on general graph and 2D/3D molecule generation tasks, we show that our method outperforms previous generative models, generating graphs with correct topology with both continuous (e.g. 3D coordinates) and discrete (e.g. atom types) features. Our code is available at https://github.com/harryjo97/GruM.

Graphs or networks are a very convenient way to represent data with lots of interaction. Recently, Machine Learning on Graph data has gained a lot of traction. In particular, vertex classification and missing edge detection have very interesting applications, ranging from drug discovery to recommender systems. To achieve such tasks, tremendous work has been accomplished to learn embedding of nodes and edges into finite-dimension vector spaces. This task is called Graph Representation Learning. However, Graph Representation Learning techniques often display prohibitive time and memory complexities, preventing their use in real-time with business size graphs. In this paper, we address this issue by leveraging a degeneracy property of Graphs - the K-Core Decomposition. We present two techniques taking advantage of this decomposition to reduce the time and memory consumption of walk-based Graph Representation Learning algorithms. We evaluate the performances, expressed in terms of quality of embedding and computational resources, of the proposed techniques on several academic datasets. Our code is available at https://github.com/SBrandeis/kcore-embedding

This study proposes a hybrid deep-learning-metaheuristic framework with a bi-level architecture for road network design problems (NDPs). We train a graph neural network (GNN) to approximate the solution of the user equilibrium (UE) traffic assignment problem and use inferences made by the trained model to calculate fitness function evaluations of a genetic algorithm (GA) to approximate solutions for NDPs. Using three test networks, two NDP variants and an exact solver as benchmark, we show that on average, our proposed framework can provide solutions within 1.5% gap of the best results in less than 0.5% of the time used by the exact solution procedure. Our framework can be utilized within an expert system for infrastructure planning to determine the best infrastructure planning and management decisions under different scenarios. Given the flexibility of the framework, it can easily be adapted to many other decision problems that can be modeled as bi-level problems on graphs. Moreover, we foreseen interesting future research directions, thus we also put forward a brief research agenda for this topic. The key observation from our research that can shape future research is that the fitness function evaluation time using the inferences made by the GNN model was in the order of milliseconds, which points to an opportunity and a need for novel heuristics that 1) can cope well with noisy fitness function values provided by deep learning models, and 2) can use the significantly enlarged efficiency of the evaluation step to explore the search space effectively (rather than efficiently). This opens a new avenue for a modern class of metaheuristics that are crafted for use with AI-powered predictors.

Learning graph generative models over latent spaces has received less attention compared to models that operate on the original data space and has so far demonstrated lacklustre performance. We present GLAD a latent space graph generative model. Unlike most previous latent space graph generative models, GLAD operates on a discrete latent space that preserves to a significant extent the discrete nature of the graph structures making no unnatural assumptions such as latent space continuity. We learn the prior of our discrete latent space by adapting diffusion bridges to its structure. By operating over an appropriately constructed latent space we avoid relying on decompositions that are often used in models that operate in the original data space. We present experiments on a series of graph benchmark datasets which clearly show the superiority of the discrete latent space and obtain state of the art graph generative performance, making GLAD the first latent space graph generative model with competitive performance. Our source code is published at: https://github.com/v18nguye/GLAD.

The DC Optimal Power Flow (DC-OPF) problem is fundamental to power system operations, requiring rapid solutions for real-time grid management. While traditional optimization solvers provide optimal solutions, their computational cost becomes prohibitive for large-scale systems requiring frequent recalculations. Machine learning approaches offer promise for acceleration but often struggle with constraint satisfaction and cost optimality. We present a novel two-stage learning framework that combines physics-informed Graph Neural Networks (GNNs) with Continuous Flow Matching (CFM) for solving DC-OPF problems. Our approach embeds fundamental physical principles--including economic dispatch optimality conditions, Kirchhoff's laws, and Karush-Kuhn-Tucker (KKT) complementarity conditions--directly into the training objectives. The first stage trains a GNN to produce feasible initial solutions by learning from physics-informed losses that encode power system constraints. The second stage employs CFM, a simulation-free continuous normalizing flow technique, to refine these solutions toward optimality through learned vector field regression. Evaluated on the IEEE 30-bus system across five load scenarios ranging from 70\% to 130\% nominal load, our method achieves near-optimal solutions with cost gaps below 0.1\% for nominal loads and below 3\% for extreme conditions, while maintaining 100\% feasibility. Our framework bridges the gap between fast but approximate neural network predictions and optimal but slow numerical solvers, offering a practical solution for modern power systems with high renewable penetration requiring frequent dispatch updates.

