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Topological Conditions for Echo Chamber Formation under the FJ model: A Cluster Consensus-based Approach

Aashi Shrinate, Twinkle Tripathy, Laxmidhar Behera

Abstract

The Friedkin-Johnsen (FJ) model is a popular opinion dynamics model that explains the disagreement that can occur even among closely interacting individuals. Cluster consensus is a special type of disagreement, where agents in a network split into subgroups such that those within a subgroup agree and those in different subgroups disagree. In large-scale social networks, users often distribute into echo chambers (i.e. groups of users with aligned views) while discussing contested issues such as electoral politics, social norms, etc. Additionally, they are exposed only to opinions and news sources that align with their existing beliefs. Hence, the interaction network plays a key role in the formation of an echo chamber. Since cluster consensus can represent echo chambers in a social network, we examine the conditions for cluster consensus under the FJ model with the objective of determining the properties of the interaction network that lead to echo chamber formation. We present topology-based necessary and sufficient conditions for cluster consensus under the FJ model, regardless of the edge weights in the network and stubbornness values (which are difficult to estimate parameters in a social network). A major advantage of the proposed results is that they are applicable to arbitrary digraphs. Moreover, using the proposed conditions, we explain the emergence of bow-tie structures which are often observed in real-world echo chambers. Finally, we also develop a computationally feasible methodology to verify the proposed conditions for cluster consensus.

Topological Conditions for Echo Chamber Formation under the FJ model: A Cluster Consensus-based Approach

Abstract

The Friedkin-Johnsen (FJ) model is a popular opinion dynamics model that explains the disagreement that can occur even among closely interacting individuals. Cluster consensus is a special type of disagreement, where agents in a network split into subgroups such that those within a subgroup agree and those in different subgroups disagree. In large-scale social networks, users often distribute into echo chambers (i.e. groups of users with aligned views) while discussing contested issues such as electoral politics, social norms, etc. Additionally, they are exposed only to opinions and news sources that align with their existing beliefs. Hence, the interaction network plays a key role in the formation of an echo chamber. Since cluster consensus can represent echo chambers in a social network, we examine the conditions for cluster consensus under the FJ model with the objective of determining the properties of the interaction network that lead to echo chamber formation. We present topology-based necessary and sufficient conditions for cluster consensus under the FJ model, regardless of the edge weights in the network and stubbornness values (which are difficult to estimate parameters in a social network). A major advantage of the proposed results is that they are applicable to arbitrary digraphs. Moreover, using the proposed conditions, we explain the emergence of bow-tie structures which are often observed in real-world echo chambers. Finally, we also develop a computationally feasible methodology to verify the proposed conditions for cluster consensus.
Paper Structure (12 sections, 7 theorems, 22 equations, 3 figures)

This paper contains 12 sections, 7 theorems, 22 equations, 3 figures.

Table of Contents

  1. INTRODUCTION
  2. Notations and Preliminaries
  3. Cluster Consensus in the FJ model
  4. FJ model
  5. Cluster Consensus
  6. Conditions for Cluster Consensus
  7. Locally Topologically persuasive (LTP) agent
  8. Formation of opinion clusters
  9. Conditions for Cluster Consensus
  10. Time complexity of identifying the LTP agents in a network
  11. Discussion
  12. Conclusions

Key Result

Lemma 1

Consider a loopy Laplacian matrix $Q \in \mathbb{R}^{p}$ with $\alpha \subset [p]$. Under Kron reduction, the following properties hold:

Figures (3)

  • Figure 1: In the given figure, agent $2$ is an LTP agent with $\mathcal{N}_2=\{3,4\}$.
  • Figure 2: This figure illustrates the formation of opinion clusters under the FJ model due to the topological properties of $\mathcal{G}$.
  • Figure 3: Bow-tie structure

Theorems & Definitions (13)

  • Lemma 1: Kron_red_digraphs
  • Lemma 2: parsegov2016novelTIAN2018213
  • Definition 1
  • Definition 2
  • Example 1
  • Lemma 3
  • Theorem 1
  • Remark 1
  • Example 2
  • Corollary 1
  • ...and 3 more