  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 entropies 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 either 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 paths in a graph, which is a relevant
measure if information flows through the graph forever. We model the
information propagation on a graph by the generalized Markov system associated
to the weighted edge-transition matrix. We estimate the volume entropy using
the stationary equation of the generalized Markov system. A prominent advantage
of using the stationary equation is that it assigns certain distribution of
weights on the edges of the brain graph, which we call the stationary
distribution. The stationary distribution shows the information capacity of
edges 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 brain imaging
data 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.
