Friedkin-Johnsen Social Influence Dynamics on Networks: A Boundary-Value Formulation and Influenceability Measures

TL;DR

This paper rigorously analyzes the Friedkin-Johnsen social influence model by framing it as a boundary-value problem, deriving solutions, sensitivity measures, and influenceability metrics, with applications to network centrality analysis.

Contribution

It introduces a boundary-value formulation for the Friedkin-Johnsen model, derives explicit solutions and sensitivity measures, and defines influenceability metrics with practical network analysis applications.

Findings

Explicit steady-state solutions for opinion dynamics.

Quantitative convergence rate estimates.

Influenceability measures related to network centralities.

Abstract

This article presents a rigorous mathematical analysis of the Friedkin--Johnsen model of social influence on networks. We frame the opinion dynamics as a discrete boundary-value problem on a network, emphasizing the role of stubborn (boundary) and susceptible (interior) agents in shaping opinion evolution. This perspective allows for a precise analysis of how network structure, stubborn agents (boundary), and susceptible agents (interior) collectively determine the evolution of opinions. We derive the transient and steady-state solutions using two distinct but related approaches: a general resolvent-based method applicable to agents with heterogeneous susceptibilities, and a spectral method valid for the special case of homogeneous susceptibility. We further establish quantitative convergence rates to the steady state, derive explicit sensitivity formulas with respect to susceptibility parameters, and prove perturbation bounds under changes in the influence matrix. Moreover, we formally define a set of influenceability measures and prove some of their basic properties. Finally, we provide a Monte Carlo illustration on the Zachary karate club graph, showing how the proposed opinion broadcasting centralities and centralizations behave under random susceptibility profiles and how they relate to classical network centralities.