Evolutionary Algorithms for Generating Graphs Matching Desired Laplacian Spectra

Abstract

Graphs with diverse structural characteristics play a central role in modelling and optimization tasks. The ability to generate different types of graphs that exhibit shared properties is likewise essential for algorithm selection and configuration. However, constructing graphs that preserve high-level properties across a broad range of graph classes remains a challenging problem. We present a novel evolutionary approach to evolve graphs based on the Laplacian graph spectra descriptor. This descriptor can be used as part of a fitness function to evaluate graphs according to their desired high-level properties. Our evolutionary algorithm evolves graphs towards this descriptor in order to obtain graphs having properties that are consistent with it but are different from each other in terms of non-spectral graph metrics, such as path length, clustering coefficient and betweenness centrality. Our experimental results show that our approach is successful for different classes of graphs and a wide range of Laplacian graph spectra.

Figures (6)

• Figure 1: Examples of target graphs and target densities for graph size $n=12$. (a) Star graph $\mathcal{S}_{11}$. (b) Circulant graph $\mathcal{C}_{12}^{1234}$. (c) Circulant graph $\mathcal{C}_{12}^{123}$. (d) Spectral densities of $\varphi_{\mathcal{T}}(x)$, see Eq. \ref{['eq:density']}, of the target graphs (a)-(c).
• Figure 2: Evolution of the spectral density $\varphi_{\mathcal{G}}(x)$ over generations $\kappa$ for $n=24$, $12$-regular initial graphs and the target graph $\mathcal{C}_{24}^{1,2,\ldots,6}$.
• Figure 3: Results for target graphs of size $n=24$. Boxplots for 30 runs and 1000 generations showing the final best fitness measured by the distance $d$, see Eq. \ref{['eq:distance']}, for eight initial graph populations and three crossover implementations. (a) Star graph $\mathcal{S}_{23}$ (b) Circulant graph $\mathcal{C}_{24}^{1,2,\ldots,8}$. (c) Circulant graph $\mathcal{C}_{24}^{1,2,\ldots,6}$.
• Figure 4: Results for target graphs of varying size. Boxplots showing the final best fitness measured by the logarithmic distance $\log(d)$ for eight initial graph populations and graph size $n=\{64, 128, 256, 512\}$. (a) Star graph $\mathcal{S}_{n-1}$ (b) Circulant graph $\mathcal{C}_{n}^{1,2,\ldots,\lfloor n/3\rfloor}$. (c) Circulant graph $\mathcal{C}_{n}^{1,2,\ldots,\lfloor n/4\rfloor}$.
• Figure 5: Scatter plots of final best fitness versus graph metrics for the circulant target graph $\mathcal{C}_{24}^{1,2,\ldots,8}$. The color code indicates different initial populations. (a) Algebraic connectivity (AC). (b) (Average) path length (PL). (c) (Global) clustering coefficient (CC). (d) (Average) betweenness centrality (BC).