Strong Convergence of a Random Actions Model in Opinion Dynamics

TL;DR

The paper analyzes the Random Actions (RA) dynamics for opinion spread, where each agent's binary action is drawn from a Bernoulli distribution with probability given by its current opinion, and updates follow $\boldsymbol{x}_{t+1}=(1-\alpha)\boldsymbol{x}_t+\alpha\boldsymbol{W}\boldsymbol{a}_t$. It provides a rigorous, self-contained proof that the RA dynamics converge to consensus almost surely and in $L^r$ for all $r>0$, and extends the result to reducible networks where irreducibility is not necessary; it also shows consensus in the presence of a stubborn agent. The authors critique Scaglione's claimed proof, identify conceptual gaps, and propose corrections, while also showing that a single maximal strongly connected component suffices for consensus in certain topologies. They discuss time-variant weights and the RA model's relation to multi-agent optimization, arguing that randomness can drive herd behavior even when deterministic models would predict stagnation or polarization. Overall, the work sharpens the theoretical understanding of stochastic consensus mechanisms in CODA-like opinion dynamics and clarifies the conditions under which global agreement emerges.

Abstract

We study an opinion dynamics model in which each agent takes a random Bernoulli distributed action whose probability is updated at each discrete time step, and we prove that this model converges almost surely to consensus. We also provide a detailed critique of a claimed proof of this result in the literature. We generalize the result by proving that the assumption of irreducibility in the original model is not necessary. Furthermore, we prove as a corollary of the generalized result that the almost sure convergence to consensus holds also in the presence of a stubborn agent which never changes its opinion. In addition, we show that the model, in both the original and generalized cases, converges to consensus also in $r$th mean.

Figures (11)

• Figure 1: A sample path $\omega = (\boldsymbol{x}_1,\boldsymbol{x}_2,\dots)$ belongs to $C_t(\delta)$ if, at time $t$, $\boldsymbol{x}_t$ lies in the shaded region. The figure illustrates the two-dimensional case ($N = 2$).
• Figure 2: A sample path $\omega = (\boldsymbol{x}_1,\boldsymbol{x}_2,\dots)$ belongs to $C_t^\mathbf{0}(\delta)$ if, at time $t$, $\boldsymbol{x}_t$ lies in the shaded corner. The figure illustrates the two-dimensional case ($N = 2$).
• Figure 3: The function $g_{\alpha,N}(\gamma)$ from Lemma \ref{['lem:prod-cont-decr']}, for $N = 6$ and where $\alpha$ is one of $12$ evenly spaced values from $0.001$ to $0.999$. The smaller $\alpha$ is, the faster the function decreases towards $0$.
• Figure 4: A network with four strongly connected components.
• Figure 5: Hasse diagram of the poset of strongly connected components obtained from the network in Figure \ref{['fig:RA-smallnetwork']}, with maximal elements $C_1,C_3$ and minimal element $C_4$.

Theorems & Definitions (20)

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