Information flow and Laplacian dynamics on local optima networks

TL;DR

The paper reframes Local Optima Networks (LONs) as dynamic objects by applying Laplacian dynamics to study information flow on fitness landscapes, using monotonic LONs constructed from QAP instances. It introduces a suite of LD-based metrics (influence and defluence variants) derived from the left/right Laplacian kernels and compares them to established LON measures, demonstrating strong correlations with and predictive power for metaheuristic performance (ILS and TS) on the Quadratic Assignment Problem. The results show that several LD metrics, notably $\\mathcal{I}_4$, $\\mathcal{I}_5$, $\\mathcal{D}_3$, and $\\mathcal{D}_5$, reliably predict performance and can outperform fractal- and pagerank-based metrics, while maintaining scalable computation via linear-algebra operations. Overall, the LD approach provides a principled, scalable framework for quantifying information flow on LONs and improving the understanding and prediction of search difficulty in combinatorial landscapes.

Abstract

We propose a new way of looking at local optima networks (LONs). LONs represent fitness landscapes; the nodes are local optima, and the edges are search transitions between them. Many metrics computed on LONs have been proposed and shown to be linked to metaheuristic search difficulty. These have typically considered LONs as describing static structures. In contrast to this, Laplacian dynamics (LD) is an approach to consider the information flow across a network as a dynamical process. We adapt and apply LD to the context of LONs. As a testbed, we consider instances from the quadratic assignment problem (QAP) library. Metrics related to LD are proposed and these are compared with existing LON metrics. The results show that certain LD metrics are strong predictors of metaheuristic performance for iterated local search and tabu search.

Figures (4)

• Figure 1: Simple example of a LON for illustrating LD
• Figure 2: Typical examples of LONs from QAP instances with both low and high perturbation
• Figure 3: Laplacian measures for LONs with different perturbation strength. Clearly differentiable clouds of data enable to classify perturbation strength. Blue circles indicate low perturbation, red circles high perturbation.
• Figure 4: Correlations between TS and ILS performance and LON metrics for both low and high perturbation. $I/T$ is the correlation between ILS and TS performance; median fractal dimension: FD, pagerank: $p_1$, $p_2$, number of SCCs: nS, Laplacian metrics: $\mathcal{I}_1$-$\mathcal{I}_5$ and $\mathcal{D}_1$-$\mathcal{D}_5$. Color code: red-TS (high), blue-TS (low), yellow-ILS (high), purple-ILS (low).