Damped random walks and the spectrum of the normalized laplacian on a graph

Let G be a weighted graph. 1 d(1) 1 d(1) e(1,2) e(1,2) e(3,0) e(3,0) 3 3

Matrix tree theorem -d(1) e(1,2) e(1,3) e(2,1) -d(2) e(2,3) = - e(3,1) e(3,2) -d(3) = T(G) e(i,j) = e(j,i) = weight of the branch ij. d(i) = (weighted) degree of node i.

-d(1)(1+λ) e(1,2)... e(1,n) e(2,1) -d(2)(1+λ)... e(2,n) = = e(n,1) e(n,2)... -d(n)(1+λ) Normalized eigenvalue equation : λ) = P( λ) =

Let G’ be the augmented graph 1 d(1) 1 d(1) e(1,2) λd(1) e(1,2) λd(1) λd(2) λd(2)

P(λ) = T(G’) P(λ) is the characteristic polynomial of a symmetric matrix, therefore the roots are all real. P(λ) is the characteristic polynomial of a symmetric matrix, therefore the roots are all real. The coefficients have the same sign, so the roots are less or equal to 0. The coefficients have the same sign, so the roots are less or equal to 0.

P(λ) = T(G’) = P(λ) = T(G’) = Σ| t | sign(t) = P(| λ |) E[sign(t)] Where, in the above expectation, the probability of getting tree t is equal to | t |/P(| λ |) The sign of t is (-1)^{# of branches incident on 0}.

Generating a random tree Start at any vertex. Start at any vertex. Do a random walk until every vertex has been visited. Do a random walk until every vertex has been visited. Add all branches along which first entries into some node have occurred. Add all branches along which first entries into some node have occurred.

Generating a random tree

Generating a random tree

Generating a random tree

Generating a random tree

Generating a random tree

Generating a random tree

Generating a random tree

Generating a random tree

Generating a random tree

p=λ/(λ-1) [λ is negative] The degree of 0 in a random tree of G’ = The degree of 0 in a random tree of G’ = The number of times a “p-damped” random walk starts on an unvisited node. The number of times a “p-damped” random walk starts on an unvisited node. Hence λ is a negative eigenvalue if and only if this number is equally likely to be even or odd.