Spectral analysis of communication networks using Dirichlet eigenvalues

Good clustering can provide critical insight into potential locations where congestion in a network may occur. A natural measure of congestion for a collection of nodes in a graph is its Cheeger ratio, defined as the ratio of the size of its boundary to its volume. Spectral methods provide effective means to estimate the smallest Cheeger ratio via the spectral gap of the graph Laplacian. Here, we compute the spectral gap of the truncated graph Laplacian, with the so-called Dirichlet boundary condition, for the graphs of a dozen communication networks at the IP-layer, which are subgraphs of the much larger global IP-layer network. We show that i) the Dirichlet spectral gap of these networks is substantially larger than the standard spectral gap and is therefore a better indicator of the true expansion properties of the graph, ii) unlike the standard spectral gap, the Dirichlet spectral gaps of progressively larger subgraphs converge to that of the global network, thus allowing properties of the global network to be efficiently obtained from them, and (iii) the (first two) eigenvectors of the Dirichlet graph Laplacian can be used for spectral clustering with arguably better results than standard spectral clustering. We first demonstrate these results analytically for finite regular trees. We then perform spectral clustering on the IP-layer networks using Dirichlet eigenvectors and show that it yields cuts near the network core, thus creating genuine single-component clusters. This is much better than traditional spectral clustering where several disjoint fragments near the network periphery are liable to be misleadingly classified as a single cluster. Since congestion in communication networks is known to peak at the core due to large-scale curvature and geometry, identification of core congestion and its localization are important steps in analysis and improved engineering of networks. Thus, spectral clustering with Dirichlet boundary condition is seen to be more effective at finding bona-fide bottlenecks and congestion than standard spectral clustering.