On the duality between consensus problems and Markov processes, with application to delay systems
Fatihcan M. Atay
TL;DR
The paper establishes a duality between deterministic consensus dynamics and Markov processes, unifying continuous- and discrete-time formulations within a common Laplacian-based framework. It extends the classic consensus model to incorporate history-dependent delays, distinguishing information propagation and processing delays via distributed delay measures and deriving explicit consensus conditions and values. Key results show that, under a simple zero-Laplacian-eigenvalue condition, consensus is achieved with a value given by a stationary-weighted average of initial data, and that the delay structure critically influences stability, with propagation delays generally preserving convergence while processing delays can destabilize for large delays. The work links deterministic consensus to stochastic process theory through Markov semigroups, and highlights implications for synchronization, time-varying graphs, and distributed control in networks.
Abstract
We consider consensus of multi-agent systems as a dual problem to Markov processes. Based on an exchange of relevant notions and results between the two fields, we present a uniform framework which admits the introduction and treatment of time delays in a common setting. We study both information propagation and information processing delays, and for each case derive conditions for reaching consensus and calculate the consensus value.