33 research outputs found

    Random Regular Graphs are not Asymptotically Gromov Hyperbolic

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    In this paper we prove that random dd--regular graphs with dβ‰₯3d\geq 3 have traffic congestion of the order O(nlog⁑dβˆ’13(n))O(n\log_{d-1}^{3}(n)) where nn is the number of nodes and geodesic routing is used. We also show that these graphs are not asymptotically Ξ΄\delta--hyperbolic for any non--negative Ξ΄\delta almost surely as nβ†’βˆžn\to\infty.Comment: 6 pages, 2 figure

    Scaling of Congestion in Small World Networks

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    In this report we show that in a planar exponentially growing network consisting of NN nodes, congestion scales as O(N2/log⁑(N))O(N^2/\log(N)) independently of how flows may be routed. This is in contrast to the O(N3/2)O(N^{3/2}) scaling of congestion in a flat polynomially growing network. We also show that without the planarity condition, congestion in a small world network could scale as low as O(N1+ϡ)O(N^{1+\epsilon}), for arbitrarily small ϡ\epsilon. These extreme results demonstrate that the small world property by itself cannot provide guidance on the level of congestion in a network and other characteristics are needed for better resolution. Finally, we investigate scaling of congestion under the geodesic flow, that is, when flows are routed on shortest paths based on a link metric. Here we prove that if the link weights are scaled by arbitrarily small or large multipliers then considerable changes in congestion may occur. However, if we constrain the link-weight multipliers to be bounded away from both zero and infinity, then variations in congestion due to such remetrization are negligible.Comment: 8 page

    Traffic Analysis in Random Delaunay Tessellations and Other Graphs

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    In this work we study the degree distribution, the maximum vertex and edge flow in non-uniform random Delaunay triangulations when geodesic routing is used. We also investigate the vertex and edge flow in Erd\"os-Renyi random graphs, geometric random graphs, expanders and random kk-regular graphs. Moreover we show that adding a random matching to the original graph can considerably reduced the maximum vertex flow.Comment: Submitted to the Journal of Discrete Computational Geometr

    Lack of Hyperbolicity in Asymptotic Erd\"os--Renyi Sparse Random Graphs

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    In this work we prove that the giant component of the Erd\"os--Renyi random graph G(n,c/n)G(n,c/n) for c a constant greater than 1 (sparse regime), is not Gromov Ξ΄\delta-hyperbolic for any positive Ξ΄\delta with probability tending to one as nβ†’βˆžn\to\infty. As a corollary we provide an alternative proof that the giant component of G(n,c/n)G(n,c/n) when c>1 has zero spectral gap almost surely as nβ†’βˆžn\to\infty.Comment: Updated version with improved results and narrativ

    Limits Laws for Geometric Means of Free Random Variables

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    Let {Tk}k=1∞\{T_{k}\}_{k=1}^{\infty} be a family of *--free identically distributed operators in a finite von Neumann algebra. In this work we prove a multiplicative version of the free central limit Theorem. More precisely, let Bn=T1βˆ—T2βˆ—...Tnβˆ—Tn...T2T1B_{n}=T_{1}^{*}T_{2}^{*}... T_{n}^{*}T_{n}... T_{2}T_{1} then BnB_{n} is a positive operator and Bn1/2nB_{n}^{1/2n} converges in distribution to an operator Ξ›\Lambda. We completely determine the probability distribution Ξ½\nu of Ξ›\Lambda from the distribution ΞΌ\mu of ∣T∣2|T|^{2}. This gives us a natural map G:M+β†’M+\mathcal{G}:\mathcal{M_{+}}\to \mathcal{M_{+}} with μ↦G(ΞΌ)=Ξ½.\mu\mapsto \mathcal{G}(\mu)=\nu. We study how this map behaves with respect to additive and multiplicative free convolution. As an interesting consequence of our results, we illustrate the relation between the probability distribution Ξ½\nu and the distribution of the Lyapunov exponents for the sequence {Tk}k=1∞\{T_{k}\}_{k=1}^{\infty} introduced in \cite{LyaV}.Comment: Published in Indiana Journal of Mathematics, vol. 59, no. 1, pp. 1-13, 201
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