On the Fractal Nature of Local Optima Networks
A Local Optima Network represents fitness landscape connectivity within the space of local optima as a mathematical graph. In certain other complex networks or graphs there have been recent observations made about inherent self-similarity. An object is said to be self-similar if it shows the same patterns when measured at different scales; another word used to convey self-similarity is fractal. The fractal dimension of an object captures how the detail observed changes with the scale at which it is measured, with a high fractal dimension being associated with complexity. We conduct a detailed study on the fractal nature of the local optima networks of a benchmark combinatorial optimisation problem (NK Landscapes). The results draw connections between fractal characteristics and performance by three prominent metaheuristics: Iterated Local Search, Simulated Annealing, and Tabu Search.
KeywordsCombinatorial fitness landscapes Local optima networks Fractal analysis NK Landscapes
This work is supported by the Leverhulme Trust (award number RPG-2015-395) and by the UK’s Engineering and Physical Sciences Research Council (grant number EP/J017515/1). We gratefully acknowledge that all network data used during this research were obtained from .
- 2.Ochoa, G., Tomassini, M., Vérel, S., Darabos, C.: A study of NK landscapes’ basins and local optima networks. In: Proceedings of the 10th Annual Conference on Genetic and Evolutionary Computation, pp. 555–562. ACM (2008)Google Scholar
- 3.Mandelbrot, B.: How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science 156(3775), 636–638 (1967). http://www.sciencemag.org/content/156/3775/636.abstractCrossRefGoogle Scholar
- 8.Zelinka, I., Zmeskal, O., Saloun, P.: Fractal analysis of fitness landscapes. In: Richter, H., Engelbrecht, A. (eds.) Recent Advances in the Theory and Application of Fitness Landscapes. ECC, vol. 6, pp. 427–456. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-41888-4_15CrossRefGoogle Scholar
- 9.Thomson, S.L., Daolio, F., Ochoa, G.: Comparing communities of optima with funnels in combinatorial fitness landscapes. In: Proceedings of the Genetic and Evolutionary Computation Conference, GECCO 2017, pp. 377–384. ACM, New York (2017). http://doi.acm.org/10.1145/3071178.3071211