[Paper Review] Digital Ecosystems: Stability of Evolving Agent Populations

Philippe De Wilde, Gerard Briscoe|ArXiv.org|Dec 26, 2007

Evolutionary Game Theory and Cooperation61 references4 citations

TL;DR

This paper extends the Chli-DeWilde stability framework to Multi-Agent Systems (MAS) with evolutionary dynamics, introducing an entropy-based measure of instability to quantify system stability in evolving agent populations. The key finding is that mutation rate critically affects stability: mutation rates above 60% significantly increase instability, while crossover has minimal impact, with high mutation rates (80%+) causing instability levels nearing 0.5 as measured by entropy of limit probabilities.

ABSTRACT

Stability is perhaps one of the most desirable features of any engineered system, given the importance of being able to predict its response to various environmental conditions prior to actual deployment. Engineered systems are becoming ever more complex, approaching the same levels of biological ecosystems, and so their stability becomes ever more important, but taking on more and more differential dynamics can make stability an ever more elusive property. The Chli-DeWilde definition of stability views a Multi-Agent System as a discrete time Markov chain with potentially unknown transition probabilities. With a Multi-Agent System being considered stable when its state, a stochastic process, has converged to an equilibrium distribution, because stability of a system can be understood intuitively as exhibiting bounded behaviour. We investigate an extension to include Multi-Agent Systems with evolutionary dynamics, focusing on the evolving agent populations of our Digital Ecosystem. We then built upon this to construct an entropy-based definition for the degree of instability (entropy of the limit probabilities), which was later used to perform a stability analysis. The Digital Ecosystem is considered to investigate the stability of an evolving agent population through simulations, for which the results were consistent with the original Chli-DeWilde definition.

Motivation & Objective

• To extend the Chli-DeWilde stability definition to Multi-Agent Systems (MAS) with evolutionary dynamics.
• To develop a macroscopic, quantifiable measure of instability for evolving agent populations in digital ecosystems.
• To analyze the impact of evolutionary parameters—particularly mutation and crossover rates—on system stability.
• To validate the extended stability framework through simulation in a Digital Ecosystem context.
• To provide a scalable, entropy-based metric applicable to diverse MAS architectures with or without evolution.

Proposed method

• Adapts the Chli-DeWilde definition of stability, treating MAS as a discrete-time Markov chain with unknown transition probabilities.
• Defines system stability as convergence to an equilibrium distribution of states, with bounded behavior indicating stability.
• Introduces an entropy-based instability metric: $ d_{ins} = -\sum_X p^{1000}_X \log_N(p^{1000}_X) $, calculated from limit probabilities at generation 1000 as a proxy for $ t \to \infty $.
• Simulates evolving agent populations in a Digital Ecosystem architecture with peer-to-peer agent migration and local evolutionary optimization.
• Varying mutation and crossover rates from 0% to 100% in 10% increments across 10,000 simulation runs to assess stability trends.
• Performs sensitivity analysis by measuring $ d_{ins} $ at the 1000th generation to evaluate long-term system behavior.

Experimental results

Research questions

• RQ1 How can the Chli-DeWilde stability definition be extended to MAS with evolutionary dynamics?
• RQ2 What macroscopic metric can quantify the degree of instability in evolving agent populations?
• RQ3 How do mutation and crossover rates affect the stability of evolving agent populations in a Digital Ecosystem?
• RQ4 Does the system converge to a single macro-state at infinite time, indicating stability, under different evolutionary parameter settings?
• RQ5 Can the entropy-based instability measure effectively distinguish between stable and unstable system behaviors?

Key findings

• Mutation rates below or equal to 60% result in zero instability ($ d_{ins} = 0 $), indicating convergence to a single macro-state at infinite time.
• Mutation rates above 60% cause a significant increase in instability, with $ d_{ins} $ values rising to 0.16 at 70% and nearing 0.5 at 80% or higher.
• Crossover rate has minimal impact on stability, as variation from crossover is limited in mature populations with highly similar agent aggregations.
• The system remains stable at a sub-optimal macro-state when mutation rate is 0%, despite showing no instability, indicating that stability does not imply optimality.
• The entropy-based instability measure successfully quantifies stability levels and confirms the extended Chli-DeWilde framework is applicable to evolving agent populations in digital ecosystems.
• Performance and stability are distinct: a system can be stable (low $ d_{ins} $) yet suboptimal, especially at low mutation rates.