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

Abstract

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.

Coevolutionary systems and PageRank

“…We present results from controlled empirical studies on CMCs operating on coevolutionary tournaments T C (V S(n) ) having specific structures [12,17,33]. They are selected based on known internal cycle structures that let us test current understanding of those structures and evaluate their impact on the visitation probabilities of vertices by the coevolutionary random walkers using quantitative measures we develop earlier.…”

“…The framework in [12] provides a formal approach to model and construct coevolutionary systems as random walks on digraphs. We consider a wide range of problems modelled as two-player strategic game C = (S, R) where S is the set of pure strategies and R corresponds to the dominance relation on S. For any game C = (S, R), its full underlying structures are captured by a coevolutionary digraph D C = (V S , A R ).…”

“…Coevolutionary digraphs D C (V S(n) ) are either reducible or irreducible. Any irreducible D C (V S(n) ) on n ≥ 3 vertices cannot be vertex-partitioned, is hamiltonian and strongly connected [12].…”

“…There is an intimate connection between qualitative characterizations of global connectivity structures in D C and quantitative characterizations of Φ [12]. A random walk on a reducible D C leads to an absorbing Φ.…”

“…Other studies have found that problem structures induced by pairwise relations can affect coevolutionary search [11]. We [12] have introduced a new framework that specifies and models coevolutionary systems formally. It uses a digraph representation of coevolutionary problem where the vertex set corresponds to the strategy set and the arc set captures preferences over all pairs of strategies.…”