Dependence on initial condition grouping also number

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Unformatted text preview: probability over time); 2) If no pair reduces number of spanning edges, swap pair that increases it the least. • Dependence on initial condition / grouping. • Also number of groups and their sizes preserved. Spectral partitioning – graph bi-section with Fiedler vector • Recall random walk state transition matrix (we started from Adjacency matrix A and row or column normalized it). 0 1 /4 B 1 /4 B B M = B 1 /4 B @ 1 /4 0 1/3 1/3 0 1/3 0 1/2 0 1 /2 0 0 1/4 1 /4 0 1 /4 1 /4 • Consider instead the Graph Laplacian, L where: No self-loops allowed Lii = di (the degree of node i) 0 Lij = −Aij for i = j . 3 −1 −1 −1 B −1 2 0 −1 B (Recall −Aij = −1 if edge B L = B −1 0 1 0 connects i and j , and 0 otherwise.) B @ −1 0 −1 0 0 0 3 −1 0 0 0 1/2 1/2 1 0 0 0 −1 1 1 C C C C C A C C C C C A Graph bisection • M , rwalk state transition matrix (Perron-Frobenious); largest eigenvalue λ = 1 (steady-state). The number of eigenvalues with λ = 1 equals number of disconnected components • L, graph Laplacian. – Related to diffusion on a network (see Newman book, sec 6.13.1) – All eigenvalues λi real and λi ≥ 0. – There will always be one λ = 0, call it λ1 (Note v1 = {1, 1, 1, · · · }) – The number of λi’s equal to zero are the number of disconnected components. – If only one λ = 0, call the next largest eigenvector λ2, the Fiedler vector. – The eigenvector v2 will have elements equal −1 or +1 only. – Elements with +1 assigned to group 1, those with -1 to group 2 (left picture:) Timescale, τ1 = 5314 to 0 5 10 20 X q q q q q qq q qq q qq 0 5 10 q qq oqooq oq oo qo o oqq q 20 X 20 5 10 + + + + ++ + ++ + ++ ++ ++ o ++ q o 0 20 5 10 0 o oo o oo oo o q q o q qq q q o o o o oq o o oq q q o q o q oo q o q qo q qq q o qo o o o q q q oq q oq q q q ooo o q q q o q o q q o q q qq q qq q q o q q q o q o Y 20 5 10 Y 0 o oo o o oo o o o oo q q o q qq q q o o o o oq o o oq q q o q o q oo q q qo oo q qo q qo q o qo o qq q q oq q q q q q q ooo o o o o o q q q o q o o oo o q qq q qq oo q o q q o q q qq oo q q o qq Y q q q qq qq q q q q q qq q qq q q qq q qq q q qq q q q q q q q q qq q qq q q q q q qq q q q q q q qq q qq qq qq q q q q qq qq q + ++ + + + ++ + + +++ + + + + ++ +++ + ++ + o+ + + ++ ++ o oo + +++ o + oo q q Timescale, τ3 = 157 to Timescale, τ2 = 1092 to + +++ + ++ + + + + +++ + ++ ++ + ++ ++ ooooo + + + ooo o ++ q q q q q q q qq q qq q qq...
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