Social networks are now ubiquitous and most of them contain interactions involving multiple actors (groups) like author collaborations, teams or emails in an organizations, etc. Hypergraphs are natural structures to effectively capture multi-actor interactions which conventional dyadic graphs fail to capture. In this work the problem of predicting collaborations is addressed while modeling the collaboration network as a hypergraph network. The problem of predicting future multi-actor collaboration is mapped to hyperedge prediction problem. Given that the higher order edge prediction is an inherently hard problem, in this work we restrict to the task of predicting edges (collaborations) that have already been observed in past. In this work, we propose a novel use of hyperincidence temporal tensors to capture time varying hypergraphs and provides a tensor decomposition based prediction algorithm. We quantitatively compare the performance of the hypergraphs based approach with the conventional dyadic graph based approach. Our hypothesis that hypergraphs preserve the information that simple graphs destroy is corroborated by experiments using author collaboration network from the DBLP dataset. Our results demonstrate the strength of hypergraph based approach to predict higher order collaborations (size>4) which is very difficult using dyadic graph based approach. Moreover, while predicting collaborations of size>2 hypergraphs in most cases provide better results with an average increase of approx. 45 in F-Score for different sizes = 3,4,5,6,7 .
Hypergraphs can easily capture the higher-order relationships while incorporating both group and node level attributes. Moreover, research has shown that several social, biological, ecological and technological systems can be better modeled using hypergraphs than using dyadic proxies @cite_21 . There is an abundant literature of hypergraph theory in past @cite_15 and many work in the spectral theory of hypergraphs recently @cite_18 @cite_14 . The past decade has also seen an increasing interest for hypergraphs in machine learning community @cite_24 @cite_13 . Hypergraphs have been used to model complex networks in different fields including biology @cite_23 , databases @cite_8 and data mining @cite_1 . In the domain of social sciences, Kapoor et a.l @cite_20 have proposed with centrality metrics for weighted hypergraphs. @cite_19 propose hypergraphs based metrics to evaluate various hypothesis, both semantic and structural, regarding team formation.
@cite_18 @cite_14 @cite_8 @cite_21 @cite_1 @cite_24 @cite_19 @cite_23 @cite_15 @cite_13 @cite_20
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Many real-world social interactions involve multiple people, for e.g., authors collaborating on a paper, email exchanges made in a company and task-oriented teams in workforce. Simple graph representation of these activities destroys the group structure present in them. Hypergraphs have recently emerged as a better tool for modeling group interactions. However, methods in social hypernetwork analysis haven't kept pace. In this work, we extend the concept of node degree centrality to hypergraphs. We validate our proposed measures using alternate measures of influence available to us using two datasets namely, the DBLP dataset of scientific collaborations and the group network in a popular Chinese multi-player online game called CR3. We discuss several schemes for assigning weights to hyperedges and compare them empirically. Finally, we define separate weak and strong tie node degree centralities which improves performance of our models. Weak tie degree centrality is found to be a better predictor of influence in hypergraphs than strong tie degree centrality. In this paper, we present a hypergraph neural networks (HGNN) framework for data representation learning, which can encode high-order data correlation in a hypergraph structure. Confronting the challenges of learning representation for complex data in real practice, we propose to incorporate such data structure in a hypergraph, which is more flexible on data modeling, especially when dealing with complex data. In this method, a hyperedge convolution operation is designed to handle the data correlation during representation learning. In this way, traditional hypergraph learning procedure can be conducted using hyperedge convolution operations efficiently. HGNN is able to learn the hidden layer representation considering the high-order data structure, which is a general framework considering the complex data correlations. We have conducted experiments on citation network classification and visual object recognition tasks and compared HGNN with graph convolutional networks and other traditional methods. Experimental results demonstrate that the proposed HGNN method outperforms recent state-of-theart methods. We can also reveal from the results that the proposed HGNN is superior when dealing with multi-modal data compared with existing methods. Driven by the importance of relational aspects of data to decision-making, graph algorithms have been developed, based on simplified pairwise relationships, to solve a variety of problems. However, evidence has shown that hypergraphs—generalizations of graphs with (hyper)edges that connect any number of vertices—can better model complex, non-pairwise relationships in data and lead to better informed decisions. In this work, we compare graph and hypergraph models in the context of spectral clustering. For these problems, we demonstrate that hypergraphs are computationally more efficient and can better model complex, non-pairwise relationships for many datasets. Many real-world domains can be