A Method to Find Community Structures Based on Information Centrality

Community structures are an important feature of many social, biological and technological networks. Here we study a variation on the method for detecting such communities proposed by Girvan and Newman and based on the idea of using centrality measures to define the community boundaries (M. Girvan and M. E. J. Newman, Community structure in social and biological networks Proc. Natl. Acad. Sci. USA 99, 7821-7826 (2002)). We develop an algorithm of hierarchical clustering that consists in finding and removing iteratively the edge with the highest information centrality. We test the algorithm on computer generated and real-world networks whose community structure is already known or has been studied by means of other methods. We show that our algorithm, although it runs to completion in a time O(n^4), is very effective especially when the communities are very mixed and hardly detectable by the other methods.Comment: 13 pages, 13 figures. Final version accepted for publication in Physical Review

Statistical Mechanics of Community Detection

Starting from a general \textit{ansatz}, we show how community detection can be interpreted as finding the ground state of an infinite range spin glass. Our approach applies to weighted and directed networks alike. It contains the \textit{at hoc} introduced quality function from \cite{ReichardtPRL} and the modularity $Q$ as defined by Newman and Girvan \cite{Girvan03} as special cases. The community structure of the network is interpreted as the spin configuration that minimizes the energy of the spin glass with the spin states being the community indices. We elucidate the properties of the ground state configuration to give a concise definition of communities as cohesive subgroups in networks that is adaptive to the specific class of network under study. Further we show, how hierarchies and overlap in the community structure can be detected. Computationally effective local update rules for optimization procedures to find the ground state are given. We show how the \textit{ansatz} may be used to discover the community around a given node without detecting all communities in the full network and we give benchmarks for the performance of this extension. Finally, we give expectation values for the modularity of random graphs, which can be used in the assessment of statistical significance of community structure

The networked seceder model: Group formation in social and economic systems

The seceder model illustrates how the desire to be different than the average can lead to formation of groups in a population. We turn the original, agent based, seceder model into a model of network evolution. We find that the structural characteristics our model closely matches empirical social networks. Statistics for the dynamics of group formation are also given. Extensions of the model to networks of companies are also discussed

A new measure for community structures through indirect social connections

Based on an expert systems approach, the issue of community detection can be conceptualized as a clustering model for networks. Building upon this further, community structure can be measured through a clustering coefficient, which is generated from the number of existing triangles around the nodes over the number of triangles that can be hypothetically constructed. This paper provides a new definition of the clustering coefficient for weighted networks under a generalized definition of triangles. Specifically, a novel concept of triangles is introduced, based on the assumption that, should the aggregate weight of two arcs be strong enough, a link between the uncommon nodes can be induced. Beyond the intuitive meaning of such generalized triangles in the social context, we also explore the usefulness of them for gaining insights into the topological structure of the underlying network. Empirical experiments on the standard networks of 500 commercial US airports and on the nervous system of the Caenorhabditis elegans support the theoretical framework and allow a comparison between our proposal and the standard definition of clustering coefficient

Detection of the elite structure in a virtual multiplex social system by means of a generalized KK-core

Elites are subgroups of individuals within a society that have the ability and means to influence, lead, govern, and shape societies. Members of elites are often well connected individuals, which enables them to impose their influence to many and to quickly gather, process, and spread information. Here we argue that elites are not only composed of highly connected individuals, but also of intermediaries connecting hubs to form a cohesive and structured elite-subgroup at the core of a social network. For this purpose we present a generalization of the $K$-core algorithm that allows to identify a social core that is composed of well-connected hubs together with their `connectors'. We show the validity of the idea in the framework of a virtual world defined by a massive multiplayer online game, on which we have complete information of various social networks. Exploiting this multiplex structure, we find that the hubs of the generalized $K$-core identify those individuals that are high social performers in terms of a series of indicators that are available in the game. In addition, using a combined strategy which involves the generalized $K$-core and the recently introduced $M$-core, the elites of the different 'nations' present in the game are perfectly identified as modules of the generalized $K$-core. Interesting sudden shifts in the composition of the elite cores are observed at deep levels. We show that elite detection with the traditional $K$-core is not possible in a reliable way. The proposed method might be useful in a series of more general applications, such as community detection.Comment: 13 figures, 3 tables, 19 pages. Accepted for publication in PLoS ON

Characterizing and Detecting Cohesive Subgroups with Applications to Social and Brain Networks

Many complex systems involve entities that interact with each other through various relationships (e.g., people in social systems, neurons in the brain). These entities and interactions are commonly represented using graphs due to several advantages. This dissertation focuses on developing theory and algorithms for novel methods in graph theory and optimization, and their applications to social and brain networks. Specifically, the major contributions of this dissertation are three fold. First, this dissertation aims not only to develop a new clique relaxation model based on a structural metric, clustering coefficient, but also to introduce a novel graph clustering algorithm using this model. Clique relaxations are used in classical models of cohesive subgroups in social network analysis. Clustering coefficient was introduced more recently as a structural feature characterizing small-world networks. Leveraging the similarities between the concepts of cohesive subgroups and small-world networks (i.e., graphs that are highly clustered with small path lengths). The first part of this dissertation introduces a new clique relaxation, α-cluster, defined by enforcing a lower bound α on the clustering coefficient in the corresponding induced subgraph. Two different definitions of the clustering coefficient are considered, namely, the local and global clustering coefficient. Certain structural properties of α-clusters are analyzed, and mathematical optimization models for determining the largest size α-clusters in a network are developed and applied to several real-life social network instances. In addition, a network clustering algorithm based on local α-cluster is introduced and successfully evaluated. Second, this dissertation explores a novel mathematical model called the maximum independent union of cliques problem (max IUC problem), which arises as a special case of α-clusters. It is an interesting problem for which both the maximum clique and maximum independent sets are feasible solutions and individually their corresponding sizes are lower bounds for the size of the IUC solution. After presenting the structural properties as well as the complexity results of different graph types (planar, unit disk graphs and claw-free graphs), an integer programming formulation is developed, followed by a branch-and-bound algorithm and several heuristic methods to approximate the maximum independent union of cliques problem. The developed methods have been empirically evaluated on many benchmark instances. Finally, this dissertation, in collaboration with Texas Institute of Preclinical Studies (TIPS), applies clique relaxation models to explore a new experimental data to understand the effect of concussion on animal brains. Our research involves cohesive and robust clustering analysis of animal brain networks utilizing a unique and novel experimental data. In collaboration with TIPS, we have analyzed multiple pairs of fMRI data about animal brains that are measured before and after a concussion. We utilize network analysis to first identify the similar regions in animal brains, and then compare how these regions as well as graph structural properties change before and after a concussion. To the best of our knowledge, this study is unique in the literature in that it not only explicitly examines the relation between concussion level and the functional unit interaction but also uses very detailed and fine-grained fMRI measurements of brain data