Spectral analysis of communication networks using Dirichlet eigenvalues

Abstract

The spectral gap of the graph Laplacian with Dirichlet boundary conditions is computed for the graphs of several communication networks at the IP-layer, which are subgraphs of the much larger global IP-layer network. We show that the Dirichlet spectral gap of these networks is substantially larger than the standard spectral gap and is likely to remain non-zero in the infinite graph limit. We first prove this result for finite regular trees, and show that the Dirichlet spectral gap in the infinite tree limit converges to the spectral gap of the infinite tree. We also perform Dirichlet spectral clustering on the IP-layer networks and show that it often yields cuts near the network core that create genuine single-component clusters. This is much better than traditional spectral clustering where several disjoint fragments near the periphery are liable to be misleadingly classified as a single cluster. Spectral clustering is often used to identify bottlenecks or congestion; since congestion in these networks is known to peak at the core, our results suggest that Dirichlet spectral clustering may be better at finding bona-fide bottlenecks.