Spreading and Structural Balance on Signed Networks

TL;DR

This paper classifies signed networks into balanced, antibalanced, and unbalanced types, analyzes their spectral properties, and explores how these classifications influence dynamics, supported by theoretical insights and numerical experiments.

Contribution

It introduces a new classification of signed networks and characterizes their spectral properties, linking network structure to dynamics in both linear and nonlinear systems.

Findings

Spectral radius is smaller in strictly unbalanced networks

Balance type influences linear and nonlinear dynamics

Two new measures for characterizing unbalanced networks

Abstract

Two competing types of interactions often play an important part in shaping system behavior, such as activatory or inhibitory functions in biological systems. Hence, signed networks, where each connection can be either positive or negative, have become popular models over recent years. However, the primary focus of the literature is on the unweighted and structurally unbalanced ones, where all cycles have an even number of negative edges. Hence here, we first introduce a classification of signed networks into balanced, antibalanced or strictly unbalanced ones, and then characterize each type of signed networks in terms of the spectral properties of the signed weighted adjacency matrix. In particular, we show that the spectral radius of the matrix with signs is smaller than that without if and only if the signed network is strictly unbalanced. These properties are important to understand the dynamics on signed networks, both linear and nonlinear ones. Specifically, we find consistent patterns in a linear and a nonlinear dynamics theoretically, depending on their type of balance. We also propose two measures to further characterize strictly unbalanced networks, motivated by perturbation theory. Finally, we numerically verify these properties through experiments on both synthetic and real networks.