A new framework for analysis of coevolutionary systems—directed graph representation and random walks

Siang Yew CHONG, Peter TIŇO, Jun HE, Xin YAO

Research output: Journal PublicationsJournal Article (refereed)peer-review

2 Citations (Scopus)


Studying coevolutionary systems in the context of simplified models (i.e., games with pairwise interactions between coevolving solutions modeled as self plays) remains an open challenge since the rich underlying structures associated with pairwise-comparison-based fitness measures are often not taken fully into account. Although cyclic dynamics have been demonstrated in several contexts (such as intransitivity in coevolutionary problems), there is no complete characterization of cycle structures and their effects on coevolutionary search. We develop a new framework to address this issue. At the core of our approach is the directed graph (digraph) representation of coevolutionary problems that fully captures structures in the relations between candidate solutions. Coevolutionary processes are modeled as a specific type of Markov chains—random walks on digraphs. Using this framework, we show that coevolutionary problems admit a qualitative characterization: a coevolutionary problem is either solvable (there is a subset of solutions that dominates the remaining candidate solutions) or not. This has an implication on coevolutionary search. We further develop our framework that provides the means to construct quantitative tools for analysis of coevolutionary processes and demonstrate their applications through case studies. We show that coevolution of solvable problems corresponds to an absorbing Markov chain for which we can compute the expected hitting time of the absorbing class. Otherwise, coevolution will cycle indefinitely and the quantity of interest will be the limiting invariant distribution of the Markov chain. We also provide an index for characterizing complexity in coevolutionary problems and show how they can be generated in a controlled manner. © 2017 Massachusetts Institute of Technology.
Original languageEnglish
Pages (from-to)195-228
Number of pages34
JournalEvolutionary Computation
Issue number2
Early online date20 Nov 2017
Publication statusPublished - Jun 2019
Externally publishedYes

Bibliographical note

This work was supported by a H2020-MSCA-IF-2014 grant (No. 657027) on “COEVOLFRAMEWORK—Unified Framework for the Analysis of Co-evolutionary Systems.”.


  • Coevolution
  • Directed graphs
  • Evolutionary computation
  • Markov chains
  • Self play


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