represented as large node-link graphs: backbone Internet routers connect with 70,000 other hosts, mid-sized Web servers handle between 20,000 and 200,000 hyperlinked documents, and dictionaries contain millions of words defined in terms of each other. Computational manipulation of such large graphs is common, but previous tools for graph visualization have been limited to datasets of a few thousand nodes. Visual depictions of graphs and networks are external representations that exploit human visual processing to reduce the cognitive load of many tasks that require understanding of global or local structure. We assert that the two key advantages of computer-based systems for information visualization over traditional paper-based visual exposition are interactivity and scalability. We also argue that designing visualization software by taking the characteristics of a target user's task domain into account leads to systems that are more effective and scale to larger datasets than previous work. This thesis contains a detailed analysis of three specialized systems for the interactive exploration of large graphs, relating the intended tasks to the spatial layout and visual encoding choices. We present two novel algorithms for specialized layout and drawing that use quite different visual metaphors. The H3 system for visualizing the hyperlink structures of web sites scales to datasets of over 100,000 nodes by using a carefully chosen spanning tree as the layout backbone, 3D hyperbolic geometry for a Focus+Context view, and provides a fluid interactive experience through guaranteed frame rate drawing. The Constellation system features a highly specialized 2D layout intended to spatially encode domain-specific information for computational linguists checking the plausibility of a large semantic network created from dictionaries. The Planet Multicast system for displaying the tunnel topology of the Internet's multicast backbone provides a literal 3D geographic layout of arcs on a globe to help MBone maintainers find misconfigured long-distance tunnels. Each of these three systems provides a very different view of the graph structure, and we evaluate their efficacy for the intended task. We generalize these findings in our analysis of the importance of interactivity and specialization for graph visualization systems that are effective and scalable. In this paper, we develop a novel paradigm, namely hypergraph shift, to find robust graph modes by probabilistic voting strategy, which are semantically sound besides the self-cohesiveness requirement in forming graph modes. Unlike the existing techniques to seek graph modes by shifting vertices based on pair-wise edges (i.e, an edge with 2 ends), our paradigm is based on shifting high-order edges (hyperedges) to deliver graph modes. Specifically, we convert the problem of seeking graph modes as the problem of seeking maximizers of a novel objective function with the aim to generate good graph modes based on sifting edges in hypergraphs. As a result, the generated graph modes based on dense subhypergraphs may more accurately capture the object semantics besides the self-cohesiveness requirement. We also formally prove that our technique is always convergent. Extensive empirical studies on synthetic and real world data sets are conducted on clustering and graph matching. They demonstrate that our techniques significantly outperform the existing techniques. A hypergraph is a generalization of the traditional graph in which the edges are arbitrary non-empty subsets of the vertex set. It has been applied successfully to capture high-order relations in various domains. In this paper, we propose a hypergraph spectral learning formulation for multi-label classification, where a hypergraph is constructed to exploit the correlation information among different labels. We show that the proposed formulation leads to an eigenvalue problem, which may be computationally expensive especially for large-scale problems. To reduce the computational cost, we propose an approximate formulation, which is shown to be equivalent to a least squares problem under a mild condition. Based on the approximate formulation, efficient algorithms for solving least squares problems can be applied to scale the formulation to very large data sets. In addition, existing regularization techniques for least squares can be incorporated into the model for improved generalization performance. We have conducted experiments using large-scale benchmark data sets, and experimental results show that the proposed hypergraph spectral learning formulation is effective in capturing the high-order relations in multi-label problems. Results also indicate that the approximate formulation is much more efficient than the original one, while keeping competitive classification performance. Relational phrases (e.g., “got married to”) and their hypernyms (e.g., “is a relative of”) are central for many tasks including question answering, open information extraction, paraphrasing, and entailment detection. This has motivated the development of several linguistic resources (e.g. DIRT, PATTY, and WiseNet) which systematically collect and organize relational phrases. These resources have demonstrable practical benefits, but are each limited due to noise, sparsity, or size. We present a new general-purpose method, RELLY, for constructing a large hypernymy graph of relational phrases with high-quality subsumptions using collective probabilistic programming techniques. Our graph induction approach integrates small highprecision knowledge bases together with large automatically curated resources, and reasons collectively to combine these