Effective target arrangement in a deterministic scale-free graph.

We study the random-walk problem on a deterministic scale-free network, in the presence of a set of static, identical targets; due to the strong inhomogeneity of the underlying structure the mean first-passage time (MFPT), meant as a measure of transport efficiency, is expected to depend sensitively on the position of targets. We consider several spatial arrangements for targets and we calculate, mainly rigorously, the related MFPT, where the average is taken over all possible starting points and over all possible paths. For all the cases studied, the MFPT asymptotically scales like ∼Nθ, being N the volume of the substrate and θ ranging from 1-log 2/log 3, for central target(s), to 1, for a single peripheral target.