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Dominant Groups and Asymmetric Polarization in Generalized Quasi-Structurally Balanced Networks

Vishnudatta Thota, Swati Priya, Twinkle Tripathy

TL;DR

This work addresses asymmetric polarization in signed networks with a dominant group. It introduces generalized quasi-structural balance (GQSB) and a generalized Laplacian flow to capture asymmetry, transforming the problem to a $z$-domain where inter-subset interactions become cooperative. The authors derive a necessary-and-sufficient condition for asymmetric polarization: the transformed network must be connected and its negative-edge spanning forest's effective-resistance matrix $\Gamma_{\mathcal{F}_{z-}}$ must be positive definite, yielding an explicit expression for the final state $\mathbf{x}_{f}$. Validation on the Highland Tribes dataset demonstrates two distinct final polarizations corresponding to the dominant group and its counterpart, highlighting practical implications for predicting and steering polarization dynamics in real-world networks.

Abstract

The paper focuses on the phenomenon of asymmetric polarization arising in the presence of a dominant group in the network. The existing works in the literature analyze polarization primarily in structurally and quasi-structurally balanced networks. In this work, we introduce generalized quasi-structurally balanced (GQSB) networks, which include both of these networks as special cases. In the presence of a dominant group, a GQSB network has a unique bipartition: the dominant group (and its allies) and the remaining agents. The dominant group's superior influence results in an asymmetry in how the inter-subset antagonistic interactions are perceived by both of the subsets. This, in turn, leads to asymmetry in the final polarized opinions. To model this behavior, we propose a generalized Laplacian flow for undirected GQSB networks with a dominant group and establish necessary and sufficient conditions for achieving asymmetric polarization. The theoretical results presented in this paper are validated through numerical simulations on the Highland Tribes real-world dataset.

Dominant Groups and Asymmetric Polarization in Generalized Quasi-Structurally Balanced Networks

TL;DR

This work addresses asymmetric polarization in signed networks with a dominant group. It introduces generalized quasi-structural balance (GQSB) and a generalized Laplacian flow to capture asymmetry, transforming the problem to a -domain where inter-subset interactions become cooperative. The authors derive a necessary-and-sufficient condition for asymmetric polarization: the transformed network must be connected and its negative-edge spanning forest's effective-resistance matrix must be positive definite, yielding an explicit expression for the final state . Validation on the Highland Tribes dataset demonstrates two distinct final polarizations corresponding to the dominant group and its counterpart, highlighting practical implications for predicting and steering polarization dynamics in real-world networks.

Abstract

The paper focuses on the phenomenon of asymmetric polarization arising in the presence of a dominant group in the network. The existing works in the literature analyze polarization primarily in structurally and quasi-structurally balanced networks. In this work, we introduce generalized quasi-structurally balanced (GQSB) networks, which include both of these networks as special cases. In the presence of a dominant group, a GQSB network has a unique bipartition: the dominant group (and its allies) and the remaining agents. The dominant group's superior influence results in an asymmetry in how the inter-subset antagonistic interactions are perceived by both of the subsets. This, in turn, leads to asymmetry in the final polarized opinions. To model this behavior, we propose a generalized Laplacian flow for undirected GQSB networks with a dominant group and establish necessary and sufficient conditions for achieving asymmetric polarization. The theoretical results presented in this paper are validated through numerical simulations on the Highland Tribes real-world dataset.

Paper Structure

This paper contains 14 sections, 4 theorems, 7 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Notations
  4. Preliminaries
  5. Problem Formulation
  6. Generalized Quasi-Structural Balance
  7. Partitions in a GQSB network
  8. Partitions in the presence of a dominant group
  9. Model Formulation
  10. Opposing and Repelling Laplacian flows
  11. Generalized Laplacian flows
  12. Convergence Analysis
  13. Simulation Results
  14. Conclusion

Key Result

Theorem 1

A signed Laplacian $L$ is positive semi-definite with a simple zero eigenvalue if and only if the underlying signed network $\mathcal{G}$ is connected and $\Gamma_{\mathcal{F}_{-}}$ is positive definite. Here, $\Gamma_{\mathcal{F}_{-}}= B^{T}_{\mathcal{F}_{-}}L^\dagger B_{\mathcal{F}_{-}}$

Figures (11)

  • Figure 1: QSB networks where the inter-subset antagonistic, intra-subset antagonistic, and cooperative interactions are represented with dashed, dotted, and solid edges, respectively.
  • Figure 2: GQSB networks where the inter-subset antagonistic, intra-subset antagonistic, and cooperative interactions are represented with dashed, dotted, and solid edges, respectively.
  • Figure 3: Set diagram of SB, QSB, GQSB, and signed networks
  • Figure : (a) $\mathcal{G}_1$
  • Figure : (a) Highland tribes network
  • ...and 6 more figures

Theorems & Definitions (9)

  • Theorem 1: chen2016characterizing_effective_resistance
  • Definition 1
  • Definition 2: altafini2012consensus
  • Definition 3: QSB_Lei_Shi_2023
  • Definition 4
  • Lemma 1
  • Remark 1
  • Lemma 2
  • Theorem 2