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Leveraging Network Topology in a Two-way Competition for Influence in the Friedkin-Johnsen Model

Aashi Shrinate, Twinkle Tripathy

TL;DR

This work addresses influence competition in Friedkin-Johnsen dynamics with two stubborn agents on strongly connected networks. It introduces topology-based edge perturbations of the form $(a,b,d)$ and leverages ${\mathcal{C}}^{1}$ topology and direct-path supporters to obtain weight-agnostic conditions that guarantee increases in the preferred agent's influence, or even convert the least influential agent into the most influential one through a sequence of perturbations. The paper provides topological characterizations of supporters, redundancy criteria (Theorems 3–5), and validates the approach on Sampson's Monastery data, demonstrating practical applicability to control social power via network structure. The significance lies in offering a principled, topology-driven mechanism to influence opinion dynamics that does not rely on tuning edge weights, with potential applications in politics, marketing, and information diffusion.

Abstract

In this paper, we consider two stubborn agents who compete for `influence' over a strongly connected group of agents. This framework represents real-world contests, such as competition among firms, two-party elections, and sports rivalries, among others. Considering stubbornness of agents to be an immutable property, we utilise the network topology alone to increase the influence of a preferred stubborn agent. We demonstrate this on a special class of strongly connected networks by identifying the supporters of each of the stubborn agents in such networks. Thereafter, we present sufficient conditions under which a network perturbation always increases the influence of the preferred stubborn agent. A key advantage of the proposed topology-based conditions is that they hold independent of the edge weights in the network. Most importantly, we assert that there exists a sequence of perturbations that can make the lesser influential stubborn agent more influential. Finally, we demonstrate our results over the Sampson's Monastery dataset.

Leveraging Network Topology in a Two-way Competition for Influence in the Friedkin-Johnsen Model

TL;DR

This work addresses influence competition in Friedkin-Johnsen dynamics with two stubborn agents on strongly connected networks. It introduces topology-based edge perturbations of the form and leverages topology and direct-path supporters to obtain weight-agnostic conditions that guarantee increases in the preferred agent's influence, or even convert the least influential agent into the most influential one through a sequence of perturbations. The paper provides topological characterizations of supporters, redundancy criteria (Theorems 3–5), and validates the approach on Sampson's Monastery data, demonstrating practical applicability to control social power via network structure. The significance lies in offering a principled, topology-driven mechanism to influence opinion dynamics that does not rely on tuning edge weights, with potential applications in politics, marketing, and information diffusion.

Abstract

In this paper, we consider two stubborn agents who compete for `influence' over a strongly connected group of agents. This framework represents real-world contests, such as competition among firms, two-party elections, and sports rivalries, among others. Considering stubbornness of agents to be an immutable property, we utilise the network topology alone to increase the influence of a preferred stubborn agent. We demonstrate this on a special class of strongly connected networks by identifying the supporters of each of the stubborn agents in such networks. Thereafter, we present sufficient conditions under which a network perturbation always increases the influence of the preferred stubborn agent. A key advantage of the proposed topology-based conditions is that they hold independent of the edge weights in the network. Most importantly, we assert that there exists a sequence of perturbations that can make the lesser influential stubborn agent more influential. Finally, we demonstrate our results over the Sampson's Monastery dataset.

Paper Structure

This paper contains 12 sections, 7 theorems, 6 equations, 6 figures.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. Friedkin-Johnsen Model
  4. PROBLEM FORMULATION
  5. Problem formulation
  6. Permissible network perturbations
  7. Key Topological Properties in a Network
  8. Global and local communicators
  9. Regions in the network $\mathcal{G}$
  10. Identification of useful edge modifications
  11. Simulation Results
  12. CONCLUSIONS AND FUTURE WORKS

Key Result

Lemma 1

In a strongly connected network $\mathcal{G}=(\mathcal{V},\mathcal{E})$, if the opinion of agents evolves by eqn. eqn:op_model and $\beta_i>0$ for an agent $i \in \mathcal{V}$, then the final opinion converge as follows: Here, $\mathbf{x}^{*}=[x_1^*,...,x_n^*]$ denotes the final opinion of agents.

Figures (6)

  • Figure 1: A network with two stubborn agents
  • Figure 2: Regions in network $\mathcal{G}$
  • Figure 3: The modified graph of type $\mathcal{C}^{1}$ constructed from Sampson's monastery dataset.
  • Figure 4: Suitable edge modifications result in initially less influential Hugh becoming more influential than Peter.
  • Figure 5: Reduced graph $G_s^1$
  • ...and 1 more figures

Theorems & Definitions (14)

  • Lemma 1: FJ_Model
  • Definition 1: Community_Cleavage
  • Example 1
  • Example 2
  • Definition 2
  • Definition 3
  • Example 3
  • Lemma 2: aashiIC
  • Remark 1
  • Theorem 1
  • ...and 4 more