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

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.

💡 Research Summary

This paper provides a rigorous mathematical treatment of the Friedkin‑Johnsen (FJ) model of opinion dynamics on networks by casting it as a discrete boundary‑value problem (IBVP). The authors partition the set of agents into interior nodes (susceptible, with susceptibility s_i∈(0,1]) and boundary nodes (stubborn, with s_i=0). By ordering the vertices accordingly, the influence matrix W is written in four blocks: interior‑interior (W_ΩΩ), interior‑boundary (W_Ω∂), boundary‑interior (W_∂Ω) and boundary‑boundary (W_∂∂). The dynamics for interior opinions become an affine recursion

v_Ω^{t+1}=S_ΩW_ΩΩ v_Ω^{t}+S_ΩW_Ω∂ ψ+(I_Ω−S_Ω)ϕ,

where ψ are the fixed boundary opinions and ϕ are the initial interior opinions.

The key existence condition is ρ(S_ΩW_ΩΩ)<1, guaranteeing that I_Ω−S_ΩW_ΩΩ is invertible. The inverse G_S=(I_Ω−S_ΩW_ΩΩ)^{-1} is interpreted as a discrete Green’s function and can be expressed as a Neumann series ∑_{k≥0}(S_ΩW_ΩΩ)^k. The unique steady state is then

v_Ω^* = G_S