Graph neural networks (GNNs) have been successfully applied to learning representation on graphs in many relational tasks. Recently, researchers study neural architecture search (NAS) to reduce the dependence of human expertise and explore better GNN architectures, but they over-emphasize entity features and ignore latent relation information concealed in the edges. To solve this problem, we incorporate edge features into graph search space and propose Edge-featured Graph Neural Architecture Search to find the optimal GNN architecture. Specifically, we design rich entity and edge updating operations to learn high-order representations, which convey more generic message passing mechanisms. Moreover, the architecture topology in our search space allows to explore complex feature dependence of both entities and edges, which can be efficiently optimized by differentiable search strategy. Experiments at three graph tasks on six datasets show EGNAS can search better GNNs with higher performance than current state-of-the-art human-designed and searched-based GNNs.

Many important optimization problems, such as the minimum spanning tree and minimum-cost flow, can be solved optimally by a greedy method. In this work, we study a learning variant of these problems, where the model of the problem is unknown and has to be learned by interacting repeatedly with the environment in the bandit setting. We formalize our learning problem quite generally, as learning how to maximize an unknown modular function on a known polymatroid. We propose a computationally efficient algorithm for solving our problem and bound its expected cumulative regret. Our gap-dependent upper bound is tight up to a constant and our gap-free upper bound is tight up to polylogarithmic factors. Finally, we evaluate our method on three problems and demonstrate that it is practical.

Graph neural networks (GNNs) emerged recently as a standard toolkit for learning from data on graphs. Current GNN designing works depend on immense human expertise to explore different message-passing mechanisms, and require manual enumeration to determine the proper message-passing depth. Inspired by the strong searching capability of neural architecture search (NAS) in CNN, this paper proposes Graph Neural Architecture Search (GNAS) with novel-designed search space. The GNAS can automatically learn better architecture with the optimal depth of message passing on the graph. Specifically, we design Graph Neural Architecture Paradigm (GAP) with tree-topology computation procedure and two types of fine-grained atomic operations (feature filtering and neighbor aggregation) from message-passing mechanism to construct powerful graph network search space. Feature filtering performs adaptive feature selection, and neighbor aggregation captures structural information and calculates neighbors' statistics. Experiments show that our GNAS can search for better GNNs with multiple message-passing mechanisms and optimal message-passing depth. The searched network achieves remarkable improvement over state-of-the-art manual designed and search-based GNNs on five large-scale datasets at three classical graph tasks. Codes can be found at https://github.com/phython96/GNAS-MP.

Graph learning has become indispensable for interpreting and harnessing relational data in diverse fields, ranging from recommendation systems to social network analysis. In this context, a variety of GNNs have emerged as promising methodologies for encoding the structural information of graphs. By effectively capturing the graph's underlying structure, these GNNs have shown great potential in enhancing performance in graph learning tasks, such as link prediction and node classification. However, despite their successes, a significant challenge persists: these advanced methods often face difficulties in generalizing to unseen graph data that significantly differs from the training instances. In this work, our aim is to advance the graph learning paradigm by developing a general graph foundation model. This model is designed to understand the complex topological patterns present in diverse graph data, enabling it to excel in zero-shot graph learning tasks across different downstream datasets. To achieve this goal, we address several key technical challenges in our OpenGraph model. Firstly, we propose a unified graph tokenizer to adapt our graph model to generalize well on unseen graph data, even when the underlying graph properties differ significantly from those encountered during training. Secondly, we develop a scalable graph transformer as the foundational encoder, which effectively captures node-wise dependencies within the global topological context. Thirdly, we introduce a data augmentation mechanism enhanced by a LLM to alleviate the limitations of data scarcity in real-world scenarios. Extensive experiments validate the effectiveness of our framework. By adapting our OpenGraph to new graph characteristics and comprehending the nuances of diverse graphs, our approach achieves remarkable zero-shot graph learning performance across various settings and domains.

Recently proposed Graph Neural Networks (GNNs) for vertex clustering are trained with an unsupervised minimum cut objective, approximated by a Spectral Clustering (SC) relaxation. However, the SC relaxation is loose and, while it offers a closed-form solution, it also yields overly smooth cluster assignments that poorly separate the vertices. In this paper, we propose a GNN model that computes cluster assignments by optimizing a tighter relaxation of the minimum cut based on graph total variation (GTV). The cluster assignments can be used directly to perform vertex clustering or to implement graph pooling in a graph classification framework. Our model consists of two core components: i) a message-passing layer that minimizes the ell_1 distance in the features of adjacent vertices, which is key to achieving sharp transitions between clusters; ii) an unsupervised loss function that minimizes the GTV of the cluster assignments while ensuring balanced partitions. Experimental results show that our model outperforms other GNNs for vertex clustering and graph classification.