resources into a consistent graph. Using RELLY, we construct a high-coverage, high-precision hypernymy graph consisting of 20K relational phrases and 35K hypernymy links. Our evaluation indicates a hypernymy link precision of 78 , and demonstrates the value of this resource for a document-relevance ranking task. Non-uniform hypergraphs appear in various domains of computer science as in the satisfiability problems and in data analysis. We analyse a general model where the probability for an edge of size t to belong to the hypergraph depends of a parameter ω t of the model. It is a natural generalization of the models of graphs used by Flajolet, Knuth and Pittel 11] and Janson, Knuth, Łuczak and Pittel 16]. The present paper follows the same general approach based on analytic combinatorics. We show that many analytic tools developed for the analysis of graphs can be extended surprisingly well to non-uniform hypergraphs. More specifically, we analyze their typical structure before and near the birth of the complex components, that are the connected components with more than one cycle. We derive the asymptotic number of sparse connected hypergraphs as their complexity, defined as the excess, increases. Although less natural than the number of edges, this parameter allows a precise description of the structure of hypergraphs. Finally, we compute some statistics of the model to link number of edges and excess. In this work, we show the deficiencies of the graph model for decomposing sparse matrices for parallel matrix-vector multiplication. Then, we propose two hypergraph models which avoid all deficiencies of the graph model. The proposed models reduce the decomposition problem to the well-known hypergraph partitioning problem widely encountered in circuit partitioning in VLSI. We have implemented fast Kernighan-Lin based graph and hypergraph partitioning heuristics and used the successful multilevel graph partitioning tool (Metis) for the experimental evaluation of the validity of the proposed hypergraph models. We have also developed a multilevel hypergraph partitioning heuristic for experimenting the performance of the multilevel approach on hypergraph partitioning. Experimental results on sparse matrices, selected from Harwell-Boeing collection and NETLIB suite, confirm both the validity of our proposed hypergraph models and appropriateness of the multilevel approach to hypergraph partitioning. Recent advances in network science allows the modeling and analysis of complex inter-related entities. These entities often interact with each other in a number of different ways. Simple graphs fail to capture these multiple types of relationships requiring more sophisticated mathematical structures. One such structure is multigraph, where entities (or nodes) can be linked to each other through multiple edges. In this paper we describe a new method to manage multiple types of relationships existing in multigraphs. Our approach is based on the concept of pair of nodes (edges) and, in particular, we study how nodes on different layers interact which each other considering the edges they share. We propose a two level strategy that summarizes global local multigraph features. The global view helps us to gain knowledge related to the characteristics of layers and how they interact while the local view provides an analysis of individual layers highlighting edge properties such as cluster structure. Our proposal is complementary to standard node-link diagram and it can be coupled with such techniques in order to intelligently explore multigraphs. The proposed visualization is tested on a real world case study and the outcomes point out the ability of our proposal to discover patterns present in the data. Given multimillion-node graphs such as "who-follows-whom", "patent-cites-patent", "user-likes-page" and "actor director-makes-movie" networks, how can we find unexpected behaviors? When companies operate on the graphs with monetary incentives to sell Twitter "Followers" and Facebook page "Likes", the graphs show strange connectivity patterns. In this paper, we study a complete graph from a large Twitter-style social network, spanning up to 3.33 billion edges. We report strange deviations from typical patterns like smooth degree distributions. We find that such deviations are often due to "lockstep behavior" that large groups of followers connect to the same groups of followees. Our first contribution is that we study strange patterns on the adjacency matrix and in the spectral subspaces with respect to several flavors of lockstep. We discover that (a) the lockstep behaviors on the graph shape dense "block" in its adjacency matrix and creates "rays" in spectral subspaces, and (b) partially overlapping of the behaviors shape "staircase" in its adjacency matrix and creates "pearls" in spectral subspaces. The second contribution is that we provide a fast algorithm, using the discovery as a guide for practitioners, to detect users who offer the lockstep behaviors in undirected directed bipartite graphs. We carry out extensive experiments on both synthetic and real datasets, as well as public datasets from IMDb and US Patent. The results demonstrate the scalability and effectiveness of our proposed algorithm. Many knowledge repositories nowadays contain billions of triplets, i.e. (head-entity, relationship, tail-entity), as relation instances. These triplets form a directed graph with entities as nodes and relationships as edges. However, this kind of symbolic and discrete storage structure makes it difficult for us to exploit the knowledge to enhance other intelligenceacquired applications (e.g. the QuestionAnswering System), as