Multifractality and dimensional determinism in local optima networks

Thomson, Sarah L.; Verel, Sébastien; Ochoa, Gabriela; Veerapen, Nadarajen; Cairns, David

Abstract

We conduct a study of local optima networks (LONs) in a search space using fractal dimensions. The fractal dimension (FD) of these networks is a complexity index which assigns a non-integer dimension to an object. We propose a fine-grained approach to obtaining the FD of LONs, using the probabilistic search transitions encoded in LON edge weights. We then apply multi-fractal calculations to LONs for the first time, comparing with mono-fractal analysis. For complex systems such as LONs, the dimensionality may be different between two sub-systems and multi-fractal analysis is needed. Here we focus on the Quadratic Assignment Problem (QAP), conducting fractal analyses on sampled LONs of reasonable size for the first time. We also include fully enumerated LONs of smaller size. Our results show that local optima spaces can be multi-fractal and that valuable information regarding probabilistic self-similarity is encoded in the edge weights of local optima networks. Links are drawn between these phenomena and the performance of two competitive metaheuristic algorithms.

Citation

Thomson, S. L., Verel, S., Ochoa, G., Veerapen, N., & Cairns, D. (2018, July). Multifractality and dimensional determinism in local optima networks. Presented at GECCO '18: Genetic and Evolutionary Computation Conference, Kyoto, Japan

• Presentation Conference Type: Conference Paper (published)
• Conference Name: GECCO '18: Genetic and Evolutionary Computation Conference
• Start Date: Jul 15, 2018
• End Date: Jul 19, 2018
• Online Publication Date: Jul 2, 2018
• Publication Date: 2018
• Deposit Date: Aug 16, 2023
• Publisher: Association for Computing Machinery (ACM)
• Pages: 371-378
• Book Title: GECCO '18: Proceedings of the Genetic and Evolutionary Computation Conference
• ISBN: 9781450356183
• DOI: https://doi.org/10.1145/3205455.3205472
• Keywords: fractal dimension, quadratic assignment problem, fitness landscapes, local optima networks
• Public URL: http://researchrepository.napier.ac.uk/Output/3169670