Statistical Behavior of Embeddedness and Communities of Overlapping Cliques in Online Social Networks

TL;DR

This paper investigates higher-order structural characteristics like embeddedness and community sizes in online social networks, providing empirical evidence, mathematical models, and demonstrating their relevance and applications.

Contribution

It introduces new empirical findings and models for higher-order network structures, particularly embeddedness and overlapping clique communities, in social networks.

Findings

Embeddedness distribution exhibits rich behavior not captured by traditional models.

A formal proof shows random k-tree models have power law embeddedness distributions.

A variant of the random k-tree model captures power law distributions of overlapping clique community sizes.

Abstract

Degree distribution of nodes, especially a power law degree distribution, has been regarded as one of the most significant structural characteristics of social and information networks. Node degree, however, only discloses the first-order structure of a network. Higher-order structures such as the edge embeddedness and the size of communities may play more important roles in many online social networks. In this paper, we provide empirical evidence on the existence of rich higherorder structural characteristics in online social networks, develop mathematical models to interpret and model these characteristics, and discuss their various applications in practice. In particular, 1) We show that the embeddedness distribution of social links in many social networks has interesting and rich behavior that cannot be captured by well-known network models. We also provide empirical results showing a clear correlation between the embeddedness distribution and the average number of messages communicated between pairs of social network nodes. 2) We formally prove that random k-tree, a recent model for complex networks, has a power law embeddedness distribution, and show empirically that the random k-tree model can be used to capture the rich behavior of higherorder structures we observed in real-world social networks. 3) Going beyond the embeddedness, we show that a variant of the random k-tree model can be used to capture the power law distribution of the size of communities of overlapping cliques discovered recently.