many AI-related algorithms prefer conducting computation on continuous data. Therefore, a series of emerging approaches have been proposed to facilitate knowledge computing via encoding the knowledge graph into a low-dimensional embedding space. TransE is the latest and most promising approach among them, and can achieve a higher performance with fewer parameters by modeling the relationship as a transitional vector from the head entity to the tail entity. Unfortunately, it is not flexible enough to tackle well with the various mapping properties of triplets, even though its authors spot the harm on performance. In this paper, we thus propose a superior model called TransM to leverage the structure of the knowledge graph via pre-calculating the distinct weight for each training triplet according to its relational mapping property. In this way, the optimal function deals with each triplet depending on its own weight. We carry out extensive experiments to compare TransM with the state-of-the-art method TransE and other prior arts. The performance of each approach is evaluated within two different application scenarios on several benchmark datasets. Results show that the model we proposed significantly outperforms the former ones with lower parameter complexity as TransE. We usually endow the investigated objects with pairwise relationships, which can be illustrated as graphs. In many real-world problems, however, relationships among the objects of our interest are more complex than pair-wise. Naively squeezing the complex relationships into pairwise ones will inevitably lead to loss of information which can be expected valuable for our learning tasks however. Therefore we consider using hypergraphs instead to completely represent complex relationships among the objects of our interest, and thus the problem of learning with hypergraphs arises. Our main contribution in this paper is to generalize the powerful methodology of spectral clustering which originally operates on undirected graphs to hypergraphs, and further develop algorithms for hypergraph embedding and transductive classification on the basis of the spectral hypergraph clustering approach. Our experiments on a number of benchmarks showed the advantages of hypergraphs over usual graphs. A hypergraph H = (V,E) is called s-orientable, if there is an assignment of each edge e ∈ E to one of its vertices v ∈ e such that no vertex is assigned more than s edges. Let Hn,m,k be a hypergraph, drawn uniformly at random from the set of all k-uniform hypergraphs with n vertices and m edges. In this paper we establish the threshold for the 1-orientability of Hn,m,k for all k ≥ 3, i.e., we determine a critical quantity ck* such that with probability 1 - o(1) the graph Hn,cn,k has a 1-orientation if c ck*. We present two applications of this result that involve the paradigm of multiple choices. First, we show how it implies sharp load thresholds for cuckoo hash tables, where each element chooses k out of n locations. Particularly, for each k ≥ 3 we prove that with probability 1 - o(1) the maximum number of elements that can be hashed is (1 - o(1))ck*n, and more items prevent the successful allocation. Second, we study random graph processes, where in each step we have the choice among any edge connecting k random vertices. Here we show the existence of a phase transition for avoiding a giant connected component, and quantify precisely the dependence on k. We present the H3 layout technique for drawing large directed graphs as node-link diagrams in 3D hyperbolic space. We can lay out much larger structures than can be handled using traditional techniques for drawing general graphs because we assume a hierarchical nature of the data. We impose a hierarchy on the graph by using domain-specific knowledge to find an appropriate spanning tree. Links which are not part of the spanning tree do not influence the layout but can be selectively drawn by user request. The volume of hyperbolic 3-space increases exponentially, as opposed to the familiar geometric increase of euclidean 3-space. We exploit this exponential amount of room by computing the layout according to the hyperbolic metric. We optimize the cone tree layout algorithm for 3D hyperbolic space by placing children on a hemisphere around the cone mouth instead of on its perimeter. Hyperbolic navigation affords a Focus+Context view of the structure with minimal visual clutter. We have successfully laid out hierarchies of over 20,000 nodes. Our implementation accommodates navigation through graphs too large to be rendered interactively by allowing the user to explicitly prune or expand subtrees. Researchers have proposed a variety of metrics to measure important graph properties, for instance, in social, biological, and computer networks. Values for a particular graph metric may capture a graph's resilience to failure or its routing efficiency. Knowledge of appropriate metric values may influence the engineering of future topologies, repair strategies in the face of failure, and understanding of fundamental properties of existing networks. Unfortunately, there are typically no algorithms to generate graphs matching one or more proposed metrics and there is little understanding of the relationships among individual metrics or their applicability to different settings. We present a new, systematic approach for analyzing network topologies. We first introduce the dK-series of probability distributions specifying all degree correlations within d-sized subgraphs of a given graph G. Increasing values of d capture progressively more properties of G at the cost of more complex representation of the probability distribution. Using