Task planning in language agents is emerging as an important research topic alongside the development of large language models (LLMs). It aims to break down complex user requests in natural language into solvable sub-tasks, thereby fulfilling the original requests. In this context, the sub-tasks can be naturally viewed as a graph, where the nodes represent the sub-tasks, and the edges denote the dependencies among them. Consequently, task planning is a decision-making problem that involves selecting a connected path or subgraph within the corresponding graph and invoking it. In this paper, we explore graph learning-based methods for task planning, a direction that is orthogonal to the prevalent focus on prompt design. Our interest in graph learning stems from a theoretical discovery: the biases of attention and auto-regressive loss impede LLMs' ability to effectively navigate decision-making on graphs, which is adeptly addressed by graph neural networks (GNNs). This theoretical insight led us to integrate GNNs with LLMs to enhance overall performance. Extensive experiments demonstrate that GNN-based methods surpass existing solutions even without training, and minimal training can further enhance their performance. The performance gain increases with a larger task graph size.

We present a new class of fast polylog-linear algorithms based on the theory of structured matrices (in particular low displacement rank) for integrating tensor fields defined on weighted trees. Several applications of the resulting fast tree-field integrators (FTFIs) are presented, including (a) approximation of graph metrics with tree metrics, (b) graph classification, (c) modeling on meshes, and finally (d) Topological Transformers (TTs) (Choromanski et al., 2022) for images. For Topological Transformers, we propose new relative position encoding (RPE) masking mechanisms with as few as three extra learnable parameters per Transformer layer, leading to 1.0-1.5%+ accuracy gains. Importantly, most of FTFIs are exact methods, thus numerically equivalent to their brute-force counterparts. When applied to graphs with thousands of nodes, those exact algorithms provide 5.7-13x speedups. We also provide an extensive theoretical analysis of our methods.

Graph neural networks (GNNs) have achieved remarkable success across a wide range of applications, such as recommendation, drug discovery, and question answering. Behind the success of GNNs lies the backpropagation (BP) algorithm, which is the de facto standard for training deep neural networks (NNs). However, despite its effectiveness, BP imposes several constraints, which are not only biologically implausible, but also limit the scalability, parallelism, and flexibility in learning NNs. Examples of such constraints include storage of neural activities computed in the forward pass for use in the subsequent backward pass, and the dependence of parameter updates on non-local signals. To address these limitations, the forward-forward algorithm (FF) was recently proposed as an alternative to BP in the image classification domain, which trains NNs by performing two forward passes over positive and negative data. Inspired by this advance, we propose ForwardGNN in this work, a new forward learning procedure for GNNs, which avoids the constraints imposed by BP via an effective layer-wise local forward training. ForwardGNN extends the original FF to deal with graph data and GNNs, and makes it possible to operate without generating negative inputs (hence no longer forward-forward). Further, ForwardGNN enables each layer to learn from both the bottom-up and top-down signals without relying on the backpropagation of errors. Extensive experiments on real-world datasets show the effectiveness and generality of the proposed forward graph learning framework. We release our code at https://github.com/facebookresearch/forwardgnn.

Since the pioneering work on the lottery ticket hypothesis for graph neural networks (GNNs) was proposed in Chen et al. (2021), the study on finding graph lottery tickets (GLT) has become one of the pivotal focus in the GNN community, inspiring researchers to discover sparser GLT while achieving comparable performance to original dense networks. In parallel, the graph structure has gained substantial attention as a crucial factor in GNN training dynamics, also elucidated by several recent studies. Despite this, contemporary studies on GLT, in general, have not fully exploited inherent pathways in the graph structure and identified tickets in an iterative manner, which is time-consuming and inefficient. To address these limitations, we introduce TEDDY, a one-shot edge sparsification framework that leverages structural information by incorporating edge-degree information. Following edge sparsification, we encourage the parameter sparsity during training via simple projected gradient descent on the ell_0 ball. Given the target sparsity levels for both the graph structure and the model parameters, our TEDDY facilitates efficient and rapid realization of GLT within a single training. Remarkably, our experimental results demonstrate that TEDDY significantly surpasses conventional iterative approaches in generalization, even when conducting one-shot sparsification that solely utilizes graph structures, without taking feature information into account.

Deep graph generative modeling has proven capable of learning the distribution of complex, multi-scale structures characterizing real-world graphs. However, one of the main limitations of existing methods is their large output space, which limits generation scalability and hinders accurate modeling of the underlying distribution. To overcome these limitations, we propose a novel approach that significantly reduces the output space of existing graph generative models. Specifically, starting from the observation that many real-world graphs have low graph bandwidth, we restrict graph bandwidth during training and generation. Our strategy improves both generation scalability and quality without increasing architectural complexity or reducing expressiveness. Our approach is compatible with existing graph generative methods, and we describe its application to both autoregressive and one-shot models. We extensively validate our strategy on synthetic and real datasets, including molecular graphs. Our experiments show that, in addition to improving generation efficiency, our approach consistently improves generation quality and reconstruction accuracy. The implementation is made available.

