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

Moses Boudourides

TL;DR

This work reframes Friedkin–Johnsen social influence on networks as a discrete boundary-value problem with boundary (stubborn) and interior (susceptible) agents, enabling a block-reduced interior dynamics analysis. It develops a resolvent/Green’s-function representation and, in the homogeneous case, a spectral solution via the Dirichlet Laplacian, yielding explicit steady-state formulas, convergence rates, and sensitivity/perturbation bounds. A comprehensive set of influenceability measures is introduced, including an all-vertex scanning framework and a broadcasting graph that supports new centralities. Monte Carlo experiments on the Zachary karate club graph demonstrate that, under random susceptibility, the proposed broadcasting indices align with classical centralities while revealing susceptibility-driven variability in influence concentration. The framework provides a rigorous, extensible toolkit for quantifying boundary-to-interior influence and centralization in networks, with potential applications to directed, multilayer, and time-varying systems.

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.

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

TL;DR

This work reframes Friedkin–Johnsen social influence on networks as a discrete boundary-value problem with boundary (stubborn) and interior (susceptible) agents, enabling a block-reduced interior dynamics analysis. It develops a resolvent/Green’s-function representation and, in the homogeneous case, a spectral solution via the Dirichlet Laplacian, yielding explicit steady-state formulas, convergence rates, and sensitivity/perturbation bounds. A comprehensive set of influenceability measures is introduced, including an all-vertex scanning framework and a broadcasting graph that supports new centralities. Monte Carlo experiments on the Zachary karate club graph demonstrate that, under random susceptibility, the proposed broadcasting indices align with classical centralities while revealing susceptibility-driven variability in influence concentration. The framework provides a rigorous, extensible toolkit for quantifying boundary-to-interior influence and centralization in networks, with potential applications to directed, multilayer, and time-varying systems.

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.
Paper Structure (22 sections, 24 theorems, 83 equations, 4 figures, 1 table)

This paper contains 22 sections, 24 theorems, 83 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries: Graph Notation and Influence Dynamics
  3. Undirected graph specialization and Laplacians
  4. The DeGroot and Friedkin--Johnsen models
  5. IBVP Formulation and Block-Matrix Reduction
  6. General Solution via Resolvent Method (Heterogeneous Susceptibility)
  7. Special Case: Spectral Solution with Homogeneous Susceptibility
  8. Influenceability Measures
  9. The Notion of Influenciability
  10. Directed distances.
  11. All--Vertex Influenceability Matrices
  12. Influence--Based Centralities on the Broadcasting Graph
  13. Broadcasting graph.
  14. Monte Carlo Illustration on the Zachary Karate Club Graph
  15. Susceptibility sampling on [0,1] (zero--inflated Beta model)
  16. ...and 7 more sections

Key Result

Theorem 2.4

Denoting by $I_\Omega$ the $|\Omega|\times|\Omega|$ identity matrix acting on interior-state vectors indexed by $\Omega$, the opinion dynamics on the interior $\Omega$ are governed by the affine recursion:

Figures (4)

  • Figure 1: Zachary karate club graph.
  • Figure 2: Nodewise comparison between classical centralities and the corresponding Monte Carlo mean broadcasting centralities (five panels: degree, closeness, betweenness, eigenvector, PageRank). Each panel includes the least-squares regression line.
  • Figure 3: Broadcasting maps on the karate club network: in each panel, node shading encodes the Monte Carlo mean of the corresponding broadcasting centrality, normalized separately within that panel (one panel per measure).
  • Figure 4: Empirical distributions of the five opinion broadcasting centralizations across $R=100{,}000$ Monte Carlo runs (five panels: degree, closeness, betweenness, eigenvector, PageRank).

Theorems & Definitions (71)

  • Definition 2.1: Random-Walk Matrix
  • Definition 2.2: Graph Laplacians
  • Definition 2.3: Dirichlet Laplacian Chung1997
  • Theorem 2.4: Interior Affine Dynamics
  • proof
  • Theorem 2.5: Existence and uniqueness of steady state Seneta2006
  • proof
  • Proposition 2.6: Necessity of boundary reachability/damping
  • proof
  • Remark 2.7: Example: a closed interior class
  • ...and 61 more