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Register a new userA Short Introduction to Local Graph Clustering Methods and Software
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Added by
Kimon Fountoulakis
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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Graph clustering has many important applications in computing, but due to the increasing sizes of graphs, even traditionally fast clustering methods can be computationally expensive for real-world graphs of interest. Scalability problems led to the development of local graph clustering algorithms that come with a variety of theoretical guarantees. Rather than return a global clustering of the entire graph, local clustering algorithms return a single cluster around a given seed node or set of seed nodes. These algorithms improve scalability because they use time and memory resources that depend only on the size of the cluster returned, instead of the size of the input graph. Indeed, for many of them, their running time grows linearly with the size of the output. In addition to scalability arguments, local graph clustering algorithms have proven to be very useful for identifying and interpreting small-scale and meso-scale structure in large-scale graphs. As opposed to heuristic operational procedures, this class of algorithms comes with strong algorithmic and statistical theory. These include statistical guarantees that prove they have implicit regularization properties. One of the challenges with the existing literature on these approaches is that they are published in a wide variety of areas, including theoretical computer science, statistics, data science, and mathematics. This has made it difficult to relate the various algorithms and ideas together into a cohesive whole. We have recently been working on unifying these diverse perspectives through the lens of optimization as well as providing software to perform these computations in a cohesive fashion. In this note, we provide a brief introduction to local graph clustering, we provide some representative examples of our perspective, and we introduce our software named Local Graph Clustering (LGC).
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Networks can represent a wide range of complex systems, such as social, biological and technological systems. Link prediction is one of the most important problems in network analysis, and has attracted much research interest recently. Many link prediction methods have been proposed to solve this problem with various technics. We can note that clustering information plays an important role in solving the link prediction problem. In previous literatures, we find node clustering coefficient appears frequently in many link prediction methods. However, node clustering coefficient is limited to describe the role of a common-neighbor in different local networks, because it can not distinguish different clustering abilities of a node to different node pairs. In this paper, we shift our focus from nodes to links, and propose the concept of asymmetric link clustering (ALC) coefficient. Further, we improve three node clustering based link prediction methods via the concept of ALC. The experimental results demonstrate that ALC-based methods outperform node clustering based methods, especially achieving remarkable improvements on food web, hamster friendship and Internet networks. Besides, comparing with other methods, the performance of ALC-based methods are very stable in both globalized and personalized top-L link prediction tasks.
The study of social networks is a burgeoning research area. However, most existing work deals with networks that simply encode whether relationships exist or not. In contrast, relationships in signed networks can be positive (like, trust) or negative (dislike, distrust). The theory of social balance shows that signed networks tend to conform to some local patterns that, in turn, induce certain global characteristics. In this paper, we exploit both local as well as global aspects of social balance theory for two fundamental problems in the analysis of signed networks: sign prediction and clustering. Motivated by local patterns of social balance, we first propose two families of sign prediction methods: measures of social imbalance (MOIs), and supervised learning using high order cycles (HOCs). These methods predict signs of edges based on triangles and ell-cycles for relatively small values of ell. Interestingly, by examining measures of social imbalance, we show that the classic Katz measure, which is used widely in unsigned link prediction, actually has a balance theoretic interpretation when applied to signed networks. Furthermore, motivated by the global structure of balanced networks, we propose an effective low rank modeling approach for both sign prediction and clustering. For the low rank modeling approach, we provide theoretical performance guarantees via convex relaxations, scale it up to large problem sizes using a matrix factorization based algorithm, and provide extensive experimental validation including comparisons with local approaches. Our experimental results indicate that, by adopting a more global viewpoint of balance structure, we get significant performance and computational gains in prediction and clustering tasks on signed networks. Our work therefore highlights the usefulness of the global aspect of balance theory for the analysis of signed networks.
In this short review we describe some aspects of $kappa$-deformation. After discussing the algebraic and geometric approaches to $kappa$-Poincare algebra we construct the free scalar field theory, both on non-commutative $kappa$-Minkowski space and on curved momentum space. Finally, we make a few remarks concerning interacting scalar field.
We discuss how to construct models of interacting anyons by generalizing quantum spin Hamiltonians to anyonic degrees of freedom. The simplest interactions energetically favor pairs of anyons to fuse into the trivial (identity) channel, similar to the quantum Heisenberg model favoring pairs of spins to form spin singlets. We present an introduction to the theory of anyons and discuss in detail how basis sets and matrix representations of the interaction terms can be obtained, using non-Abelian Fibonacci anyons as example. Besides discussing the golden chain, a one-dimensional system of anyons with nearest neighbor interactions, we also present the derivation of more complicated interaction terms, such as three-anyon interactions in the spirit of the Majumdar-Ghosh spin chain, longer range interactions and two-leg ladders. We also discuss generalizations to anyons with general non-Abelian su(2)_k statistics. The k to infinity limit of the latter yields ordinary SU(2) spin chains.
Recently, there has been considerable research interest in graph clustering aimed at data partition using the graph information. However, one limitation of the most of graph-based methods is that they assume the graph structure to operate is fixed and reliable. And there are inevitably some edges in the graph that are not conducive to graph clustering, which we call spurious edges. This paper is the first attempt to employ graph pooling technique for node clustering and we propose a novel dual graph embedding network (DGEN), which is designed as a two-step graph encoder connected by a graph pooling layer to learn the graph embedding. In our model, it is assumed that if a node and its nearest neighboring node are close to the same clustering center, this node is an informative node and this edge can be considered as a cluster-friendly edge. Based on this assumption, the neighbor cluster pooling (NCPool) is devised to select the most informative subset of nodes and the corresponding edges based on the distance of nodes and their nearest neighbors to the cluster centers. This can effectively alleviate the impact of the spurious edges on the clustering. Finally, to obtain the clustering assignment of all nodes, a classifier is trained using the clustering results of the selected nodes. Experiments on five benchmark graph datasets demonstrate the superiority of the proposed method over state-of-the-art algorithms.
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