Graph Structure Learning (GSL) has recently garnered considerable attention due to its ability to optimize both the parameters of Graph Neural Networks (GNNs) and the computation graph structure simultaneously. Despite the proliferation of GSL methods developed in recent years, there is no standard experimental setting or fair comparison for performance evaluation, which creates a great obstacle to understanding the progress in this field. To fill this gap, we systematically analyze the performance of GSL in different scenarios and develop a comprehensive Graph Structure Learning Benchmark (GSLB) curated from 20 diverse graph datasets and 16 distinct GSL algorithms. Specifically, GSLB systematically investigates the characteristics of GSL in terms of three dimensions: effectiveness, robustness, and complexity. We comprehensively evaluate state-of-the-art GSL algorithms in node- and graph-level tasks, and analyze their performance in robust learning and model complexity. Further, to facilitate reproducible research, we have developed an easy-to-use library for training, evaluating, and visualizing different GSL methods. Empirical results of our extensive experiments demonstrate the ability of GSL and reveal its potential benefits on various downstream tasks, offering insights and opportunities for future research. The code of GSLB is available at: https://github.com/GSL-Benchmark/GSLB.

We propose JFR, a Bellman-Ford-based optimization framework leveraging frontier contraction and abstract multi-hop jump propagation to accelerate shortest-path computation while strictly preserving correctness. JFR achieves substantial reductions in relaxation operations, ranging from 25 to 99 percent, across sparse, dense, and negative-edge graphs, ensuring robust performance even under adversarial or highly connected topologies. On ultra-large graphs with up to N=20,000 nodes and 295 million edges, JFR maintains strong operational reductions and comparable or improved runtime relative to SPFA-SLF, demonstrating consistent robustness across graph size and density. Lower relaxation counts imply reduced memory-access overheads and computational effort; this normalized work reduction highlights JFR's suitability for scenarios requiring high throughput or energy-conscious operation. Future work focuses on integrating high-performance queue structures, adaptive frontier strategies, and cache-aware techniques to further reduce constant-factor overheads and fully realize JFR's practical runtime potential.

A subgraph is constructed by using a subset of vertices and edges of a given graph. There exist many graph properties that are hereditary for subgraphs. Hence, researchers from different communities have paid a great deal of attention in studying numerous subgraph problems, on top of the ordinary graph problems. Many algorithms are proposed in studying subgraph problems, where one common approach is by extracting the patterns and structures of a given graph. Due to the complex structures of certain types of graphs and to improve overall performances of the existing frameworks, machine learning techniques have recently been employed in dealing with various subgraph problems. In this article, we present a comprehensive review on five well known subgraph problems that have been tackled by using machine learning methods. They are subgraph isomorphism (both counting and matching), maximum common subgraph, community detection and community search problems. We provide an outline of each proposed method, and examine its designs and performances. We also explore non-learning-based algorithms for each problem and a brief discussion is given. We then suggest some promising research directions in this area, hoping that relevant subgraph problems can be tackled by using a similar strategy. Since there is a huge growth in employing machine learning techniques in recent years, we believe that this survey will serve as a good reference point to relevant research communities.

We study a class of optimization problems motivated by automating the design and update of AI systems like coding assistants, robots, and copilots. We propose an end-to-end optimization framework, Trace, which treats the computational workflow of an AI system as a graph akin to neural networks, based on a generalization of back-propagation. Optimization of computational workflows often involves rich feedback (e.g. console output or user's responses), heterogeneous parameters (e.g. prompts, hyper-parameters, codes), and intricate objectives (beyond maximizing a score). Moreover, its computation graph can change dynamically with the inputs and parameters. We frame a new mathematical setup of iterative optimization, Optimization with Trace Oracle (OPTO), to capture and abstract these properties so as to design optimizers that work across many domains. In OPTO, an optimizer receives an execution trace along with feedback on the computed output and updates parameters iteratively. Trace is the tool to implement OPTO in practice. Trace has a Python interface that efficiently converts a computational workflow into an OPTO instance using a PyTorch-like interface. Using Trace, we develop a general-purpose LLM-based optimizer called OptoPrime that can effectively solve OPTO problems. In empirical studies, we find that OptoPrime is capable of first-order numerical optimization, prompt optimization, hyper-parameter tuning, robot controller design, code debugging, etc., and is often competitive with specialized optimizers for each domain. We believe that Trace, OptoPrime and the OPTO framework will enable the next generation of interactive agents that automatically adapt using various kinds of feedback. Website: https://microsoft.github.io/Trace

Graph neural networks (GNNs) are the most widely adopted model in graph-structured data oriented learning and representation. Despite their extraordinary success in real-world applications, understanding their working mechanism by theory is still on primary stage. In this paper, we move towards this goal from the perspective of generalization. To be specific, we first establish high probability bounds of generalization gap and gradients in transductive learning with consideration of stochastic optimization. After that, we provide high probability bounds of generalization gap for popular GNNs. The theoretical results reveal the architecture specific factors affecting the generalization gap. Experimental results on benchmark datasets show the consistency between theoretical results and empirical evidence. Our results provide new insights in understanding the generalization of GNNs.

