Curvature and entropy of a graph

On the brain network, various entropies have been proposed and studied. They are used as measures of how efficiently a signal is transmitted over the whole brain, but they are insufficient in some respects. In order to compensate them, we propose an entropy for a graph based on Boltzmann’s entropy for the heat flow, which shows how efficiently a signal is transmitted at each time. We show that our entropy is closely related to the curvature of a graph. •We propose the entropy Hδ(t) based on Boltzmann’s entropy which is a function of time t.•By series expansions of Hδ(t), we study relations between Hδ(t) and geometric structures such as curvature.•We show that Hδ(t) can be a measure for the transmission efficiency.