Woong Kook (국웅), A Combinatorial Formula for Information Flow in a Network

A Combinatorial Formula for Information Flow in a Network
Woong Kook (국웅)
Department of Mathematics, University of Rhode Island, Kingston, Rhode Island, U.S.A.
2010/04/09 Fri 4PM-5PM

In 1989, Stephenson and Zelen derived an elegant formula for the information Iab contained in all possible paths between two nodes a and b in a network, which is described as follows. Given a finite weighted graph G and its Laplacian matrix L, the combinatorial Green’s function \(\mathcal{G}\), of G is the inverse of L+J, where J is the all 1’s matrix. Then, it was shown that Iab=(gaa+gbb-2gab)-1, where gij is the (i,j)-th entry of \(\mathcal{G}\). In this talk, we prove an intriguing combinatorial formula for Iab:

\(I_{ab}=\kappa(G)/\kappa(G_{a\ast b})\),

where \(\kappa(G)\) is the complexity, or tree-number, of G, and Ga*b is obtained from G by identifying two vertices a and b. We will discuss several implications of this new formula, including the equivalence of Iab and the effective conductance between two nodes in electrical networks.

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