Volume entropy and information flow in a brain graph

Entropy is a classical measure to quantify the amount of information or complexity of a system. Various entropy-based measures such as functional and spectral entropy have been proposed in brain network analysis. However, they are less widely used than traditional graph theoretic measures such as global and local efficiencies because they are not well-defined on a graph or difficult to interpret its biological meaning. In this paper, we propose a new entropy-based graph invariant, called volume entropy. It measures the exponential growth rate of the number of graph paths, based on the assumption that information flows through a graph forever. We model the information propagation on a brain graph by the generalized Markov system associated to a new edge transition matrix. The volume entropy is estimated by the stationary equation of the generalized Markov system. Moreover, its stationary distribution shows the information capacity of edge and the direction of information flow on a brain graph. The simulation results show that the volume entropy distinguishes the underlying graph topology and geometry better than the existing graph measures. In the clinical application, the volume entropy of brain graphs was significantly related to healthy normal aging from 20s to 60s. In addition, the stationary distribution of information propagation gives a new insight into the information flow of functional brain graph.