1303.6203 (Ernesto Estrada et al.)

Walk Entropies in Quantum Networks    [PDF]

Ernesto Estrada, Jose A. de la Pena, Naomichi Hatano

Walk entropies are defined for thermal Green's function on a graph. The definition is based on the summation over diagonal and off-diagonal elements of the thermal Green's function also known as communicability, in contrast to the Shannon-like entropy based on the summation over eigenmodes or the von Neumann one based on normalized eigenvalues of the Laplacian matrix. The walk entropies are strongly related to the walk regularity of graphs; the vertex-walk entropy takes its maximum for 0-walk regular graphs and the edge-walk entropy does for 1-walk regular ones. The walk entropy is not biased by graph size as in the cases of the von Neumann and Shannon entropies. It has significantly better correlation with the inverse participation ratio of the eigenmodes of the adjacency matrix than the other mentioned entropies. The temperature dependence of the walk entropies is also discussed. In particular, the vertex-walk entropy is shown to be non-monotonic for regular but non-0-walk regular graphs in contrast to non-regular graphs.

View original: http://arxiv.org/abs/1303.6203