Finding overlapping communities in social networks

Abstract

A community in a social network is usually understood to be a group of nodes more densely connected with each other than with the rest of the network. This is an important concept in most domains where networks arise: social, technological, biological, etc. For many years algorithms for finding communities implicitly assumed communities are nonoverlapping (leading to use of clustering-based approaches) but there is increasing interest in finding overlapping communities. A barrier to finding communities is that the solution concept is often defined in terms of an NP-complete problem such as Clique or Hierarchical Clustering.This paper seeks to initiate a rigorous approach to the problem of finding overlapping communities, where "rigorous" means that we clearly state the following: (a) the object sought by our algorithm (b) the assumptions about the underlying network (c) the (worst-case) running time.Our assumptions about the network lie between worst-case and average-case. An averagecase analysis would require a precise probabilistic model of the network, on which there is currently no consensus. However, some plausible assumptions about network parameters can be gleaned from a long body of work in the sociology community spanning five decades focusing on the study of individual communities and ego-centric networks (in graph theoretic terms, this is the subgraph induced on a node's neighborhood). Thus our assumptions are somewhat "local" in nature. Nevertheless they suffice to permit a rigorous analysis of running time of algorithms that recover global structure.Our algorithms use random sampling similar to that in property testing and algorithms for dense graphs. We note however that our networks are not necessarily dense graphs, not even in local neighborhoods.Our algorithms explore a local-global relationship between ego-centric and socio-centric networks that we hope will provide a fruitful framework for future work both in computer science and sociology.

Low-distortion Inference of Latent Similarities from a Multiplex Social Network

“…More importantly, to the extent that precise distributions have been empirically tested, remarkable fits have been found [3,5,34] with Kleinberg's inverse polynomial distribution [27,28]. 4 Furthermore, our main constant-distortion result holds for a much more general class of distributions, including logit-linear distributions.…”

“…Both choices (Euclidean and near-uniform) are ubiquitous in past work 5 [20,23,24,25,27,31,41,43], and are made mostly for technical convenience; they allow us to separate the conceptual difficulty of teasing apart different metrics and inferring distances with low distortion from the technical difficulty of dealing with arbitrary metric spaces. We believe that future work 4 However, links that appear long could plausibly be short in another metric; whether inverse polynomial distributions remain prevalent when multiple metrics are considered is an interesting -although difficult -direction for future empirical work. 5 In many respects, our kind of latent space models deteriorate if node densities can be highly non-uniform [19].…”

On the Advantage of Overlapping Clusters for Minimizing Conductance

“…Following the conference version of this paper, there have been various attempts to study the overlapping clustering problem from a theoretical perspective, and present a rigorous study of computing overlapping communities in social networks. For example, Arora et al [4] study the problem of overlapping communities, present an axiomatic approach for defining overlapping social communities, and design approximation algorithms for discovering such communities in a set of randomly generated graphs. Moreover, Balcan et al [8] considered the problem of enumerating and identifying communities generalizing and refining the definition of communi-ties by Mishra et al [29].…”

“…Moreover, Balcan et al [8] considered the problem of enumerating and identifying communities generalizing and refining the definition of communi-ties by Mishra et al [29]. These papers do not explicitly attempt to optimize the cut conductance, but they consider other related notions for quality of clusters like the cut value and density of clustersl [4,8]. All the recent research about the topic of overlapping clustering imply the need for exploring this topic from a more algorithmic and theoretical perspectivel [4,8].…”

“…These papers do not explicitly attempt to optimize the cut conductance, but they consider other related notions for quality of clusters like the cut value and density of clustersl [4,8]. All the recent research about the topic of overlapping clustering imply the need for exploring this topic from a more algorithmic and theoretical perspectivel [4,8].…”