A Study of NK Landscapes' Basins and Local Optima Networks (0810.3484v1)

Abstract: We propose a network characterization of combinatorial fitness landscapes by adapting the notion of inherent networks proposed for energy surfaces (Doye, 2002). We use the well-known family of $NK$ landscapes as an example. In our case the inherent network is the graph where the vertices are all the local maxima and edges mean basin adjacency between two maxima. We exhaustively extract such networks on representative small NK landscape instances, and show that they are 'small-worlds'. However, the maxima graphs are not random, since their clustering coefficients are much larger than those of corresponding random graphs. Furthermore, the degree distributions are close to exponential instead of Poissonian. We also describe the nature of the basins of attraction and their relationship with the local maxima network.

• The paper adapts inherent networks analysis to map NK landscape local optima and basins, viewing complexity through a network lens.
• The analysis reveals small-world network topology for local optima and a positive correlation between basin size and the fitness of their respective peaks.
• Understanding basin size distribution and network structure provides insights for designing more efficient search algorithms, particularly as landscape ruggedness increases with K.

A Network Approach to Analyzing Basins and Local Optima in NK Landscapes

The paper explores the intricate structure of combinatorial fitness landscapes by employing network analysis techniques. The core of this research revolves around the adaptation and extension of the "inherent networks" concept, initially proposed for analyzing energy surfaces, to the $NK$ landscape models—a well-established family of synthetic landscapes defined by the parameters $N$ (number of elements) and $K$ (interactions between elements). The emphasis is placed on mapping the landscape's local maxima and the corresponding basins of attraction into a graphical network, aiming to gain insights into the landscape's complexity and difficulty.

Main Findings and Numerical Insights

1. Network Topology: The authors exhaustively construct the inherent network of local optima for small $NK$ landscape instances. These networks exhibit a "small-world" topology characterized by short average path lengths and a much higher clustering coefficient than expected from random graphs. Interestingly, the degree distributions of these networks tend towards exponential rather than Poisson, indicating the presence of non-random structures within the landscapes.
2. Basins of Attraction: The size and distribution of basins of attraction—crucial determinants of search difficulty—are analyzed. The research shows that large basins correlate positively with high fitness values at their corresponding local maxima. Moreover, while the number of basins grows with $K$, the size of the global optimum's basin decreases, suggesting increased landscape ruggedness and, hence, higher search difficulty with increasing $K$.
3. Algorithmic Implications: Findings from the network and basin analyses can assist in devising more efficient search algorithms. The positive correlation between basin size and maximum fitness implies that reaching a higher fitness peak is often associated with traversing broader basins, potentially simplifying the search for global optima under some conditions.

Theoretical Implications

The paper advances the theoretical understanding of $NK$ landscapes by portraying them not just as isolated optimization problems but as complex interconnected networks. This network perspective allows for analyzing landscape structure, offering a clearer picture of problem difficulty and search dynamics in evolutionary computation and other heuristic optimization methods.

Future Directions

The research opens avenues for further exploration. There is scope for extending the analysis to larger $NK$ landscape instances using efficient sampling techniques, examining landscapes with neutrality, and investigating other combinatorial landscapes. Additionally, practical applications could benefit from these insights by informing the design of adaptive or self-tuning search heuristics tailored to exploit specific network features of the landscape.

In summary, this work provides a comprehensive framework for understanding complex landscapes through the lens of network theory. It highlights the importance of analyzing the structural properties of local optima networks to inform the design and assessment of search algorithms in combinatorial optimization problems.