Optimal scale-free network with a minimum scaling of transport efficiency for random walks with a perfect trap

TL;DR

This paper investigates the trapping problem in hierarchical scale-free networks, deriving formulas for average trapping time and demonstrating that minimal scaling can be achieved, informing the design of highly efficient transport networks.

Contribution

It provides explicit formulas and demonstrates minimal scaling of average trapping time in scale-free networks with a perfect trap, advancing understanding of optimal network design.

Findings

Derived closed-form formulas for ATT in four trap locations.

Showed that ATT scaling reaches the theoretical minimum in studied networks.

Deepened understanding of trapping efficiency in scale-free networks.

Abstract

Average trapping time (ATT) is central in the trapping problem since it is a key indicator characterizing the efficiency of the problem. Previous research has provided the scaling of a lower bound of the ATT for random walks in general networks with a deep trap. However, it is still not well understood in which networks this minimal scaling can be reached. Particularly, explicit quantitative results for ATT in such networks, even in a specific network, are lacking, in spite that such networks shed light on the design for optimal networks with the highest trapping efficiency. In this paper, we study the trapping problem taking place on a hierarchical scale-free network with a perfect trap. We focus on four representative cases with the immobile trap located at the root, a peripheral node, a neighbor of the root with a single connectivity, and a farthest node from the root, respectively. For all the four cases, we obtain the closed-form formulas for the ATT, as well as its leading scalings. We show that for all the four cases of trapping problems, the dominating scalings of ATT can reach the predicted minimum scalings. This work deepens the understanding of behavior of trapping in scale-free networks, and is helpful for designing networks with the most efficient transport process.