Emergent structures in coupled opinion and network dynamics

TL;DR

This work analyzes a coupled, continuous-time model of opinion formation on adaptive networks, where each individual's opinion $x_i(t)\in[-1,1]$ evolves by interactions weighted by $w_{ij}(t)$ and strength $\phi(x_j-x_i)$, and the network co-evolves via $dw_{ij}/dt$ governed by memory or logistic weight rules. Two main regimes are studied: interaction with full support ($R=2$) and bounded-confidence interactions with compact support ($R<2$), revealing that adaptive networks generate community structure that mirrors emerging opinion clusters; memory-weight dynamics tend to drive consensus and full connectivity, while logistic-weight dynamics can sustain clustering depending on initial topology and parameters. For full support, a general result shows finite clustering with cross-cluster edges vanishing under logistic weights, and consensus is recovered under certain parameter ranges; for bounded confidence, $R$-chains form under memory dynamics and, under logistic dynamics, simulations across ER/WS/BA/SBM networks show transitions from many clusters to polarisation to consensus as $R$ grows, with network structure strongly influencing outcomes. The paper also derives short-time approximations for early weight dynamics, linking initial distances and velocity of opinion differences to transient edge evolution, and validates these with simulations, highlighting transient clustering before eventual convergence. Overall, the study extends long-standing fixed-network results to adaptive networks, elucidating how the interaction function and edge-weight dynamics jointly shape the emergence of opinion clusters and community structure with potential implications for real-world social systems.

Abstract

This paper investigates a model of opinion formation on an adaptive social network, consisting of a system of coupled ordinary differential equations for individuals' opinions and corresponding network edge weights. A key driver of the system's behaviour is the form of the interaction function, which determines the strength of interactions based on the distance between individuals' opinions and appears in both opinion and network dynamics. Two cases are examined: in the first the interaction function is always positive and in the second case the interaction function is of bounded-confidence type. In both cases there is positive feedback between opinion clustering and the emergence of community structure in the social network. This is confirmed through analytical results on long-term behaviour, extending existing results for a fixed network, as well as through numerical simulations. Transient network dynamics are also examined through a short-time approximation that captures the `typical' early network dynamics. Each approach improves some aspect of our understanding of the interplay between opinion and network evolution.

Figures (10)

• Figure 1: Demonstration of opinion clustering under logistic weight dynamics. In the absence of weight dynamics the interaction function having full support would lead to consensus. Here, the network is instead reorganised according to individuals' opinions, leading to the formation of communities that are then mirrored in the formation of opinion clusters.
• Figure 2: Diagram showing the scenario described in Remark \ref{['Remark: Not possible to identify cluster locations']}. Under logistic weight dynamics two clusters ($\mathcal{C}_1$ and $\mathcal{C}_2$) may exist at any distance apart due to an initial network structure in which they are only connected via a distant individual ($x_M$) from whom they both become disconnected.
• Figure 3: Diagram showing the scenario described in extending Remark \ref{['Remark: Not possible to identify cluster locations']} to FOAF weight dynamics. Under FOAF weight dynamics two clusters ($\mathcal{C}_1$ and $\mathcal{C}_2$) may exist at any distance apart due to an initial network structure in which they are only connected via two distant connected individuals (both at $x_M$) from whom the clusters become disconnected.
• Figure 4: Heatmaps showing the order parameter \ref{['eqn: order parameter']} at steady state for opinion formation with a fixed network (left) and logistic weight dynamics (right) using different random network models to generate $w(0)$. Blue areas show opinion clustering, grey areas indicate polarisation while yellow means the population has reached consensus. In each heatmap the cutoff value $R$ for a bounded confidence interaction function is varied on the vertical axis, while a model parameter specific to each random network is varied along the horizontal axis. In general, logistic weight dynamics makes consensus more common but cannot prevent clustering when $R$ is very low.
• Figure 5: Demonstration of the estimates \ref{['eqn: v_ij estimates']} using an exponential interaction function. A population size of $N=100$ was used, with $100$ different, random initial conditions generated to allow accurate calculation of the mean and variance. A representative $10,000$ data points are plotted, coloured using a kernel density estimate. The estimates capture the distribution well.

Theorems & Definitions (25)

• Definition 2.1
• Proposition 3.1
• proof
• Corollary 3.1
• proof
• Remark 3.1
• Proposition 3.2
• proof
• Theorem 3.1
• proof