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Opinion Clustering under the Friedkin-Johnsen Model: Agreement in Disagreement

Aashi Shrinate, Twinkle Tripathy

TL;DR

The paper addresses how network topology governs opinion clustering in the Friedkin-Johnsen model by introducing Locally Topologically Persuasive (LTP) agents. Using Kron reduction and a carefully defined topological structure, it proves that an LTP agent and the agents it persuades form an opinion cluster independent of edge weights and agent stubbornness, extending prior results to arbitrarily connected digraphs. The framework enables designing networks to realize a desired number and composition of opinion clusters, with simulations illustrating practical network design for targeted clustering. This topology-aware approach offers a scalable tool for shaping collective opinions in social networks and related multi-agent systems.

Abstract

The convergence of opinions in the Friedkin-Johnsen (FJ) framework is well studied, but the topological conditions leading to opinion clustering remain less explored. To bridge this gap, we examine the role of topology in the emergence of opinion clusters within the network. The key contribution of the paper lies in the introduction of the notion of topologically prominent agents, referred to as Locally Topologically Persuasive (LTP) agents. Interestingly, each LTP agent is associated with a unique set of (non-influential) agents in its vicinity. Using them, we present conditions to obtain opinion clusters in the FJ framework in any arbitrarily connected digraph. A key advantage of the proposed result is that the resulting opinion clusters are independent of the edge weights and the stubbornness of the agents. Finally, we demonstrate using simulation results that, by suitably placing LTP agents, one can design networks that achieve any desired opinion clustering.

Opinion Clustering under the Friedkin-Johnsen Model: Agreement in Disagreement

TL;DR

The paper addresses how network topology governs opinion clustering in the Friedkin-Johnsen model by introducing Locally Topologically Persuasive (LTP) agents. Using Kron reduction and a carefully defined topological structure, it proves that an LTP agent and the agents it persuades form an opinion cluster independent of edge weights and agent stubbornness, extending prior results to arbitrarily connected digraphs. The framework enables designing networks to realize a desired number and composition of opinion clusters, with simulations illustrating practical network design for targeted clustering. This topology-aware approach offers a scalable tool for shaping collective opinions in social networks and related multi-agent systems.

Abstract

The convergence of opinions in the Friedkin-Johnsen (FJ) framework is well studied, but the topological conditions leading to opinion clustering remain less explored. To bridge this gap, we examine the role of topology in the emergence of opinion clusters within the network. The key contribution of the paper lies in the introduction of the notion of topologically prominent agents, referred to as Locally Topologically Persuasive (LTP) agents. Interestingly, each LTP agent is associated with a unique set of (non-influential) agents in its vicinity. Using them, we present conditions to obtain opinion clusters in the FJ framework in any arbitrarily connected digraph. A key advantage of the proposed result is that the resulting opinion clusters are independent of the edge weights and the stubbornness of the agents. Finally, we demonstrate using simulation results that, by suitably placing LTP agents, one can design networks that achieve any desired opinion clustering.

Paper Structure

This paper contains 9 sections, 6 theorems, 9 equations, 4 figures.

Table of Contents

  1. INTRODUCTION
  2. Notations and Preliminaries
  3. FJ model and emergent behaviours
  4. Topology-based opinion clustering
  5. The role of network topology on opinion formation
  6. Main Results
  7. Conclusions
  8. Proof of Theorem-1
  9. Proof of Corollary \ref{['cor:cluster_consensus']}

Key Result

Lemma 1

Consider a network $\mathcal{G}$ with $m\geq 1$ stubborn agents such that Assumption assump:ns holds. The opinions evolving under the FJ model eq:FJ_opinion_dynamics converge to, where $\mathbf{x}^*=[\mathbf{x}_1^*,\mathbf{x}_2^*]$ denotes the final opinions and $W_{11}^*=\lim_{k \to \infty}W_{11}^k$.

Figures (4)

  • Figure 1: Formation of opinion clusters. Throughout this paper, we represent the stubborn and non-stubborn agents by red and orange nodes, respectively.
  • Figure 2: Each path from the stubborn agents to $4$ and $5$ traverses $3$.
  • Figure 3: The network $\mathcal{G}(R)$ derived from $\mathcal{G}$.
  • Figure 4: Suitable design of $\mathcal{G}$ for desired opinion clusters.

Theorems & Definitions (20)

  • Lemma 1: parsegov2016novelTIAN2018213
  • Definition 1
  • Example 1
  • Remark 1
  • Definition 2
  • Example 2
  • Remark 2
  • Lemma 2
  • proof
  • Remark 3
  • ...and 10 more