In this paper, we establish a connection between the parameterization of flow-based and energy-based generative models, and present a new flow-based modeling approach called energy-based normalizing flow (EBFlow). We demonstrate that by optimizing EBFlow with score-matching objectives, the computation of Jacobian determinants for linear transformations can be entirely bypassed. This feature enables the use of arbitrary linear layers in the construction of flow-based models without increasing the computational time complexity of each training iteration from O(D^2L) to O(D^3L) for an L-layered model that accepts D-dimensional inputs. This makes the training of EBFlow more efficient than the commonly-adopted maximum likelihood training method. In addition to the reduction in runtime, we enhance the training stability and empirical performance of EBFlow through a number of techniques developed based on our analysis of the score-matching methods. The experimental results demonstrate that our approach achieves a significant speedup compared to maximum likelihood estimation while outperforming prior methods with a noticeable margin in terms of negative log-likelihood (NLL).

This note describes the development of an exact solver for Minimal Directed Feedback Vertex Set as part of the PACE 2022 competition. The solver is powered largely by aggressively trying to reduce the DFVS problem to a Minimal Cover problem, and applying reduction rules adapted from Vertex Cover literature. The resulting problem is solved as an Integer Linear Program (ILP) using SCIP. The resulting solver performed the second-best in the competition, although a bug at submission time disqualified it. As an additional note, we describe a new vertex cover reduction generalizing the Desk reduction rule.

Hypergraphs offer a powerful framework for modeling higher-order interactions that traditional pairwise graphs cannot fully capture. However, practical constraints often lead to their simplification into projected graphs, resulting in substantial information loss and ambiguity in representing higher-order relationships. In this work, we propose MARIOH, a supervised approach for reconstructing the original hypergraph from its projected graph by leveraging edge multiplicity. To overcome the difficulties posed by the large search space, MARIOH integrates several key ideas: (a) identifying provable size-2 hyperedges, which reduces the candidate search space, (b) predicting the likelihood of candidates being hyperedges by utilizing both structural and multiplicity-related features, and (c) not only targeting promising hyperedge candidates but also examining less confident ones to explore alternative possibilities. Together, these ideas enable MARIOH to efficiently and effectively explore the search space. In our experiments using 10 real-world datasets, MARIOH achieves up to 74.51% higher reconstruction accuracy compared to state-of-the-art methods.

Test-Time Scaling (TTS) improves large language models (LLMs) by allocating additional computation during inference, typically through parallel, sequential, or hybrid scaling. However, prior studies often assume fixed collaboration architectures (e.g., topologies) and single-model usage, overlooking that optimal architectures and model combinations can vary across tasks. Therefore, we study the novel problem of searching for compute-optimal model combinations and architectures in TTS under a fixed budget. We formalize it as a multi-LLM collaboration graph, where nodes encode roles and LLM model assignments, and edges capture information flow. This problem is challenging because (i) the combinatorial search space is prohibitively large, and (ii) task-specific requirements demand tailored designs. To address these, we reformulate the problem as probabilistic graph optimization and, through pilot experiments, derive three empirical insights into TTS collaboration graphs. Guided by these insights, we propose Agent-REINFORCE, an LLM-agent-augmented framework that mirrors the REINFORCE pipeline by mapping sampling-gradient-update to sampling-feedback-update, where feedback serves as a textual gradient to update the probabilistic graph and efficiently search for optimal multi-LLM collaboration graphs. Experiments show that Agent-REINFORCE outperforms both traditional and LLM-based baselines in sample efficiency and search performance, and effectively identifies optimal graphs under joint objectives of accuracy and inference latency.

Transformers often fail to learn generalizable algorithms, instead relying on brittle heuristics. Using graph connectivity as a testbed, we explain this phenomenon both theoretically and empirically. We consider a simplified Transformer architecture, the disentangled Transformer, and prove that an L-layer model has capacity to solve for graphs with diameters up to exactly 3^L, implementing an algorithm equivalent to computing powers of the adjacency matrix. We analyze the training-dynamics, and show that the learned strategy hinges on whether most training instances are within this model capacity. Within-capacity graphs (diameter leq 3^L) drive the learning of a correct algorithmic solution while beyond-capacity graphs drive the learning of a simple heuristic based on node degrees. Finally, we empirically demonstrate that restricting training data within a model's capacity leads to both standard and disentangled transformers learning the exact algorithm rather than the degree-based heuristic.