this series, we can quantitatively measure the distance between two graphs and construct random graphs that accurately reproduce virtually all metrics proposed in the literature. The nature of the dK-series implies that it will also capture any future metrics that may be proposed. Using our approach, we construct graphs for d=0, 1, 2, 3 and demonstrate that these graphs reproduce, with increasing accuracy, important properties of measured and modeled Internet topologies. We find that the d=2 case is sufficient for most practical purposes, while d=3 essentially reconstructs the Internet AS-and router-level topologies exactly. We hope that a systematic method to analyze and synthesize topologies offers a significant improvement to the set of tools available to network topology and protocol researchers. Many networks can be modeled as signed graphs. These include social networks, and relationships interactions networks. Detecting sub-structures in such networks helps us understand user behavior, predict links, and recommend products. In this paper, we detect dense sub-structures from a signed graph, called quasi antagonistic communities (QACs). An antagonistic community consists of two groups of users expressing positive relationships within each group but negative relationships across groups. Instead of requiring complete set of negative links across its groups, a QAC allows a small number of inter-group negative links to be missing. We propose an algorithm, Mascot, to find all maximal quasi antagonistic communities (MQACs). Mascot consists of two stages: pruning and enumeration stages. Based on the properties of QAC, we propose four pruning rules to reduce the size of candidate graphs in the pruning stage. We use an enumeration tree to enumerate all strongly connected subgraphs in a top---down fashion in the second stage before they are used to construct MQACs. We have conducted extensive experiments using synthetic signed graphs and two real networks to demonstrate the efficiency and accuracy of the Mascot algorithm. We have also found that detecting MQACs helps us to predict the signs of links. In the framework of structural representations for applications in image understanding, we establish links between similarities, hypergraph theory and mathematical morphology. We propose new similarity measures and pseudo-metrics on lattices of hypergraphs based on morphological operators. New forms of these operators on hypergraphs are introduced as well. Some examples based on various dilations and openings on hypergraphs illustrate the relevance of our approach. We present a structured learning approach to inducing hypernym taxonomies using a probabilistic graphical model formulation. Our model incorporates heterogeneous relational evidence about both hypernymy and siblinghood, captured by semantic features based on patterns and statistics from Web n-grams and Wikipedia abstracts. For efficient inference over taxonomy structures, we use loopy belief propagation along with a directed spanning tree algorithm for the core hypernymy factor. To train the system, we extract sub-structures of WordNet and discriminatively learn to reproduce them, using adaptive subgradient stochastic optimization. On the task of reproducing sub-hierarchies of WordNet, our approach achieves a 51 error reduction over a chance baseline, including a 15 error reduction due to the non-hypernym-factored sibling features. On a comparison setup, we find up to 29 relative error reduction over previous work on ancestor F1. Trust networks, where people leave trust and distrust feedback, are becoming increasingly common. These networks may be regarded as signed graphs, where a positive edge weight captures the degree of trust while a negative edge weight captures the degree of distrust. Analysis of such signed networks has become an increasingly important research topic. One important analysis task is that of sign inference, i.e., infer unknown (or future) trust or distrust relationships given a partially observed signed network. Most state-of-the-art approaches consider the notion of structural balance in signed networks, building inference algorithms based on information about links, triads, and cycles in the network. In this paper, we first show that the notion of weak structural balance in signed networks naturally leads to a global low-rank model for the network. Under such a model, the sign inference problem can be formulated as a low-rank matrix completion problem. We show that we can perfectly recover missing relationships, under certain conditions, using state-of-the-art matrix completion algorithms. We also propose the use of a low-rank matrix factorization approach with generalized loss functions as a practical method for sign inference - this approach yields high accuracy while being scalable to large signed networks, for instance, we show that this analysis can be performed on a synthetic graph with 1.1 million nodes and 120 million edges in 10 minutes. We further show that the low-rank model can be used for other analysis tasks on signed networks, such as user segmentation through signed graph clustering, with theoretical guarantees. Experiments on synthetic as well as real data show that our low rank model substantially improves accuracy of sign inference as well as clustering. As an example, on the largest real dataset available to us (Epinions data with 130K nodes and 840K edges), our matrix factorization approach yields 94.6 accuracy on the sign inference task as compared to 90.8 accuracy using a state-of-the-art cycle-based method - moreover, our method runs in 40 seconds as compared to 10,000 seconds for the cycle-based method.