A New Framework for Analysis of Coevolutionary Systems-Directed Graph Representation and Random Walks

Area of Science:

• Artificial Intelligence
• Computational Biology
• Evolutionary Computation

Background:

• Coevolutionary systems, often modeled as games with self-play, present challenges in fully capturing underlying structures.
• Existing research demonstrates cyclic dynamics but lacks a complete characterization of cycle structures and their impact on coevolutionary search.

Purpose of the Study:

• To develop a novel framework for analyzing coevolutionary systems by fully accounting for relational structures.
• To provide a qualitative characterization of coevolutionary problems and develop quantitative tools for analyzing their dynamics.

Main Methods:

• Representing coevolutionary problems using directed graphs (digraphs) to capture solution relationships.
• Modeling coevolutionary processes as Markov chains (random walks on digraphs).
• Analyzing solvable problems as absorbing Markov chains and cyclic problems using invariant distributions.

Main Results:

• Coevolutionary problems are qualitatively characterized as either solvable (possessing a dominant subset of solutions) or not.
• Solvable problems correspond to absorbing Markov chains, allowing computation of expected hitting times.
• Unsolvable (cyclic) problems exhibit indefinite cycling, analyzed via limiting invariant distributions.

Conclusions:

• The digraph-based Markov chain framework offers a comprehensive approach to understanding coevolutionary dynamics.
• The framework enables quantitative analysis of coevolutionary search, including complexity characterization and controlled generation of problems.
• This work provides crucial insights into the behavior and solvability of complex coevolutionary systems.