Polynomial filters, a kind of Graph Neural Networks, typically use a predetermined polynomial basis and learn the coefficients from the training data. It has been observed that the effectiveness of the model is highly dependent on the property of the polynomial basis. Consequently, two natural and fundamental questions arise: Can we learn a suitable polynomial basis from the training data? Can we determine the optimal polynomial basis for a given graph and node features? In this paper, we propose two spectral GNN models that provide positive answers to the questions posed above. First, inspired by Favard's Theorem, we propose the FavardGNN model, which learns a polynomial basis from the space of all possible orthonormal bases. Second, we examine the supposedly unsolvable definition of optimal polynomial basis from Wang & Zhang (2022) and propose a simple model, OptBasisGNN, which computes the optimal basis for a given graph structure and graph signal. Extensive experiments are conducted to demonstrate the effectiveness of our proposed models.

Network embedding has been widely used in social recommendation and network analysis, such as recommendation systems and anomaly detection with graphs. However, most of previous approaches cannot handle large graphs efficiently, due to that (i) computation on graphs is often costly and (ii) the size of graph or the intermediate results of vectors could be prohibitively large, rendering it difficult to be processed on a single machine. In this paper, we propose an efficient and effective distributed algorithm for network embedding on large graphs using Apache Spark, which recursively partitions a graph into several small-sized subgraphs to capture the internal and external structural information of nodes, and then computes the network embedding for each subgraph in parallel. Finally, by aggregating the outputs on all subgraphs, we obtain the embeddings of nodes in a linear cost. After that, we demonstrate in various experiments that our proposed approach is able to handle graphs with billions of edges within a few hours and is at least 4 times faster than the state-of-the-art approaches. Besides, it achieves up to 4.25% and 4.27% improvements on link prediction and node classification tasks respectively. In the end, we deploy the proposed algorithms in two online games of Tencent with the applications of friend recommendation and item recommendation, which improve the competitors by up to 91.11% in running time and up to 12.80% in the corresponding evaluation metrics.

Multi-agent coordination is crucial for reliable multi-robot navigation in shared spaces such as automated warehouses. In regions of dense robot traffic, local coordination methods may fail to find a deadlock-free solution. In these scenarios, it is appropriate to let a central unit generate a global schedule that decides the passing order of robots. However, the runtime of such centralized coordination methods increases significantly with the problem scale. In this paper, we propose to leverage Graph Neural Network Variational Autoencoders (GNN-VAE) to solve the multi-agent coordination problem at scale faster than through centralized optimization. We formulate the coordination problem as a graph problem and collect ground truth data using a Mixed-Integer Linear Program (MILP) solver. During training, our learning framework encodes good quality solutions of the graph problem into a latent space. At inference time, solution samples are decoded from the sampled latent variables, and the lowest-cost sample is selected for coordination. Finally, the feasible proposal with the highest performance index is selected for the deployment. By construction, our GNN-VAE framework returns solutions that always respect the constraints of the considered coordination problem. Numerical results show that our approach trained on small-scale problems can achieve high-quality solutions even for large-scale problems with 250 robots, being much faster than other baselines. Project page: https://mengyuest.github.io/gnn-vae-coord

Graph neural networks aim to learn representations for graph-structured data and show impressive performance, particularly in node classification. Recently, many methods have studied the representations of GNNs from the perspective of optimization goals and spectral graph theory. However, the feature space that dominates representation learning has not been systematically studied in graph neural networks. In this paper, we propose to fill this gap by analyzing the feature space of both spatial and spectral models. We decompose graph neural networks into determined feature spaces and trainable weights, providing the convenience of studying the feature space explicitly using matrix space analysis. In particular, we theoretically find that the feature space tends to be linearly correlated due to repeated aggregations. Motivated by these findings, we propose 1) feature subspaces flattening and 2) structural principal components to expand the feature space. Extensive experiments verify the effectiveness of our proposed more comprehensive feature space, with comparable inference time to the baseline, and demonstrate its efficient convergence capability.

