Network-based representations of fitness landscapes have grown in popularity in the past decade; this is probably because of growing interest in explainability for optimisation algorithms. Local optima networks (LONs) have been especially dominant in the literature and capture an approximation of local optima and their connectivity in the landscape. However, thus far, LONs have been constructed according to a strict definition of what a local optimum is: the result of local search. Many evolutionary approaches do not include this, however. Popular algorithms such as CMA-ES have therefore never been subject to LON analysis. Search trajectory networks (STNs) offer a possible alternative: nodes can be any search space location. However, STNs are not typically modelled in such a way that models temporal stalls: that is, a region in the search space where an algorithm fails to find a better solution over a defined period of time. In this work, we approach this by systematically analysing a special case of STN which we name attractor networks. These offer a coarse-grained view of algorithm behaviour with a singular focus on stall locations. We construct attractor networks for CMA-ES, differential evolution, and random search for 24 noiseless black-box optimisation benchmark problems. The properties of attractor networks are systematically explored. They are also visualised and compared to traditional LONs and STN models. We find that attractor networks facilitate insights into algorithm behaviour which other models cannot, and we advocate for the consideration of attractor analysis even for algorithms which do not include local search.

翻译：基于网络的适应度景观表示在过去十年中日益流行；这可能是由于对优化算法可解释性的关注度不断提升。局部最优网络（LONs）在文献中尤其占据主导地位，它捕捉了景观中局部最优解及其连通性的近似表示。然而，迄今为止，LONs的构建一直基于对局部最优解的严格定义：即局部搜索的结果。然而，许多进化方法并不包含这一过程。因此，像CMA-ES这样的流行算法从未接受过LON分析。搜索轨迹网络（STNs）提供了一个可能的替代方案：节点可以是搜索空间中的任意位置。然而，STNs通常未被建模以描述时间性停滞现象：即算法在搜索空间中某个区域，在定义的时间段内未能找到更优解的情况。在本研究中，我们通过系统分析一种特殊的STN变体——我们称之为吸引子网络——来探讨这一问题。这些网络提供了算法行为的粗粒度视图，并聚焦于停滞位置。我们针对24个无噪声黑盒优化基准问题，为CMA-ES、差分进化和随机搜索构建了吸引子网络。我们系统性地探索了吸引子网络的特性，并将其可视化，与传统LONs和STN模型进行了比较。我们发现吸引子网络能够揭示其他模型无法提供的算法行为洞见，因此我们主张即使对于不包含局部搜索的算法，也应考虑进行吸引子分析。