Characters and patterns of communities in networks

TL;DR

This paper introduces new notions to characterize network communities, revealing their core structures, heterogeneity, and a common three-degree separation property across real networks.

Contribution

It proposes internal and external dominating sets, internal and external slopes, and demonstrates their significance in understanding community structure and properties.

Findings

Most communities have small internal and external dominating sets.

Communities exhibit heterogeneity with core/periphery structures.

Average community distance is approximately three in real networks.

Abstract

In this paper, we propose some new notions to characterize and analyze the communities. The new notions are general characters of the communities or local structures of networks. At first, we introduce the notions of internal dominating set and external dominating set of a community. We show that most communities in real networks have a small internal dominating set and a small external dominating set, and that the internal dominating set of a community keeps much of the information of the community. Secondly, based on the notions of the internal dominating set and the external dominating set, we define an internal slope (ISlope, for short) and an external slope (ESlope, for short) to measure the internal heterogeneity and external heterogeneity of a community respectively. We show that the internal slope (ISlope) of a community largely determines the structure of the community, that most communities in real networks are heterogeneous, meaning that most of the communities have a core/periphery structure, and that both ISlopes and ESlopes (reflecting the structure of communities) of all the communities of a network approximately follow a normal distribution. Therefore typical values of both ISolpes and ESoples of all the communities of a given network are in a narrow interval, and there is only a small number of communities having ISlopes or ESlopes out of the range of typical values of the ISlopes and ESlopes of the network. Finally, we show that all the communities of the real networks we studied, have a three degree separation phenomenon, that is, the average distance of communities is approximately 3, implying a general property of true communities for many real networks, and that good community finding algorithms find communities that amplify clustering coefficients of the networks, for many real networks.