We deal with the combinatorial problem of learning directed acyclic graph (DAG) structure from observational data adhering to a linear structural equation model (SEM). Leveraging advances in differentiable, nonconvex characterizations of acyclicity, recent efforts have advocated a continuous constrained optimization paradigm to efficiently explore the space of DAGs. Most existing methods employ lasso-type score functions to guide this search, which (i) require expensive penalty parameter retuning when the unknown SEM noise variances change across problem instances; and (ii) implicitly rely on limiting homoscedasticity assumptions. In this work, we propose a new convex score function for sparsity-aware learning of linear DAGs, which incorporates concomitant estimation of scale and thus effectively decouples the sparsity parameter from the exogenous noise levels. Regularization via a smooth, nonconvex acyclicity penalty term yields CoLiDE (Concomitant Linear DAG Estimation), a regression-based criterion amenable to efficient gradient computation and closed-form estimation of noise variances in heteroscedastic scenarios. Our algorithm outperforms state-of-the-art methods without incurring added complexity, especially when the DAGs are larger and the noise level profile is heterogeneous. We also find CoLiDE exhibits enhanced stability manifested via reduced standard deviations in several domain-specific metrics, underscoring the robustness of our novel linear DAG estimator.

Subgraph matching is vital in knowledge graph (KG) question answering, molecule design, scene graph, code and circuit search, etc. Neural methods have shown promising results for subgraph matching. Our study of recent systems suggests refactoring them into a unified design space for graph matching networks. Existing methods occupy only a few isolated patches in this space, which remains largely uncharted. We undertake the first comprehensive exploration of this space, featuring such axes as attention-based vs. soft permutation-based interaction between query and corpus graphs, aligning nodes vs. edges, and the form of the final scoring network that integrates neural representations of the graphs. Our extensive experiments reveal that judicious and hitherto-unexplored combinations of choices in this space lead to large performance benefits. Beyond better performance, our study uncovers valuable insights and establishes general design principles for neural graph representation and interaction, which may be of wider interest.

Hypergraphs are a powerful abstraction for representing higher-order interactions between entities of interest. To exploit these relationships in making downstream predictions, a variety of hypergraph neural network architectures have recently been proposed, in large part building upon precursors from the more traditional graph neural network (GNN) literature. Somewhat differently, in this paper we begin by presenting an expressive family of parameterized, hypergraph-regularized energy functions. We then demonstrate how minimizers of these energies effectively serve as node embeddings that, when paired with a parameterized classifier, can be trained end-to-end via a supervised bilevel optimization process. Later, we draw parallels between the implicit architecture of the predictive models emerging from the proposed bilevel hypergraph optimization, and existing GNN architectures in common use. Empirically, we demonstrate state-of-the-art results on various hypergraph node classification benchmarks. Code is available at https://github.com/yxzwang/PhenomNN.

Subgraphs of a larger global graph may be distributed across multiple devices, and only locally accessible due to privacy restrictions, although there may be links between subgraphs. Recently proposed subgraph Federated Learning (FL) methods deal with those missing links across local subgraphs while distributively training Graph Neural Networks (GNNs) on them. However, they have overlooked the inevitable heterogeneity between subgraphs comprising different communities of a global graph, consequently collapsing the incompatible knowledge from local GNN models. To this end, we introduce a new subgraph FL problem, personalized subgraph FL, which focuses on the joint improvement of the interrelated local GNNs rather than learning a single global model, and propose a novel framework, FEDerated Personalized sUBgraph learning (FED-PUB), to tackle it. Since the server cannot access the subgraph in each client, FED-PUB utilizes functional embeddings of the local GNNs using random graphs as inputs to compute similarities between them, and use the similarities to perform weighted averaging for server-side aggregation. Further, it learns a personalized sparse mask at each client to select and update only the subgraph-relevant subset of the aggregated parameters. We validate our FED-PUB for its subgraph FL performance on six datasets, considering both non-overlapping and overlapping subgraphs, on which it significantly outperforms relevant baselines. Our code is available at https://github.com/JinheonBaek/FED-PUB.

Graph Neural Networks (GNNs) are neural models that leverage the dependency structure in graphical data via message passing among the graph nodes. GNNs have emerged as pivotal architectures in analyzing graph-structured data, and their expansive application in sensitive domains requires a comprehensive understanding of their decision-making processes -- necessitating a framework for GNN explainability. An explanation function for GNNs takes a pre-trained GNN along with a graph as input, to produce a `sufficient statistic' subgraph with respect to the graph label. A main challenge in studying GNN explainability is to provide fidelity measures that evaluate the performance of these explanation functions. This paper studies this foundational challenge, spotlighting the inherent limitations of prevailing fidelity metrics, including Fid_+, Fid_-, and Fid_Delta. Specifically, a formal, information-theoretic definition of explainability is introduced and it is shown that existing metrics often fail to align with this definition across various statistical scenarios. The reason is due to potential distribution shifts when subgraphs are removed in computing these fidelity measures. Subsequently, a robust class of fidelity measures are introduced, and it is shown analytically that they are resilient to distribution shift issues and are applicable in a wide range of scenarios. Extensive empirical analysis on both synthetic and real datasets are provided to illustrate that the proposed metrics are more coherent with gold standard metrics. The source code is available at https://trustai4s-lab.github.io/fidelity.

