Designing Laplacian flows for opinion clustering in structurally balanced and unbalanced networks

TL;DR

The paper develops a Laplacian-flow model $\dot{x}=-L_x x$ with a modified out-degree $\theta_x$ to induce controllable opinion clustering on unsigned and structurally balanced signed graphs, plus a special structurally unbalanced case. By transforming coordinates with $z=Px$ and enforcing $A_z=PAP^{-1}\ge 0$, it derives a design procedure for $L_x=\theta_x-A$ and proves existence of suitable $P$ in broad graph classes; anti-balanced graphs are shown to admit no such $P$. Stability and final-state formulas hinge on the nullspace of $-L_z$, yielding either consensus or clustering depending on graph structure. Numerical simulations illustrate polarization versus clustering and validate the theoretical conditions for final-state clustering. The work offers a controllable mechanism to mitigate polarization via a tailored Laplacian, with future work extending to more graph classes and non-diagonal transforms.

Abstract

In this work, we consider a group of n agents whose interactions can be represented using unsigned or signed structurally balanced graphs or a special case of structurally unbalanced graphs. A Laplacian-based model is proposed to govern the evolution of opinions. The objective of the paper is to analyze the proposed opinion model on the opinion evolution of the agents. Further, we also determine the conditions required to apply the proposed Laplacian-based opinion model. Finally, some numerical results are shown to validate these results.

Figures (5)

• Figure 1: Graph topology 1
• Figure 2: Polarisation of opinion states
• Figure 3: Different final opinion states
• Figure 4: Graph topology 2
• Figure 5: Final opinion states for structurally unbalanced graph

Theorems & Definitions (15)

• Lemma 1
• proof
• Theorem 2
• proof
• Corollary 3
• proof
• Theorem 4
• proof
• Theorem 5
• proof