High degree nodes slow down spreading plain old

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Unformatted text preview: ted networks. – Local spatial coordination enhances spreading (having a spatial metric; graph embeddable in small dimension). – High-degree nodes slow down spreading. Plain old diffusion: The graph Laplacian Diffusion of a substance φ on a network with Adjacency Matrix A • dφi dt =C j Aij (φj − φi) =C j Aij φj − Cφi =C j Aij φj − Cφiki =C j (Aij φj − δij ki) φj . • In matrix form: dφ dt j Aij = C (A − D)φ = C Lφ • Graph Laplacian: L, where matrix D has zero entries except for diagonal with is degree of node: Dij = ki if i = j and 0 otherwise. Plain old diffusion The graph Laplacian • L has real positive eigenvalues 0 = λ1 ≤ λ2 ≤ · · · ≤ λN . • Number of eigenvalues equal to 0 is the number of distinct, disconnected components of a graph (for the random-walk state transition vector, it is the number of λ’s equal to 1). • If λ2 = 0 the graph is fully connected. The bigger the value of λ2 the more connected (less modular) the graph. Part I. Ensemble approaches • A. Net...
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This document was uploaded on 03/12/2014 for the course CSCI 289 at UC Davis.

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