Abstract
Many real world networks are reported to have hierarchically modular organization. However, there exists no algorithm-independent metric to characterize hierarchical modularity in a complex system. The main results of the paper are a set of methods to address this problem. First, classical results from random matrix theory are used to derive the spectrum of a typical stochastic block model hierarchical modular network form. Second, it is shown that hierarchical modularity can be fingerprinted using the spectrum of its largest eigenvalues and gaps between clusters of closely spaced eigenvalues that are well separated from the bulk distribution of eigenvalues around the origin. Third, some well-known results on fingerprinting non-hierarchical modularity in networks automatically follow as special cases, threreby unifying these previously fragmented results. Finally, using these spectral results, it is found that the limits of detection of modularity can be empirically established by studying the mean values of the largest eigenvalues and the limits of the bulk distribution of eigenvalues for an ensemble of networks. It is shown that even when modularity and hierarchical modularity are present in a weak form in the network, they are impossible to detect, because some of the leading eigenvalues fall within the bulk distribution. This provides a threshold for the detection of modularity. Eigenvalue distributions of some technological, social, and biological networks are studied, and the implications of detecting hierarchical modularity in real world networks are discussed.
Highlights
Many real world networks have been reported to have modular or hierarchical modular organization, including social networks [1], collaboration networks [1], biological networks such as structural and functional brain networks [2,3,4,5], metabolic networks [6], and gene expression networks [7], and technological networks such as the Internet, the World Wide Web, and the global air transportation network [1]
We empirically show that when probability parameters for instantiating edges in networks are varied, there is a threshold set by the probabilities and the limits of the bulk distribution of eigenvalues around the origin beyond which hierarchical modularity and modularity cannot be detected even if present
We address the problem of characterizing the hierarchical modularity of a network
Summary
Many real world networks have been reported to have modular or hierarchical modular organization, including social networks [1], collaboration networks [1], biological networks such as structural and functional brain networks [2,3,4,5], metabolic networks [6], and gene expression networks [7], and technological networks such as the Internet, the World Wide Web, and the global air transportation network [1]. A modularity detection algorithm that is based on strict graph partitioning techniques will fail to find overlaps between communities and hierarchical organization, unless modified. The computation of Q requires that the network first be divided into modules before it can be evaluated, and provides no information on the uniqueness of the postulated modules; i.e., which solution should be preferred if two solutions have the same Q value. No such well accepted metric exists for measurement of hierarchical modularity in networks, there exist some modularity detection algorithms based on quantifying the quality of hierarchical modularity and partitions in network structure [10]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
