In this paper, we investigate the graphs in which all balls are convex and the groups acting on them geometrically (which we call CB-graphs and CB-groups). These graphs have been introduced and characterized by Soltan and Chepoi (1983) and Farber and Jamison (1987). CB-graphs and CB-groups generalize systolic (alias bridged) and weakly systolic graphs and groups, which play an important role in geometric group theory. We present metric and local-to-global characterizations of CB-graphs. Namely, we characterize CB-graphs $G$ as graphs whose triangle-pentagonal complexes $X(G)$ are simply connected and balls of radius at most $3$ are convex. Similarly to systolic and weakly systolic graphs, we prove a dismantlability result for CB-graphs $G$: we show that their squares $G^2$ are dismantlable. This implies that the Rips complexes of CB-graphs are contractible. Finally, we adapt and extend the approach of Januszkiewicz and Swiatkowski (2006) for systolic groups and of Chalopin et al. (2020) for Helly groups, to show that the CB-groups are biautomatic.
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The monotone variational inequality is a central problem in mathematical programming that unifies and generalizes many important settings such as smooth convex optimization, two-player zero-sum games, convex-concave saddle point problems, etc. The extragradient method by Korpelevich [1976] is one of the most popular methods for solving monotone variational inequalities. Despite its long history and intensive attention from the optimization and machine learning community, the following major problem remains open. What is the last-iterate convergence rate of the extragradient method for monotone and Lipschitz variational inequalities with constraints? We resolve this open problem by showing a tight $O\left(\frac{1}{\sqrt{T}}\right)$ last-iterate convergence rate for arbitrary convex feasible sets, which matches the lower bound by Golowich et al. [2020]. Our rate is measured in terms of the standard gap function. The technical core of our result is the monotonicity of a new performance measure -- the tangent residual, which can be viewed as an adaptation of the norm of the operator that takes the local constraints into account. To establish the monotonicity, we develop a new approach that combines the power of the sum-of-squares programming with the low dimensionality of the update rule of the extragradient method. We believe our approach has many additional applications in the analysis of iterative methods.
A directed graph is oriented if it can be obtained by orienting the edges of a simple, undirected graph. For an oriented graph $G$, let $\beta(G)$ denote the size of a minimum feedback arc set, a smallest subset of edges whose deletion leaves an acyclic subgraph. A simple consequence of a result of Berger and Shor is that any oriented graph $G$ with $m$ edges satisfies $\beta(G) = m/2 - \Omega(m^{3/4})$. We observe that if an oriented graph $G$ has a fixed forbidden subgraph $B$, the upper bound of $\beta(G) = m/2 - \Omega(m^{3/4})$ is best possible as a function of the number of edges if $B$ is not bipartite, but the exponent $3/4$ in the lower order term can be improved if $B$ is bipartite. We also show that for every rational number $r$ between $3/4$ and $1$, there is a finite collection of digraphs $\mathcal{B}$ such that every $\mathcal{B}$-free digraph $G$ with $m$ edges satisfies $\beta(G) = m/2 - \Omega(m^r)$, and this bound is best possible up to the implied constant factor. The proof uses a connection to Tur\'an numbers and a result of Bukh and Conlon. Both of our upper bounds come equipped with randomized linear-time algorithms that construct feedback arc sets achieving those bounds. Finally, we give a characterization of quasirandom directed graphs via minimum feedback arc sets.
Over the years, many graph problems specifically those in NP-complete are studied by a wide range of researchers. Some famous examples include graph colouring, travelling salesman problem and subgraph isomorphism. Most of these problems are typically addressed by exact algorithms, approximate algorithms and heuristics. There are however some drawback for each of these methods. Recent studies have employed learning-based frameworks such as machine learning techniques in solving these problems, given that they are useful in discovering new patterns in structured data that can be represented using graphs. This research direction has successfully attracted a considerable amount of attention. In this survey, we provide a systematic review mainly on classic graph problems in which learning-based approaches have been proposed in addressing the problems. We discuss the overview of each framework, and provide analyses based on the design and performance of the framework. Some potential research questions are also suggested. Ultimately, this survey gives a clearer insight and can be used as a stepping stone to the research community in studying problems in this field.
Multigrid is a powerful solver for large-scale linear systems arising from discretized partial differential equations. The convergence theory of multigrid methods for symmetric positive definite problems has been well developed over the past decades, while, for nonsymmetric problems, such theory is still not mature. As a foundation for multigrid analysis, two-grid convergence theory plays an important role in motivating multigrid algorithms. Regarding two-grid methods for nonsymmetric problems, most previous works focus on the spectral radius of iteration matrix or rely on convergence measures that are typically difficult to compute in practice. Moreover, the existing results are confined to two-grid methods with exact solution of the coarse-grid system. In this paper, we analyze the convergence of a two-grid method for nonsymmetric positive definite problems (e.g., linear systems arising from the discretizations of convection-diffusion equations). In the case of exact coarse solver, we establish an elegant identity for characterizing two-grid convergence factor, which is measured by a smoother-induced norm. The identity can be conveniently used to derive a class of optimal restriction operators and analyze how the convergence factor is influenced by restriction. More generally, we present some convergence estimates for an inexact variant of the two-grid method, in which both linear and nonlinear coarse solvers are considered.
