Conventional network data has largely focused on pairwise interactions between two entities, yet multi-way interactions among multiple entities have been frequently observed in real-life hypergraph networks. In this article, we propose a novel method for detecting community structure in general hypergraph networks, uniform or non-uniform. The proposed method introduces a null vertex to augment a non-uniform hypergraph into a uniform multi-hypergraph, and then embeds the multi-hypergraph in a low-dimensional vector space such that vertices within the same community are close to each other. The resultant optimization task can be efficiently tackled by an alternative updating scheme. The asymptotic consistencies of the proposed method are established in terms of both community detection and hypergraph estimation, which are also supported by numerical experiments on some synthetic and real-life hypergraph networks.
The University of Texas at Austin（得克萨斯大学奥斯汀分校）；The University of Tehran（德黑兰大学）；Stanford University（斯坦福大学）
Many problems such as node classification and link prediction in network data can be solved using graph embeddings. However, it is difficult to use graphs to capture non-binary relations such as communities of nodes. These kinds of complex relations are expressed more naturally as hypergraphs. While hypergraphs are a generalization of graphs, state-of-the-art graph embedding techniques are not adequate for solving prediction and classification tasks on large hypergraphs accurately in reasonable time. In this paper, we introduce HyperNetVec, a novel hierarchical framework for scalable unsupervi…
Problems in scientific computing, such as distributing large sparse matrix operations, have analogous formulations as hypergraph partitioning problems. A hypergraph is a generalization of a traditional graph wherein "hyperedges" may connect any number of nodes. As a result, hypergraph partitioning is an NP-Hard problem to both solve or approximate. State-of-the-art algorithms that solve this problem follow the multilevel paradigm, which begins by iteratively "coarsening" the input hypergraph to smaller problem instances that share key structural features. Once identifying an approximate proble…
Istanbul Technical University（伊斯坦布尔理工大学）；University of Dundee（邓迪大学）
-Background. Network neuroscience examines the brain as a complex system represented by a network (or connectome), providing deeper insights into the brain morphology and function, allowing the identification of atypical brain connectivity alterations, which can be used as diagnostic markers of neurological disorders. -Existing Methods. Graph embedding methods which map data samples (e.g., brain networks) into a low dimensional space have been widely used to explore the relationship between samples for classification or prediction tasks. However, the majority of these works are based on modeli…
Detecting communities on graphs has received significant interest in recent literature. Current state-of-the-art community embedding approach called \textit{ComE} tackles this problem by coupling graph embedding with community detection. Considering the success of hyperbolic representations of graph-structured data in last years, an ongoing challenge is to set up a hyperbolic approach for the community detection problem. The present paper meets this challenge by introducing a Riemannian equivalent of \textit{ComE}. Our proposed approach combines hyperbolic embeddings with Riemannian K-means or…
Recently, graph neural networks have attracted great attention and achieved prominent performance in various research fields. Most of those algorithms have assumed pairwise relationships of objects of interest. However, in many real applications, the relationships between objects are in higher-order, beyond a pairwise formulation. To efficiently learn deep embeddings on the high-order graph-structured data, we introduce two end-to-end trainable operators to the family of graph neural networks, i.e., hypergraph convolution and hypergraph attention. Whilst hypergraph convolution defines the basi…
Stability and Generalization of Hypergraph Collaborative Networks
Institute of Data Science, The University of Hong Kong（香港大学数据科学学院）；College of Information Science and Technology, Jinan University（暨南大学信息科学技术学院）；The University of Hong Kong（香港大学）
Learning from Heterogeneity: A Dynamic Learning Framework for Hypergraphs
School of Computer Science and Technology, Tongji University（同济大学计算机科学与技术学院）；Ant Group（蚂蚁集团）；School of Computer Science and Artificial Intelligence, Wuhan University of Technology（武汉理工大学计算机科学与人工智能学院）；School of Computer Science, Fudan University（复旦大学计算机学院）；School of Science, Computing and Engineering Technologies, Swinburne University of Technology（斯威本科技大学科学、计算与工程技术学院）
IBM Research（IBM研究院）；Indian Institute of Science（印度科学院）
Network (or graph) embedding is the task to map the nodes of a graph to a lower dimensional vector space, such that it preserves the graph properties and facilitates the downstream network mining tasks. Real world networks often come with (community) outlier nodes, which behave differently from the regular nodes of the community. These outlier nodes can affect the embedding of the regular nodes, if not handled carefully. In this paper, we propose a novel unsupervised graph embedding approach (called DMGD) which integrates outlier and community detection with node embedding. We extend the idea …
