This preview shows page 1. Sign up to view the full content.
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
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...
View Full Document
This document was uploaded on 03/12/2014 for the course CSCI 289 at UC Davis.
- Winter '11