"Preparatory work starts with formulating a unified, principled framework for representing coevolutionary systems. The first task starts with the specification of coevolutionary systems: (1) coevolutionary problem - the set of all alternatives (strategies in a two-player game) and a pairwise preference relation indicating the superior alternative (when one strategy defeats another one), and (2) coevolutionary process - the population of alternatives undergoing a process of selection and variation guided only by their interaction outcomes. Such specified coevolutionary systems can be formulated through: (1) digraph representation of coevolutionary problem that fully captures structures in the pairwise relations between strategies, and (2) Markov chain modelling of coevolutionary processes as random walks on digraphs. Then, macro-scale indices for precise characterizations of coevolutionary systems are developed. This involves qualitative characterizations of coevolutionary problems through cycle structures by examining the presence of strong connectivities in the digraphs. Solvable coevolutionary problems are associated with reducible digraphs that can be vertex-partitioned whereby a subset of strategies dominate the rest. Non-solvable cooevolutionary problems are associated with irreducible (strongly connected) digraphs that cannot be vertex-partitioned.
This qualitative result is used on the second task to develop quantitative characterizations of coevolutionary processes modelled as Markov chains. Coevolution of a solvable problem corresponds to an absorbing Markov chain where one can precisely formulate the expected hitting times to the absorbing class (subset of dominant strategies). Coevolution of a non-solvable problem corresponds to an irreducible Markov chain, in which case the quantity of interest is its limiting invariant distribution. Both theoretical and computational case studies have been carried out to demonstrate how the analysis can be performed.
The third task involves quantifications of coevolved solutions generated by the coevolutionary system. The main idea involves introducing random restart to the associated coevolutionary process. One can then calculate the visitation probabilities of vertices (strategies) by the coevolutionary random walker for both reducible and irreducible digraphs. Theoretical groundings and empirical support for such quantifications have been provided in this task. The visitation probabilities correspond to performance measure of coevolved solutions and they are mathematically shown to exist. Changes to those results due to changes in restart probabilities can be quantified precisely.
The fourth, final task involves comparison of dynamics in coevolutionary processes. This requires evaluating outcomes from coevolutionary random walks over a benchmark of coevolutionary digraphs with different levels of complexity. A benchmark of digraphs of known cycle structures from digraph theory and those obtained from a new generative methodology from statistical physics has been developed. The network-growth-based approach can produce a hierarchy of random coevolutionary digraphs where a control parameter is used to move between two complexity extremes (irreducible digraphs and reducible digraphs with prominent transitive structures). Coevolutionary digraph complexity is quantified precisely with a newly developed, digraph-theoretic index that count the number of strong components. Computational studies have uncovered how these structures affect the coevolutionary search process.
The outcomes of first, second and fourth tasks have been published as ""A New Framework for Analysis of Coevolutionary Systems - Directed Graph Representation and Random Walks,"" Evolutionary Computation, MIT Press (
https://doi.org/10.1162/evco_a_00218(se abrirá en una nueva ventana)). Outcomes from the third task have been submitted for a high-impact journal publication."
