A recent trend in data mining has explored (hyper)graph clustering algorithms for data with categorical relationship types. Such algorithms have applications in the analysis of social, co-authorship, and protein interaction networks, to name a few. Many such applications naturally have some overlap between clusters, a nuance which is missing from current combinatorial models. Additionally, existing models lack a mechanism for handling noise in datasets. We address these concerns by generalizing Edge-Colored Clustering, a recent framework for categorical clustering of hypergraphs. Our generalizations allow for a budgeted number of either (a) overlapping cluster assignments or (b) node deletions. For each new model we present a greedy algorithm which approximately minimizes an edge mistake objective, as well as bicriteria approximations where the second approximation factor is on the budget. Additionally, we address the parameterized complexity of each problem, providing FPT algorithms and hardness results.
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In cut sparsification, all cuts of a hypergraph $H=(V,E,w)$ are approximated within $1\pm\epsilon$ factor by a small hypergraph $H'$. This widely applied method was generalized recently to a setting where the cost of cutting each $e\in E$ is provided by a splitting function, $g_e: 2^e\to\mathbb{R}_+$. This generalization is called a submodular hypergraph when the functions $\{g_e\}_{e\in E}$ are submodular, and it arises in machine learning, combinatorial optimization, and algorithmic game theory. Previous work focused on the setting where $H'$ is a reweighted sub-hypergraph of $H$, and measured size by the number of hyperedges in $H'$. We study such sparsification, and also a more general notion of representing $H$ succinctly, where size is measured in bits. In the sparsification setting, where size is the number of hyperedges, we present three results: (i) all submodular hypergraphs admit sparsifiers of size polynomial in $n=|V|$; (ii) monotone-submodular hypergraphs admit sparsifiers of size $O(\epsilon^{-2} n^3)$; and (iii) we propose a new parameter, called spread, to obtain even smaller sparsifiers in some cases. In the succinct-representation setting, we show that a natural family of splitting functions admits a succinct representation of much smaller size than via reweighted subgraphs (almost by factor $n$). This large gap is surprising because for graphs, the most succinct representation is attained by reweighted subgraphs. Along the way, we introduce the notion of deformation, where $g_e$ is decomposed into a sum of functions of small description, and we provide upper and lower bounds for deformation of common splitting functions.
We consider hypergraph network design problems where the goal is to construct a hypergraph satisfying certain properties. In graph network design problems, the number of edges in an arbitrary solution is at most the square of the number of vertices. In contrast, in hypergraph network design problems, the number of hyperedges in an arbitrary solution could be exponential in the number of vertices and hence, additional care is necessary to design polynomial-time algorithms. The central theme of this work is to show that certain hypergraph network design problems admit solutions with polynomial number of hyperedges and moreover, can be solved in strongly polynomial time. Our work improves on the previous fastest pseudo-polynomial run-time for these problems. In addition, we develop algorithms that return (near-)uniform hypergraphs as solutions. The hypergraph network design problems that we focus upon are splitting-off operation in hypergraphs, connectivity augmentation using hyperedges, and covering skew-supermodular functions using hyperedges. Our definition of the splitting-off operation in hypergraphs and our proof showing the existence of the operation using a strongly polynomial-time algorithm to compute it are likely to be of independent graph-theoretical interest.
We consider the problem of distributed lossless computation of a function of two sources by one common user. To do so, we first build a bipartite graph, where two disjoint parts denote the individual source outcomes. We then project the bipartite graph onto each source to obtain an edge-weighted characteristic graph (EWCG), where edge weights capture the function's structure, by how much the source outcomes are to be distinguished, generalizing the classical notion of characteristic graphs. Via exploiting the notions of characteristic graphs, the fractional coloring of such graphs, and edge weights, the sources separately build multi-fold graphs that capture vector-valued source sequences, determine vertex colorings for such graphs, encode these colorings, and send them to the user that performs minimum-entropy decoding on its received information to recover the desired function in an asymptotically lossless manner. For the proposed EWCG compression setup, we characterize the fundamental limits of distributed compression, verify the communication complexity through an example, contrast it with traditional coloring schemes, and demonstrate that we can attain compression gains higher than $\% 30$ over traditional coloring.
Knowledge graph reasoning (KGR), aiming to deduce new facts from existing facts based on mined logic rules underlying knowledge graphs (KGs), has become a fast-growing research direction. It has been proven to significantly benefit the usage of KGs in many AI applications, such as question answering and recommendation systems, etc. According to the graph types, the existing KGR models can be roughly divided into three categories, \textit{i.e.,} static models, temporal models, and multi-modal models. The early works in this domain mainly focus on static KGR and tend to directly apply general knowledge graph embedding models to the reasoning task. However, these models are not suitable for more complex but practical tasks, such as inductive static KGR, temporal KGR, and multi-modal KGR. To this end, multiple works have been developed recently, but no survey papers and open-source repositories comprehensively summarize and discuss models in this important direction. To fill the gap, we conduct a survey for knowledge graph reasoning tracing from static to temporal and then to multi-modal KGs. Concretely, the preliminaries, summaries of KGR models, and typical datasets are introduced and discussed consequently. Moreover, we discuss the challenges and potential opportunities. The corresponding open-source repository is shared on GitHub: //github.com/LIANGKE23/Awesome-Knowledge-Graph-Reasoning.
