Close or connected? Distance and connectivity effects on transport in networks

We develop an analytical approach which provides the dependence of the mean first-passage time (MFPT) for random walks on complex networks both on the target connectivity and on the source-target distance. Our approach puts forward two strongly different behaviors depending on the type - compact or non compact - of the random walk. In the case of non compact exploration, we show that the MFPT scales linearly with the inverse connectivity of the target, and is largely independent of the starting point. On the contrary, in the compact case the MFPT is controlled by the source-target distance, and we find that unexpectedly the target connectivity becomes irrelevant for remote targets.