Large Language Models (LLMs) have demonstrated impressive reasoning capabilities, yet their direct application to NP-hard combinatorial problems (CPs) remains underexplored. In this work, we systematically investigate the reasoning abilities of LLMs on a variety of NP-hard combinatorial optimization tasks and introduce ACCORD: Autoregressive Constraint-satisfying generation for COmbinatorial optimization with Routing and Dynamic attention. ACCORD features a novel dataset representation and model architecture that leverage the autoregressive nature of LLMs to dynamically enforce feasibility constraints, coupled with attention-based routing to activate problem-specific LoRA modules. We also present the ACCORD-90k supervised dataset, covering six NP-hard combinatorial problems: TSP, VRP, Knapsack, FlowShop, JSSP, and BinPacking. Extensive experiments demonstrate that our ACCORD model, built on an 8B-parameter Llama backbone, consistently outperforms standard prompting and input-output methods, even when compared to much larger LLMs, such as gpt-4. Ablation studies further show that our output structure enhances solution feasibility. To the best of our knowledge, this is the first large-scale, end-to-end framework for exploring the applications of LLMs to a broad spectrum of combinatorial optimization problems. The codes are publicly available at https://github.com/starjob42/ACCORD

The notion of shortcut partition, introduced recently by Chang, Conroy, Le, Milenkovi\'c, Solomon, and Than [CCLMST23], is a new type of graph partition into low-diameter clusters. Roughly speaking, the shortcut partition guarantees that for every two vertices u and v in the graph, there exists a path between u and v that intersects only a few clusters. They proved that any planar graph admits a shortcut partition and gave several applications, including a construction of tree cover for arbitrary planar graphs with stretch 1+varepsilon and O(1) many trees for any fixed varepsilon in (0,1). However, the construction heavily exploits planarity in multiple steps, and is thus inherently limited to planar graphs. In this work, we breach the "planarity barrier" to construct a shortcut partition for K_r-minor-free graphs for any r. To this end, we take a completely different approach -- our key contribution is a novel deterministic variant of the cop decomposition in minor-free graphs [And86, AGG14]. Our shortcut partition for K_r-minor-free graphs yields several direct applications. Most notably, we construct the first optimal distance oracle for K_r-minor-free graphs, with 1+varepsilon stretch, linear space, and constant query time for any fixed varepsilon in (0,1). The previous best distance oracle [AG06] uses O(nlog n) space and O(log n) query time, and its construction relies on Robertson-Seymour structural theorem and other sophisticated tools. We also obtain the first tree cover of O(1) size for minor-free graphs with stretch 1+varepsilon, while the previous best (1+varepsilon)-tree cover has size O(log^2 n) [BFN19].

Graph Neural Networks (GNNs) have shown great promise in learning node embeddings for link prediction (LP). While numerous studies aim to improve the overall LP performance of GNNs, none have explored its varying performance across different nodes and its underlying reasons. To this end, we aim to demystify which nodes will perform better from the perspective of their local topology. Despite the widespread belief that low-degree nodes exhibit poorer LP performance, our empirical findings provide nuances to this viewpoint and prompt us to propose a better metric, Topological Concentration (TC), based on the intersection of the local subgraph of each node with the ones of its neighbors. We empirically demonstrate that TC has a higher correlation with LP performance than other node-level topological metrics like degree and subgraph density, offering a better way to identify low-performing nodes than using cold-start. With TC, we discover a novel topological distribution shift issue in which newly joined neighbors of a node tend to become less interactive with that node's existing neighbors, compromising the generalizability of node embeddings for LP at testing time. To make the computation of TC scalable, We further propose Approximated Topological Concentration (ATC) and theoretically/empirically justify its efficacy in approximating TC and reducing the computation complexity. Given the positive correlation between node TC and its LP performance, we explore the potential of boosting LP performance via enhancing TC by re-weighting edges in the message-passing and discuss its effectiveness with limitations. Our code is publicly available at https://github.com/YuWVandy/Topo_LP_GNN.

Applications of machine learning techniques for materials modeling typically involve functions known to be equivariant or invariant to specific symmetries. While graph neural networks (GNNs) have proven successful in such tasks, they enforce symmetries via the model architecture, which often reduces their expressivity, scalability and comprehensibility. In this paper, we introduce (1) a flexible framework relying on stochastic frame-averaging (SFA) to make any model E(3)-equivariant or invariant through data transformations. (2) FAENet: a simple, fast and expressive GNN, optimized for SFA, that processes geometric information without any symmetrypreserving design constraints. We prove the validity of our method theoretically and empirically demonstrate its superior accuracy and computational scalability in materials modeling on the OC20 dataset (S2EF, IS2RE) as well as common molecular modeling tasks (QM9, QM7-X). A package implementation is available at https://faenet.readthedocs.io.