Graph embedding has become a key component of many data mining and analysis systems. Current graph embedding approaches either sample a large number of node pairs from a graph to learn node embeddings via stochastic optimization or factorize a high-order proximity/adjacency matrix of the graph via computationally expensive matrix factorization techniques. These approaches typically require significant resources for the learning process and rely on multiple parameters, which limits their applicability in practice. Moreover, most of the existing graph embedding techniques operate effectively in …
In this paper, we study the information theoretic bounds for exact recovery in sub-hypergraph models for community detection. We define a general model called the $m-$uniform sub-hypergraph stochastic block model ($m-$ShSBM). Under the $m-$ShSBM, we use Fano's inequality to identify the region of model parameters where any algorithm fails to exactly recover the planted communities with a large probability. We also identify the region where a Maximum Likelihood Estimation (MLE) algorithm succeeds to exactly recover the communities with high probability. Our bounds are tight and pertain to the c…
Luddy School of Informatics, Computing and Engineering, Indiana University（印第安纳大学勒迪信息学、计算与工程学院）；King Abdulaziz University（阿卜杜勒阿齐兹国王大学）；Indiana University Network Science Institute (IUNI)（印第安纳大学网络科学研究所）
Graph embedding methods are becoming increasingly popular in the machine learning community, where they are widely used for tasks such as node classification and link prediction. Embedding graphs in geometric spaces should aid the identification of network communities as well, because nodes in the same community should be projected close to each other in the geometric space, where they can be detected via standard data clustering algorithms. In this paper, we test the ability of several graph embedding techniques to detect communities on benchmark graphs. We compare their performance against t…
University of Washington（华盛顿大学）；University of California, San Diego（加州大学圣地亚哥分校）
We consider the community detection problem in sparse random hypergraphs. Angelini et al. (2015) conjectured the existence of a sharp threshold on model parameters for community detection in sparse hypergraphs generated by a hypergraph stochastic block model. We solve the positive part of the conjecture for the case of two blocks: above the threshold, there is a spectral algorithm which asymptotically almost surely constructs a partition of the hypergraph correlated with the true partition. Our method is a generalization to random hypergraphs of the method developed by Massoulié (2014) for spa…
We propose a theoretical framework of multi-way similarity to model real-valued data into hypergraphs for clustering via spectral embedding. For graph cut based spectral clustering, it is common to model real-valued data into graph by modeling pairwise similarities using kernel function. This is because the kernel function has a theoretical connection to the graph cut. For problems where using multi-way similarities are more suitable than pairwise ones, it is natural to model as a hypergraph, which is generalization of a graph. However, although the hypergraph cut is well-studied, there is not…
Hypergraph data appear and are hidden in many places in the modern age. They are data structure that can be used to model many real data examples since their structures contain information about higher order relations among data points. One of the main contributions of our paper is to introduce a new topological structure to hypergraph data which bears a resemblance to a usual metric space structure. Using this new topological space structure of hypergraph data, we propose several approaches to study community detection problem, detecting persistent features arising from homological structure …
North Dakota State University（北达科他州立大学）；North China University of Water Resources and Electric Power（华北水利水电大学）
Community detection refers to the problem of clustering the nodes of a network (either graph or hypergrah) into groups. Various algorithms are available for community detection and all these methods apply to uncensored networks. In practice, a network may has censored (or missing) values and it is shown that censored values have non-negligible effect on the structural properties of a network. In this paper, we study community detection in censored $m$-uniform hypergraph from information-theoretic point of view. We derive the information-theoretic threshold for exact recovery of the community s…
We derive a Matern Gaussian process (GP) on the vertices of a hypergraph. This enables estimation of regression models of observed or latent values associated with the vertices, in which the correlation and uncertainty estimates are informed by the hypergraph structure. We further present a framework for embedding the vertices of a hypergraph into a latent space using the hypergraph GP. Finally, we provide a scheme for identifying a small number of representative inducing vertices that enables scalable inference through sparse GPs. We demonstrate the utility of our framework on three challengi…
