Community detectability and structural balance dynamics in signed networks

TL;DR

This paper analyzes the spectral properties and evolution of signed networks with community structure, revealing detectability and sociality transitions, and models their dynamics under structural balance theory.

Contribution

It introduces spectral analysis of stochastic block models for signed networks and links detectability and sociality transitions to structural balance dynamics.

Findings

Detectability transition occurs when community eigenvalues exit the spectral band.

Spectral analysis predicts regimes of community structure and sociality.

Structural balance dynamics lead to three distinct outcome regimes.

Abstract

We investigate signed networks with community structure with respect to their spectrum and their evolution under a dynamical model of structural balance, a prominent theory of signed social networks. The spectrum of the adjacency matrix generated by a stochastic block model with two equal size communities shows detectability transitions in which the community structure becomes manifest when its signal eigenvalue appears outside the main spectral band. The spectrum also exhibits "sociality" transitions involving the homogeneous structure representing the average tie value. We derive expressions for the eigenvalues associated with the community and homogeneous structure as well as the transition boundaries, all in good agreement with numerical results. Using the stochastically-generated networks as initial conditions for a simple model of structural balance dynamics yields three outcome regimes: two hostile factions that correspond with the initial communities, two hostile factions uncorrelated with those communities, and a single harmonious faction of all nodes. The detectability transition predicts the boundary between the assortative and mixed two-faction states and the sociality transition predicts that between the mixed and harmonious states. Our results may yield insight into the dynamics of cooperation and conflict among actors with distinct social identities.