Mean first-passage time calculations: comparison of the deterministic Hill’s algorithm with Monte Carlo simulations

Abstract

Accurate determination of mean first-passage times (MFPTs) between any two states of a complex network still attracts considerable attention. Appropriate methods should take into account the discrepancy in MFPTs when a random walker moves first from a source to a target and then in the opposite direction. In addition, it is desirable to allow fast evaluation of mean first-passage times when transition probabilities are allowed to vary over time. For our calculations we make use of Hill’s algorithm, which enables the exact calculation of MFPTs. As we show in this work, when given a fixed distance to travel, the calculation of a particular MFPT depends on the choice of source and target points and their relative position on a lattice. We also demonstrate, when the network contains a relatively low number of cycles, that this deterministic technique provides exact results much faster in comparison to the more standard, but computationally demanding, stochastic Monte Carlo simulation method, where only approximate results can be obtained that are highly dependent on the number of walkers. Therefore, our specific implementation of Hill’s algorithm should facilitate efficient and accurate computation of MFPTs on a variety of network topologies of practical interest to the broad scientific community.