Effective target arrangement in a deterministic scale-free graph

TL;DR

This paper analyzes how the placement of static targets affects the efficiency of random walks on a deterministic scale-free network, revealing that target position significantly influences mean first-passage times.

Contribution

It provides a rigorous analysis of MFPT scaling in relation to target placement on a deterministic scale-free graph, highlighting the impact of inhomogeneity.

Findings

MFPT scales as N^{theta} with theta between (1 - log 2/log3) and 1.

Central targets yield lower MFPT exponents, indicating higher efficiency.

Peripheral targets result in MFPT scaling close to linear with network size.

Abstract

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 N^{theta}, being N the volume of the substrate and theta ranging from (1 - log 2/log3), for central target(s), to 1, for a single peripheral target.