The adaptive processing of structured data is a long-standing research topic in machine learning that investigates how to automatically learn a mapping from a structured input to outputs of various nature. Recently, there has been an increasing interest in the adaptive processing of graphs, which led to the development of different neural network-based methodologies. In this thesis, we take a different route and develop a Bayesian Deep Learning framework for graph learning. The dissertation begins with a review of the principles over which most of the methods in the field are built, followed by a study on graph classification reproducibility issues. We then proceed to bridge the basic ideas of deep learning for graphs with the Bayesian world, by building our deep architectures in an incremental fashion. This framework allows us to consider graphs with discrete and continuous edge features, producing unsupervised embeddings rich enough to reach the state of the art on several classification tasks. Our approach is also amenable to a Bayesian nonparametric extension that automatizes the choice of almost all model's hyper-parameters. Two real-world applications demonstrate the efficacy of deep learning for graphs. The first concerns the prediction of information-theoretic quantities for molecular simulations with supervised neural models. After that, we exploit our Bayesian models to solve a malware-classification task while being robust to intra-procedural code obfuscation techniques. We conclude the dissertation with an attempt to blend the best of the neural and Bayesian worlds together. The resulting hybrid model is able to predict multimodal distributions conditioned on input graphs, with the consequent ability to model stochasticity and uncertainty better than most works. Overall, we aim to provide a Bayesian perspective into the articulated research field of deep learning for graphs.
This manuscript portrays optimization as a process. In many practical applications the environment is so complex that it is infeasible to lay out a comprehensive theoretical model and use classical algorithmic theory and mathematical optimization. It is necessary as well as beneficial to take a robust approach, by applying an optimization method that learns as one goes along, learning from experience as more aspects of the problem are observed. This view of optimization as a process has become prominent in varied fields and has led to some spectacular success in modeling and systems that are now part of our daily lives.
Deep graph neural networks (GNNs) have achieved excellent results on various tasks on increasingly large graph datasets with millions of nodes and edges. However, memory complexity has become a major obstacle when training deep GNNs for practical applications due to the immense number of nodes, edges, and intermediate activations. To improve the scalability of GNNs, prior works propose smart graph sampling or partitioning strategies to train GNNs with a smaller set of nodes or sub-graphs. In this work, we study reversible connections, group convolutions, weight tying, and equilibrium models to advance the memory and parameter efficiency of GNNs. We find that reversible connections in combination with deep network architectures enable the training of overparameterized GNNs that significantly outperform existing methods on multiple datasets. Our models RevGNN-Deep (1001 layers with 80 channels each) and RevGNN-Wide (448 layers with 224 channels each) were both trained on a single commodity GPU and achieve an ROC-AUC of $87.74 \pm 0.13$ and $88.14 \pm 0.15$ on the ogbn-proteins dataset. To the best of our knowledge, RevGNN-Deep is the deepest GNN in the literature by one order of magnitude. Please visit our project website //www.deepgcns.org/arch/gnn1000 for more information.
Graph Neural Networks (GNNs) have recently become increasingly popular due to their ability to learn complex systems of relations or interactions arising in a broad spectrum of problems ranging from biology and particle physics to social networks and recommendation systems. Despite the plethora of different models for deep learning on graphs, few approaches have been proposed thus far for dealing with graphs that present some sort of dynamic nature (e.g. evolving features or connectivity over time). In this paper, we present Temporal Graph Networks (TGNs), a generic, efficient framework for deep learning on dynamic graphs represented as sequences of timed events. Thanks to a novel combination of memory modules and graph-based operators, TGNs are able to significantly outperform previous approaches being at the same time more computationally efficient. We furthermore show that several previous models for learning on dynamic graphs can be cast as specific instances of our framework. We perform a detailed ablation study of different components of our framework and devise the best configuration that achieves state-of-the-art performance on several transductive and inductive prediction tasks for dynamic graphs.
In this paper we provide a comprehensive introduction to knowledge graphs, which have recently garnered significant attention from both industry and academia in scenarios that require exploiting diverse, dynamic, large-scale collections of data. After a general introduction, we motivate and contrast various graph-based data models and query languages that are used for knowledge graphs. We discuss the roles of schema, identity, and context in knowledge graphs. We explain how knowledge can be represented and extracted using a combination of deductive and inductive techniques. We summarise methods for the creation, enrichment, quality assessment, refinement, and publication of knowledge graphs. We provide an overview of prominent open knowledge graphs and enterprise knowledge graphs, their applications, and how they use the aforementioned techniques. We conclude with high-level future research directions for knowledge graphs.
Graph Neural Networks (GNNs) for representation learning of graphs broadly follow a neighborhood aggregation framework, where the representation vector of a node is computed by recursively aggregating and transforming feature vectors of its neighboring nodes. Many GNN variants have been proposed and have achieved state-of-the-art results on both node and graph classification tasks. However, despite GNNs revolutionizing graph representation learning, there is limited understanding of their representational properties and limitations. Here, we present a theoretical framework for analyzing the expressive power of GNNs in capturing different graph structures. Our results characterize the discriminative power of popular GNN variants, such as Graph Convolutional Networks and GraphSAGE, and show that they cannot learn to distinguish certain simple graph structures. We then develop a simple architecture that is provably the most expressive among the class of GNNs and is as powerful as the Weisfeiler-Lehman graph isomorphism test. We empirically validate our theoretical findings on a number of graph classification benchmarks, and demonstrate that our model achieves state-of-the-art performance.
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