Network Efficiency under Nonconservative Diffusion

The standard definition of network efficiency assumes in an ad hoc way that network navigability occurs by shortest paths only. This is obviously not the case for the many diffusive processes occurring in real-world complex systems. Here we propose from first principles a network efficiency measure that accounts for network navigability in diffusive terms. In particular, we prove that this efficiency index is based on nonconservative (NC) diffusion processes on the network, which are ubiquitous in the real-world. We then investigate analytically several properties of this efficiency index and provide computational examples of its effectivity for the analysis of complex systems. In particular, we show that the new efficiency index of a network does not necessarily change monotonically with the removal of edges, like it trivially happens with the index based on shortest paths. It also does not change trivially with the network density, and it indeed predicts the rate of convergence to the steady state of NC diffusion.