Mean first-passage time for random walks in general graphs with a deep trap

TL;DR

This paper derives an explicit formula for the mean first-passage time in general graphs with a trap, providing bounds and scaling behaviors, especially in scale-free networks, to deepen understanding of trapping phenomena.

Contribution

It introduces a formula for GMFPT based on Laplacian eigenvalues and eigenvectors, and establishes bounds and scaling laws for different graph types.

Findings

Explicit GMFPT formula in terms of Laplacian eigenvalues

Lower bound for GMFPT depending on graph size and trap degree

Upper bound for GMFPT scaling as N^3 in worst-case graphs

Abstract

We provide an explicit formula for the global mean first-passage time (GMFPT) for random walks in a general graph with a perfect trap fixed at an arbitrary node, where GMFPT is the average of mean first-passage time to the trap over all starting nodes in the whole graph. The formula is expressed in terms of eigenvalues and eigenvectors of Laplacian matrix for the graph. We then use the formula to deduce a tight lower bound for the GMFPT in terms of only the numbers of nodes and edges, as well as the degree of the trap, which can be achieved in both complete graphs and star graphs. We show that for a large sparse graph the leading scaling for this lower bound is proportional to the system size and the reciprocal of the degree for the trap node. Particularly, we demonstrate that for a scale-free graph of size $N$ with a degree distribution $P(d)\sim d^{-\gamma}$ characterized by $\gamma$, when the trap is placed on a most connected node, the dominating scaling of the lower bound becomes $N^{1-1/\gamma}$, which can be reached in some scale-free graphs. Finally, we prove that the leading behavior of upper bounds for GMFPT on any graph is at most $N^{3}$ that can be reached in the bar-bell graphs. This work provides a comprehensive understanding of previous results about trapping in various special graphs with a trap located at a specific location.