Structural balance and interpersonal appraisals dynamics: Beyond all-to-all and two-faction networks

Abstract

Structural balance theory characterizes stable configurations of signed interpersonal appraisal networks. Existing models explaining the convergence of appraisal networks to structural balance either diverge in finite time, or could get stuck in jammed states, or converge to only complete graphs. In this paper, we study the open problem of how non-complete appraisal networks converge to structural balance while their graph topologies remain unchanged. Firstly, we present two established definitions of structural balance: the local balance and the global balance, which are equivalent in complete graphs but not in graphs with arbitrary topologies. The notion of local balance is characterized as “Friends’ friends are not enemies; Friends’ enemies are not friends; Enemies’ friends are not friends; Enemies’ enemies are not enemies.”, which is a straightforward generalization of the famous Heider’s axioms in structural balance theory. On the other hand, the global balance, widely adopted in the modeling of opinion dynamics, means that the appraisal network can be divided into two antagonistic factions. We propose two mechanisms: the symmetry-influence-homophily (SIH) dynamics and the symmetry-influence-opinion-homophily (SIOH) dynamics, in which the appraisal networks almost surely achieve the local balance (for certain initial conditions) and the global balance (for any initial condition) in finite time respectively. Along these dynamics, the connection patterns of the appraisal networks remain unchanged and thereby avoid evolving into complete graphs. Furthermore, we provide certain graph-theoretical conditions on which the local balance is equivalent to the global balance. In addition, numerical studies of our dynamics indicate some insightful take-home messages, e.g., whether multilateral relations reduce or exacerbate conflicts.