Graph clustering, which aims to divide the nodes in the graph into several distinct clusters, is a fundamental and challenging task. In recent years, deep graph clustering methods have been increasingly proposed and achieved promising performance. However, the corresponding survey paper is scarce and it is imminent to make a summary in this field. From this motivation, this paper makes the first comprehensive survey of deep graph clustering. Firstly, the detailed definition of deep graph clustering and the important baseline methods are introduced. Besides, the taxonomy of deep graph clustering methods is proposed based on four different criteria including graph type, network architecture, learning paradigm, and clustering method. In addition, through the careful analysis of the existing works, the challenges and opportunities from five perspectives are summarized. At last, the applications of deep graph clustering in four domains are presented. It is worth mentioning that a collection of state-of-the-art deep graph clustering methods including papers, codes, and datasets is available on GitHub. We hope this work will serve as a quick guide and help researchers to overcome challenges in this vibrant field.
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.
Graph Convolutional Networks (GCNs) have been widely applied in various fields due to their significant power on processing graph-structured data. Typical GCN and its variants work under a homophily assumption (i.e., nodes with same class are prone to connect to each other), while ignoring the heterophily which exists in many real-world networks (i.e., nodes with different classes tend to form edges). Existing methods deal with heterophily by mainly aggregating higher-order neighborhoods or combing the immediate representations, which leads to noise and irrelevant information in the result. But these methods did not change the propagation mechanism which works under homophily assumption (that is a fundamental part of GCNs). This makes it difficult to distinguish the representation of nodes from different classes. To address this problem, in this paper we design a novel propagation mechanism, which can automatically change the propagation and aggregation process according to homophily or heterophily between node pairs. To adaptively learn the propagation process, we introduce two measurements of homophily degree between node pairs, which is learned based on topological and attribute information, respectively. Then we incorporate the learnable homophily degree into the graph convolution framework, which is trained in an end-to-end schema, enabling it to go beyond the assumption of homophily. More importantly, we theoretically prove that our model can constrain the similarity of representations between nodes according to their homophily degree. Experiments on seven real-world datasets demonstrate that this new approach outperforms the state-of-the-art methods under heterophily or low homophily, and gains competitive performance under homophily.
Deep learning models on graphs have achieved remarkable performance in various graph analysis tasks, e.g., node classification, link prediction and graph clustering. However, they expose uncertainty and unreliability against the well-designed inputs, i.e., adversarial examples. Accordingly, various studies have emerged for both attack and defense addressed in different graph analysis tasks, leading to the arms race in graph adversarial learning. For instance, the attacker has poisoning and evasion attack, and the defense group correspondingly has preprocessing- and adversarial- based methods. Despite the booming works, there still lacks a unified problem definition and a comprehensive review. To bridge this gap, we investigate and summarize the existing works on graph adversarial learning tasks systemically. Specifically, we survey and unify the existing works w.r.t. attack and defense in graph analysis tasks, and give proper definitions and taxonomies at the same time. Besides, we emphasize the importance of related evaluation metrics, and investigate and summarize them comprehensively. Hopefully, our works can serve as a reference for the relevant researchers, thus providing assistance for their studies. More details of our works are available at //github.com/gitgiter/Graph-Adversarial-Learning.
Attributed graph clustering is challenging as it requires joint modelling of graph structures and node attributes. Recent progress on graph convolutional networks has proved that graph convolution is effective in combining structural and content information, and several recent methods based on it have achieved promising clustering performance on some real attributed networks. However, there is limited understanding of how graph convolution affects clustering performance and how to properly use it to optimize performance for different graphs. Existing methods essentially use graph convolution of a fixed and low order that only takes into account neighbours within a few hops of each node, which underutilizes node relations and ignores the diversity of graphs. In this paper, we propose an adaptive graph convolution method for attributed graph clustering that exploits high-order graph convolution to capture global cluster structure and adaptively selects the appropriate order for different graphs. We establish the validity of our method by theoretical analysis and extensive experiments on benchmark datasets. Empirical results show that our method compares favourably with state-of-the-art methods.
In many real-world network datasets such as co-authorship, co-citation, email communication, etc., relationships are complex and go beyond pairwise. Hypergraphs provide a flexible and natural modeling tool to model such complex relationships. The obvious existence of such complex relationships in many real-world networks naturaly motivates the problem of learning with hypergraphs. A popular learning paradigm is hypergraph-based semi-supervised learning (SSL) where the goal is to assign labels to initially unlabeled vertices in a hypergraph. Motivated by the fact that a graph convolutional network (GCN) has been effective for graph-based SSL, we propose HyperGCN, a novel GCN for SSL on attributed hypergraphs. Additionally, we show how HyperGCN can be used as a learning-based approach for combinatorial optimisation on NP-hard hypergraph problems. We demonstrate HyperGCN's effectiveness through detailed experimentation on real-world hypergraphs.
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