Fairness and Consensus in an Asynchronous Opinion Model for Social Networks (Technical Report)

TL;DR

This work addresses consensus in asynchronous DeGroot-style opinion dynamics over social networks, using Opinion Transition Systems (OTS) to model one-agent-at-a-time updates. It introduces a hierarchy of fairness concepts, showing that standard strong fairness does not guarantee consensus, but a new $m$-bounded fairness does in strongly-connected, puppet-free graphs; it further extends these results to dynamic, bounded-influence settings. The key contributions are the formal OTS framework, the $m$-bounded fairness consensus theorem, and the dynamic-influence extension that preserves convergence under reasonable bounds. The results provide constructive liveness properties, random-inclusivity for random runs, and links to distributed averaging and gossip algorithms, offering a tractable and realistic lens on opinion formation in social networks.

Abstract

We introduce a DeGroot-based model for opinion dynamics in social networks. A community of agents is represented as a weighted directed graph whose edges indicate how much agents influence one another. The model is formalized using labeled transition systems, henceforth called opinion transition systems (OTS), whose states represent the agents' opinions and whose actions are the edges of the influence graph. If a transition labeled $(i,j)$ is performed, agent $j$ updates their opinion taking into account the opinion of agent $i$ and the influence $i$ has over $j$. We study (convergence to) opinion consensus among the agents of strongly-connected graphs with influence values in the interval $(0,1)$. We show that consensus cannot be guaranteed under the standard strong fairness assumption on transition systems. We derive that consensus is guaranteed under a stronger notion from the literature of concurrent systems; bounded fairness. We argue that bounded-fairness is too strong of a notion for consensus as it almost surely rules out random runs and it is not a constructive liveness property. We introduce a weaker fairness notion, called $m$-bounded fairness, and show that it guarantees consensus. The new notion includes almost surely all random runs and it is a constructive liveness property. Finally, we consider OTS with dynamic influence and show convergence to consensus holds under $m$-bounded fairness if the influence changes within a fixed interval $[L,U]$ with $0<L<U<1$. We illustrate OTS with examples and simulations, offering insights into opinion formation under fairness and dynamic influence.

Figures (5)

• Figure 1: Run examples for OTS in Fig. \ref{['fig:ex1']} and randomly-generated OTS in Fig. \ref{['fig:introEx3']}
• Figure 2: Run examples for OTS in Fig. \ref{['fig:ex1']} and Fig. \ref{['fig:strConnCounterexGraph']}
• Figure 3: Run examples for OTS in Fig. \ref{['fig:ex1']}.
• Figure 4: Examples of an $m$-bounded fair runs. In Fig. \ref{['fig:mCBFCounterexPlot']} and \ref{['fig:mCBFexPlot']}, each plot corresponds to the opinion of the agent with the same color in Fig. \ref{['fig:mCBFCounterexNetwork']}.
• Figure 5: Plots for DOTS in Fig. \ref{['fig:dynamicCounterexNetwork1']} and Fig. \ref{['fig:dynamicCounterexNetwork2']} with $U=0.8$ and $L=0.2$.

Theorems & Definitions (49)

• Definition 1: Influence Graph
• Definition 2: OTS
• Remark 3
• Definition 4: e-path, runs and words
• Remark 5
• Definition 6: Consensus
• Example 7
• Definition 8: Strong fairness
• Proposition 9
• Proposition 10