Generative Graph Models based on Laplacian Spectra?

Alana Shine, David Kempe

Abstract: We present techniques for generating random graphs whose Laplacian spectrum approximately matches that of a given input graph. The motivation for matching the Laplacian spectrum is that it naturally encodes high-level connectivity information about the input graph; most existing models (e.g., variants of the Configuration Model, Stochastic Block Model, or Kronecker Graphs) focus on local structure or limited high-level partitions. Our techniques succeed in matching the spectrum of the input graph more closely than the benchmark models. We also evaluate our generative model using other global and local properties, including shortest path distances, betweenness centrality, degree distribution, and clustering coefficients. The graphs produced by our model almost always match the input graph better than those produced by the benchmark models with respect to shortest path distance and clustering coefficient distributions. The performance on betweenness centrality is comparable to the benchmarks, while a worse match on the degree distribution is a price our method pays for more global similarity. Our results suggest that focusing on spectral properties may lead to good performance for other global properties, at a modest loss in local similarity. Since global connectivity patterns are usually more important than local features for processes such as information flow, spread of epidemics, routing, etc., our main goal is to advocate for a shift in focus from graph generative models matching local properties to those matching global connectivity patterns.