Motif Clustering and Overlapping Clustering for Social Network Analysis

Motivated by applications in social network community analysis, we introduce a new clustering paradigm termed motif clustering. Unlike classical clustering, motif clustering aims to minimize the number of clustering errors associated with both edges and certain higher order graph structures (motifs) that represent "atomic units" of social organizations. Our contributions are two-fold: We first introduce motif correlation clustering, in which the goal is to agnostically partition the vertices of a weighted complete graph so that certain predetermined "important" social subgraphs mostly lie within the same cluster, while "less relevant" social subgraphs are allowed to lie across clusters. We then proceed to introduce the notion of motif covers, in which the goal is to cover the vertices of motifs via the smallest number of (near) cliques in the graph. Motif cover algorithms provide a natural solution for overlapping clustering and they also play an important role in latent feature inference of networks. For both motif correlation clustering and its extension introduced via the covering problem, we provide hardness results, algorithmic solutions and community detection results for two well-studied social networks

Clustering and the hyperbolic geometry of complex networks

Clustering is a fundamental property of complex networks and it is the mathematical expression of a ubiquitous phenomenon that arises in various types of self-organized networks such as biological networks, computer networks or social networks. In this paper, we consider what is called the global clustering coefficient of random graphs on the hyperbolic plane. This model of random graphs was proposed recently by Krioukov et al. as a mathematical model of complex networks, under the fundamental assumption that hyperbolic geometry underlies the structure of these networks. We give a rigorous analysis of clustering and characterize the global clustering coefficient in terms of the parameters of the model. We show how the global clustering coefficient can be tuned by these parameters and we give an explicit formula for this function.Comment: 51 pages, 1 figur

Spectral clustering and the high-dimensional stochastic blockmodel

Networks or graphs can easily represent a diverse set of data sources that are characterized by interacting units or actors. Social networks, representing people who communicate with each other, are one example. Communities or clusters of highly connected actors form an essential feature in the structure of several empirical networks. Spectral clustering is a popular and computationally feasible method to discover these communities. The stochastic blockmodel [Social Networks 5 (1983) 109--137] is a social network model with well-defined communities; each node is a member of one community. For a network generated from the Stochastic Blockmodel, we bound the number of nodes "misclustered" by spectral clustering. The asymptotic results in this paper are the first clustering results that allow the number of clusters in the model to grow with the number of nodes, hence the name high-dimensional. In order to study spectral clustering under the stochastic blockmodel, we first show that under the more general latent space model, the eigenvectors of the normalized graph Laplacian asymptotically converge to the eigenvectors of a "population" normalized graph Laplacian. Aside from the implication for spectral clustering, this provides insight into a graph visualization technique. Our method of studying the eigenvectors of random matrices is original.Comment: Published in at http://dx.doi.org/10.1214/11-AOS887 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Random Networks with Tunable Degree Distribution and Clustering

We present an algorithm for generating random networks with arbitrary degree distribution and Clustering (frequency of triadic closure). We use this algorithm to generate networks with exponential, power law, and poisson degree distributions with variable levels of clustering. Such networks may be used as models of social networks and as a testable null hypothesis about network structure. Finally, we explore the effects of clustering on the point of the phase transition where a giant component forms in a random network, and on the size of the giant component. Some analysis of these effects is presented.Comment: 9 pages, 13 figures corrected typos, added two references, reorganized reference

Why social networks are different from other types of networks

We argue that social networks differ from most other types of networks, including technological and biological networks, in two important ways. First, they have non-trivial clustering or network transitivity, and second, they show positive correlations, also called assortative mixing, between the degrees of adjacent vertices. Social networks are often divided into groups or communities, and it has recently been suggested that this division could account for the observed clustering. We demonstrate that group structure in networks can also account for degree correlations. We show using a simple model that we should expect assortative mixing in such networks whenever there is variation in the sizes of the groups and that the predicted level of assortative mixing compares well with that observed in real-world networks.Comment: 9 pages, 2 figure

Disassortative mixing in online social networks

The conventional wisdom is that social networks exhibit an assortative mixing pattern, whereas biological and technological networks show a disassortative mixing pattern. However, the recent research on the online social networks modifies the widespread belief, and many online social networks show a disassortative or neutral mixing feature. Especially, we found that an online social network, Wealink, underwent a transition from degree assortativity characteristic of real social networks to degree disassortativity characteristic of many online social networks, and the transition can be reasonably elucidated by a simple network model that we propose. The relations among network assortativity, clustering, and modularity are also discussed in the paper.Comment: 6 pages, 5 figures, 1 tabl