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.

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一种新的协同进化系统分析框架 - 有向图表示和随机游走

在简化模型的背景下研究共同进化系统（即，将共同进化解决方案之间成对相互作用建模为自我对弈的游戏）仍然是一个开放的挑战，因为与基于成对比较的适应度测量相关的丰富基础结构通常没有被完全考虑在内。尽管循环动力学已在多种情况下得到证明（例如共同进化问题中的不传递性），但没有完整的循环结构特征及其对共同进化搜索的影响。我们开发了一个新框架来解决这个问题。我们方法的核心是共同进化问题的有向图（有向图）表示，它完全捕获了候选解决方案之间关系的结构。共同进化过程被建模为一种特定类型的马尔可夫链——有向图上的随机游走。使用这个框架，我们证明了共同进化问题承认一个定性特征：共同进化问题要么是可解决的（有一个解决方案的子集支配剩余的候选解决方案）要么不可解决。这对协同进化搜索有影响。我们进一步开发了我们的框架，该框架提供了构建用于分析协同进化过程的定量工具的方法，并通过案例研究展示了它们的应用。我们表明可解决问题的协同进化对应于吸收马尔可夫链，我们可以计算吸收类的预期命中时间。否则，共同进化将无限循环，感兴趣的数量将是马尔可夫链的极限不变分布。我们还提供了表征协同进化问题复杂性的指标，并展示了如何以受控方式生成它们。