Stochastic consensus dynamics for decentralized decision systems

Abstract

A fundamental mechanism underlying collective phenomena in social, technological, and economic systems is decentralized decision-making, in which the behavior of individual agents follows from local interactions in the absence of central coordination. Consensus formation is a central feature of such systems, with relevance to blockchain networks, distributed artificial intelligence, autonomous multi-agent systems, distributed control, and collective decision networks. We investigate consensus formation using a stochastic consensus model on random networks and examine how initial conditions, network connectivity, and system size shape the emergence of unanimous states. We show that the stochastic dynamics strongly amplify small initial majorities, progressively suppress the competing state, and drive the system toward a predictable collective outcome. Network connectivity primarily controls the efficiency of this process: increasing connectivity accelerates the propagation of local agreement and reduces the likelihood that fluctuations reverse the initially dominant state, although these gains gradually saturate in highly connected networks. System size produces a complementary effect. Larger networks require more individual updates to reach unanimity, but they are also increasingly reliable in selecting the state favored by the initial majority. Finite-size analysis shows that the range of initial conditions associated with uncertain outcomes becomes progressively narrower as the network grows, decreasing approximately with the inverse square root of the system size. These results reveal a collective amplification mechanism by which weak initial asymmetries become increasingly decisive in large decentralized networks, and they provide a simple framework for understanding the efficiency, predictability, and reliability of consensus formation in distributed decision systems.