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Register a new userSpectral analysis of communication networks using Dirichlet eigenvalues
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Added by
Alexander Tsiatas
Publication date
2011
fields
Informatics Engineering
and research's language is
English
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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.
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Let $H_{0, D}$ (resp., $H_{0,N}$) be the Schroedinger operator in constant magnetic field on the half-plane with Dirichlet (resp., Neumann) boundary conditions, and let $H_ell : = H_{0, ell} - V$, $ell =D,N$, where the scalar potential $V$ is non negative, bounded, does not vanish identically, and decays at infinity. We compare the distribution of the eigenvalues of $H_D$ and $H_N$ below the respective infima of the essential spectra. To this end, we construct effective Hamiltonians which govern the asymptotic behaviour of the discrete spectrum of $H_ell$ near $inf sigma_{ess}(H_ell) = inf sigma(H_{0,ell})$, $ell = D,N$. Applying these Hamiltonians, we show that $sigma_{disc}(H_D)$ is infinite even if $V$ has a compact support, while $sigma_{disc}(H_N)$ could be finite or infinite depending on the decay rate of $V$.
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We study the spectrum of the Dirichlet Laplacian on an unbounded twisted tube with twisting velocity exploding to infinity. If the tube cross section does not intersect the axis of rotation, then its spectrum is purely discrete under some additional conditions on the twisting velocity (D.Krejcirik, 2015). In the current work we prove a Berezin type upper bound for the eigenvalue moments.
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