Controlling the efficiency of trapping in treelike fractals

TL;DR

This paper investigates how to control trapping efficiency in directed treelike fractals by analyzing eigenvalues and mean first-passage time, revealing that the process can be finely tuned through edge weight parameters.

Contribution

It introduces a method to precisely control trapping efficiency in directed fractals by deriving eigenvalues and relating them to the mean first-passage time, which was not previously established.

Findings

MFPT can be tuned to scale superlinearly, linearly, or sublinearly with system size.

Eigenvalues are obtained through recursive relations based on fractal self-similarity.

The trapping process is controllable via the directed edge weight parameter.

Abstract

Efficiently controlling the trapping process, especially the trapping efficiency, is central in the study of trap problem in complex systems, since it is a fundamental mechanism for diverse other dynamic processes. Thus, it is of theoretical and practical significance to study the control technique for trapping problem. In this paper, we study the trapping problem in a family of proposed directed fractals with a deep trap at a central node. The directed fractals are a generalization of previous undirected fractals by introducing the directed edge weights dominated by a parameter. We characterize all the eigenvalues and their degeneracies for an associated matrix governing the trapping process. The eigenvalues are provided through an exact recursive relation deduced from the self-similar structure of the fractals. We also obtain the expressions for the smallest eigenvalue and the mean first-passage time (MFPT) as a measure of trapping efficiency, which is the expected time for the walker to first visit the trap. The MFPT is evaluated according to the proved fact that it is approximately equal to reciprocal of the smallest eigenvalue. We show that the MFPT is controlled by the weight parameter, by modifying which, the MFPT can scale superlinealy, linearly, or sublinearly with the system size. Thus, this work paves a way to delicately controlling the trapping process in the fractals.