Hierarchical Fuzzy Spectral Clustering in Social Networks using Spectral Characterization

An important aspect of community analysis is not only determining the communities within the network, but also sub-communities and hierarchies. We present an approach for finding hierarchies in social networks that uses work from random matrix theory to estimate the number of clusters. The method analyzes the spectral fingerprint of the network to determine the level of hierarchy in the network. Using this information to inform the choice of clusters, the network is broken into successively smaller communities that are attached to their parents via Jaccard similarity. The efficacy of the approach is examined on two well known real world social networks as well as a political social network derived from campaign finance data. Introduction Many real world networks are characterized by dense subnetworks that are commonly referred to as communities and are generally composed of groups of nodes that have elements in common with each other. Examples of networks that have community structure can be drawn from social (Fortunato 2009), biological (Power et al. 2011), gene expression (Zhang and Horvath 2005), and many other types of networks. Since the communities can represent fundamental properties of the network, their discovery is important for understanding the nature of the networks (Newman and Girvan 2004),(Flake et al. 2002). The primary focus of this paper is on social networks. To best represent the communities, a classification of the nodes into clusters should satisfy two important realities of many social networks: overlap and hierarchy. For the first, nodes within the network may belong to multiple communities. Much like in human social groups, an individual may belong to more than one community or have multiple affiliations (Zhang, Wang, and Zhang 2007). Hierarchy is another important aspect of some social networks wherein smaller communities together make up larger ones. Military, business, and political hierarchies are all examples where individual smaller groups combine into a larger group. ∗also affiliated with the National Institute on Money in State Politics Copyright c © 2015, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. There is already a wealth of research on finding communities within networks. Some initial work focused on crisp splits of the network into non-overlapping, non-hierarchical communities (Newman and Girvan 2004), (Newman 2006). As part of this, a method for evaluating the quality of a partitioning of the data into clusters was developed called modularity. The idea behind modularity is to determine how well a community split describes the likelihood of the community as it relates to a null model, where each node keeps the same degree but is connected at random to other nodes. This is defined by Q = ∑