University of British Columbia（英属哥伦比亚大学）；Element AI
Knowledge graphs store facts using relations between two entities. In this work, we address the question of link prediction in knowledge hypergraphs where relations are defined on any number of entities. While techniques exist (such as reification) that convert non-binary relations into binary ones, we show that current embedding-based methods for knowledge graph completion do not work well out of the box for knowledge graphs obtained through these techniques. To overcome this, we introduce HSimplE and HypE, two embedding-based methods that work directly with knowledge hypergraphs. In both mod…
We study hypergraph clustering in the weighted $d$-uniform hypergraph stochastic block model ($d$\textsf{-WHSBM}), where each edge consisting of $d$ nodes from the same community has higher expected weight than the edges consisting of nodes from different communities. We propose a new hypergraph clustering algorithm, called \textsf{CRTMLE}, and provide its performance guarantee under the $d$\textsf{-WHSBM} for general parameter regimes. We show that the proposed method achieves the order-wise optimal or the best existing results for approximately balanced community sizes. Moreover, our results…
Community detection in hypergraphs is explored. Under a generative hypergraph model called "d-wise hypergraph stochastic block model" (d-hSBM) which naturally extends the Stochastic Block Model from graphs to d-uniform hypergraphs, the asymptotic minimax mismatch ratio is characterized. For proving the achievability, we propose a two-step polynomial time algorithm that achieves the fundamental limit. The first step of the algorithm is a hypergraph spectral clustering method which achieves partial recovery to a certain precision level. The second step is a local refinement method which leverage…
Hypergraphs have gained increasing attention in the machine learning community lately due to their superiority over graphs in capturing super-dyadic interactions among entities. In this work, we propose a novel approach for the partitioning of k-uniform hypergraphs. Most of the existing methods work by reducing the hypergraph to a graph followed by applying standard graph partitioning algorithms. The reduction step restricts the algorithms to capturing only some weighted pairwise interactions and hence loses essential information about the original hypergraph. We overcome this issue by utilizi…
We study the community detection problem on a Gaussian mixture model, in which vertices are divided into $k\geq 2$ distinct communities. The major difference in our model is that the intensities for Gaussian perturbations are different for different entries in the observation matrix, and we do not assume that every community has the same number of vertices. We explicitly find the threshold for the exact recovery of the maximum likelihood estimation. Applications include the community detection on hypergraphs.
Heterogeneous Hypergraph Embedding for Recommendation Systems
Griffith University（格里菲斯大学）；Hanoi University of Science and Technology（河内科技大学）；Ecole Polytechnique Federale de Lausanne（洛桑联邦理工学院）；The University of Queensland（昆士兰大学）
Graph matching involves combinatorial optimization based on edge-to-edge affinity matrix, which can be generally formulated as Lawler's Quadratic Assignment Problem (QAP). This paper presents a QAP network directly learning with the affinity matrix (equivalently the association graph) whereby the matching problem is translated into a constrained vertex classification task. The association graph is learned by an embedding network for vertex classification, followed by Sinkhorn normalization and a cross-entropy loss for end-to-end learning. We further improve the embedding model on association g…
Recent research on network embedding in hyperbolic space have proven successful in several applications. However, nodes in real world networks tend to interact through several distinct channels. Simple aggregation or ignorance of this multiplexity will lead to misleading results. On the other hand, there exists redundant information between different interaction patterns between nodes. Recent research reveals the analogy between the community structure and the hyperbolic coordinate. To learn each node's effective embedding representation while reducing the redundancy of multiplex network, we t…
Hypergraphs are a natural modeling paradigm for a wide range of complex relational systems. A standard analysis task is to identify clusters of closely related or densely interconnected nodes. Many graph algorithms for this task are based on variants of the stochastic blockmodel, a random graph with flexible cluster structure. However, there are few models and algorithms for hypergraph clustering. Here, we propose a Poisson degree-corrected hypergraph stochastic blockmodel (DCHSBM), a generative model of clustered hypergraphs with heterogeneous node degrees and edge sizes. Approximate maximum-…
