Spectral Analysis of Communication Networks Using Dirichlet Eigenvalues

We study spectral characteristics of networks that represent the IP layer connectivity of communication systems as measured and documented by previous researchers. Our goal is to understand the behavior of these networks as truncated samples of in- nite graphs. As such, the existence of a spectral gap and positive Cheeger constant in the extrapolated innite variety would provide insight into their basic geometry. We apply Dirichlet boundary conditions to the computation of the eigenvalues to show that unlike standard spectral techniques, the gap and the Cheeger constant of nite truncations of regular trees provide accurate estimates of the corresponding parameters for the innite tree. Having shown the effectiveness of the Dirichlet spectrum for trees, we compute spectral decompositions via Dirichlet eigenvectors for the communication networks. We show that Dirichlet eigenvectors provide a strong means to separate clusters and conclude that the said networks exhibit characteristics common in hyperbolic networks.