Asynchronous Opinion Dynamics in Social Networks

TL;DR

This work analyzes asynchronous opinion dynamics in Hegselmann-Krause systems on arbitrary social networks and proves convergence to a $\\delta$-stable state for random updates. It introduces a potential-based analysis and a projection technique that reduces the problem to 1D, enabling a tight bound $O(Φ(S_0)\, n \\lvert E\\rvert / δ^2) = O(n \\lvert E\\rvert^2 (\\varepsilon/δ)^2)$ on convergence time, with a significantly stronger bound for complete graphs, $O(n^3(n^2+(\\varepsilon/δ)^2))$, improving the previous $O(n^9(\\varepsilon/δ)^2)$. The authors establish near-tightness via lower-bound constructions and simulations, including a matching $\\Omega(n^4)$ lower bound for 1D instances. The results advance understanding of asynchronous HK dynamics on sparse networks and open avenues for studying directed/weighted networks and external influences.

Abstract

Opinion spreading in a society decides the fate of elections, the success of products, and the impact of political or social movements. The model by Hegselmann and Krause is a well-known theoretical model to study such opinion formation processes in social networks. In contrast to many other theoretical models, it does not converge towards a situation where all agents agree on the same opinion. Instead, it assumes that people find an opinion reasonable if and only if it is close to their own. The system converges towards a stable situation where agents sharing the same opinion form a cluster, and agents in different clusters do not \mbox{influence each other.} We focus on the social variant of the Hegselmann-Krause model where agents are connected by a social network and their opinions evolve in an iterative process. When activated, an agent adopts the average of the opinions of its neighbors having a similar opinion. By this, the set of influencing neighbors of an agent may change over time. To the best of our knowledge, social Hegselmann-Krause systems with asynchronous opinion updates have only been studied with the complete graph as social network. We show that such opinion dynamics with random agent activation are guaranteed to converge for any social network. We provide an upper bound of $\mathcal{O}(n|E|^2 (\varepsilon/δ)^2)$ on the expected number of opinion updates until convergence, where $|E|$ is the number of edges of the social network. For the complete social network we show a bound of $\mathcal{O}(n^3(n^2 + (\varepsilon/δ)^2))$ that represents a major improvement over the previously best upper bound of $\mathcal{O}(n^9 (\varepsilon/δ)^2)$. Our bounds are complemented by simulations that indicate asymptotically matching lower bounds.

Figures (2)

• Figure 1: A state $S$ of a HKS with $\Phi(S) = \Theta(n^2 \varepsilon)$ and an expected potential drop of $\Theta(\varepsilon^2/n^3)$. Only edges in $\mathcal{E}_{0}$ are presented, and $\hat{m} = \varepsilon/(n^2/16 +5n/4-1)$ represents the equal available movement of all nodes. Note that the state $S$ is a one-dimensional instance and the position of all nodes of the cliques $C_{\ell}$ and $C_r$ have the same position, respectively. We use the second dimension only for a better illustration of the influencing network. We call the state $S$ with its social network reduced to the edges in $\mathcal{E}_{0}$ a Dumbbell instance.
• Figure 2: The plot shows the normalized convergence time: the number of agent activations until a $\delta$-stable state has been reached, divided by $n^3$. The data indicate that the convergence time on Path instances with equal distances scales as $n^3$ and on Dumbbell instances it scales as $n^4$.

Theorems & Definitions (31)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4
• proof
• Lemma 5
• proof
• Corollary 6
• proof
• Lemma 6