[Paper Review] Modelling Gossip Interactions in Open Multi-Agent Systems

Charles Monnoyer de Galland, Samuel Martín+1 more|arXiv (Cornell University)|Sep 7, 2020

Evolutionary Game Theory and Cooperation4 citations

TL;DR

This paper proposes a continuous-time modeling framework for open multi-agent systems with dynamic agent arrivals and departures, using scale-independent descriptors to characterize the evolution of system moments (mean and variance) via fixed-size linear dynamical systems. It derives exact results for fixed-size systems and an upper bound on expected variance for variable-size systems, showing that variance decays to zero when gossip interactions are frequent relative to departures.

ABSTRACT

We consider open multi-agent systems, which are systems subject to frequent arrivals and departures of agents while the studied process takes place. We study the behavior of all-to-all pairwise gossip interactions in such open systems. Arrivals and departures of agents imply that the composition and size of the system evolve with time, and in particular prevent convergence. We describe the expected behavior of the system by showing that the evolution of scale-independent quantities can be characterized exactly by a fixed-size linear dynamical system. We apply this approach to characterize the evolution of the two first moments (and thus also of the variance) for open systems of fixed and variable size. Our approach is based on the continuous-time modelling of random asynchronous events impacting the systems (gossip steps, arrivals, departures, and replacements), and can be extended to other types of events.

Motivation & Objective

• To address the challenge of analyzing open multi-agent systems where agent arrivals and departures prevent asymptotic convergence.
• To develop a general methodology for characterizing system behavior through scale-independent descriptors, avoiding reliance on fixed system size.
• To model the evolution of statistical moments—particularly mean and variance—under random, asynchronous gossip interactions, arrivals, and departures.
• To provide exact results for fixed-size systems and upper bounds for variable-size systems, enabling performance analysis in realistic open environments.
• To extend classical consensus and averaging models to open systems by incorporating stochastic, time-varying dynamics using continuous-time Markov processes.

Proposed method

• Models agent interactions, arrivals, and departures as continuous-time Markov jump processes with rates λg (gossip), λa (arrival), and λd (departure).
• Introduces 'descriptors'—scale-independent system quantities like mean and variance—that evolve according to fixed-size linear dynamical systems.
• Uses the ergodicity of the system size process (modeled as a birth-death process) to derive stationary distributions for the number of agents.
• Applies Grönwall's lemma and moment dynamics to bound the expected variance in time-varying systems, yielding an upper bound dependent on λg/λd.
• Derives exact evolution equations for the first two moments in fixed-size systems with replacements, leveraging the linear structure of the descriptor dynamics.
• Employs the stationary distribution πj* = (n̄^j / j!) e^(-n̄) for system size n(t), with n̄ = λa/λd, to compute long-term expectations.

Experimental results

Research questions

• RQ1 How can the behavior of open multi-agent systems with dynamic agent populations be characterized when convergence is impossible?
• RQ2 What is the exact evolution of the first two moments (mean and variance) in fixed-size open systems under random gossip and replacement events?
• RQ3 How can the expected variance be bounded in variable-size open systems where arrivals and departures are decoupled?
• RQ4 What role does the ratio of gossip frequency to departure rate (γ = λg/λd) play in the long-term stability of the system variance?
• RQ5 Can continuous-time modeling provide a general framework for analyzing open systems beyond pairwise gossip, including other interaction types and topologies?

Key findings

• For fixed-size systems with replacements, the expected evolution of the first two moments (and thus variance) is characterized exactly via a linear dynamical system, enabling precise prediction of system behavior.
• In variable-size systems, the expected asymptotic variance is bounded by (1 - e^(-n̄))/γ × σ², where n̄ = λa/λd and γ = λg/λd, showing decay to zero when γ is large.
• The bound is inversely proportional to γ, indicating that frequent gossip interactions relative to departures lead to stable, low-variance system states.
• When λd → 0 (no departures), the expected variance converges to zero, confirming that only growth without loss leads to consensus-like behavior.
• The theoretical bound is approximately four times larger than simulated results, indicating room for tighter analysis, though the framework remains robust and generalizable.
• The system size process is ergodic when λa/λd < ∞, ensuring the existence of a stationary distribution πj* = (n̄^j / j!) e^(-n̄), which underpins the long-term analysis.