Covariance, correlation matrix and the multi-scale community structure of networks

TL;DR

This paper introduces a correlation matrix approach for detecting multi-scale community structures in networks, demonstrating its superiority over covariance and modularity matrices through extensive tests on real and artificial networks.

Contribution

It proposes a correlation matrix method based on rescaling transformations to better identify multi-scale community structures in heterogeneous networks.

Findings

Correlation matrix outperforms covariance and modularity matrices in community detection.

Rescaling transformation is crucial for identifying multi-scale structures.

The approach provides a new perspective for dimension reduction in network analysis.

Abstract

Empirical studies show that real world networks often exhibit multiple scales of topological descriptions. However, it is still an open problem how to identify the intrinsic multiple scales of networks. In this article, we consider detecting the multi-scale community structure of network from the perspective of dimension reduction. According to this perspective, a covariance matrix of network is defined to uncover the multi-scale community structure through the translation and rotation transformations. It is proved that the covariance matrix is the unbiased version of the well-known modularity matrix. We then point out that the translation and rotation transformations fail to deal with the heterogeneous network, which is very common in nature and society. To address this problem, a correlation matrix is proposed through introducing the rescaling transformation into the covariance matrix. Extensive tests on real world and artificial networks demonstrate that the correlation matrix significantly outperforms the covariance matrix, identically the modularity matrix, as regards identifying the multi-scale community structure of network. This work provides a novel perspective to the identification of community structure and thus various dimension reduction methods might be used for the identification of community structure. Through introducing the correlation matrix, we further conclude that the rescaling transformation is crucial to identify the multi-scale community structure of network, as well as the translation and rotation transformations.