Lecture : Modeling graphs with electrical networks

So far, we have been adopting the usual approach to spectral graph theory: understand graphs via the eigenvectors and eigenvalues of associated matrices. For example, given a graph G = (V,E), we defined an adjacency matrix A and considered the eigensystem Av = λv, and we also defined the Laplacian matrix L = D−A and considered the Laplacian quadratic form xTLx− ∑ (ij)∈E(xi−xj). There are other ways to think about spectral graph methods that, while related, are different in important ways. In particular, one can draw from physical intuition and define physical-based models from the graph G, and one can also consider more directly vectors that are obtained from various diffusions and random walks on G. We will do the former today, and we will do the latter next time.