Doubly Stochastic Pairwise Interactions for Agreement and Alignment

TL;DR

This paper introduces a simple stochastic pairwise interaction model for agents that leads to consensus or geometric alignment, with analysis showing convergence in finite expected time for various state types.

Contribution

It proposes a novel doubly stochastic interaction process for agents that guarantees convergence to agreement or alignment, expanding understanding of emergent phenomena in multi-agent systems.

Findings

Processes lead to consensus in finite expected time for unconstrained states.

Processes lead to alignment in finite expected time for unit vector states.

The model applies to diverse systems like swarm robotics and social dynamics.

Abstract

Random pairwise encounters often occur in large populations, or groups of mobile agents, and various types of local interactions that happen at encounters account for emergent global phenomena. In particular, in the fields of swarm robotics, sociobiology, and social dynamics, several types of local pairwise interactions were proposed and analysed leading to spatial gathering or clustering and agreement in teams of robotic agents coordinated motion, in animal herds, or in human societies. We here propose a very simple stochastic interaction at encounters that leads to agreement or geometric alignment in swarms of simple agents, and analyse the process of converging to consensus. Consider a group of agents whose "states" evolve in time by pairwise interactions: the state of an agent is either a real value (a randomly initialised position within an interval) or a vector that is either unconstrained (e.g. the location of the agent in the plane) or constrained to have unit length (e.g. the direction of the agent's motion). The interactions are doubly stochastic, in the sense that, at discrete time steps, pairs of agents are randomly selected and their new states are independently and uniformly set at random in (local) domains or intervals defined by the states of the interacting pair. We show that such processes lead, in finite expected time (measured by the number of interactions that occurred) to agreement in case of unconstrained states and alignment when the states are unit vectors.