Exact and approximate mean first passage times on trees and other necklace structures: a local equilibrium approach

In this work we propose a novel method to calculate mean first-passage times (MFPTs) for random walks on graphs, based on a dimensionality reduction technique for Markov State Models, known as local-equilibrium (LE). We show that for a broad class of graphs, which includes trees, LE coarse-graining preserves the MFPTs between certain nodes, upon making a suitable choice of the coarse-grained states (or clusters). We prove that this relation is exact for graphs that can be coarse-grained into a one-dimensional lattice where each cluster connects to the lattice only through a single node of the original graph. A side result of the proof generalises the well-known essential edge lemma (EEL), which is valid for reversible random walks, to irreversible walkers. Such a generalised EEL leads to explicit formulae for the MFPTs between certain nodes in this class of graphs. For graphs that do not fall in this class, the generalised EEL provides useful approximations if the graph allows a one-dimensional coarse-grained representation and the clusters are sparsely interconnected. We first demonstrate our method for the simple random walk on the $c$-ary tree, then we consider other graph structures and more general random walks, including irreversible random walks.

[1]  Rajeev Motwani,et al.  The PageRank Citation Ranking : Bringing Order to the Web , 1999, WWW 1999.

[2]  D. Aldous.  Probability Approximations via the Poisson Clumping Heuristic , 1988 .

[3]  Edina Rosta,et al.  Correlation functions, mean first passage times, and the Kemeny constant. , 2019, The Journal of chemical physics.

[5]  Kathryn B. Laskey,et al.  Stochastic blockmodels: First steps , 1983 .

[6]  Edward A. Codling,et al.  Random walk models in biology , 2008, Journal of The Royal Society Interface.

[7]  R. Pastor-Satorras,et al.  Random walks on complex trees. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  Sergey Brin,et al.  The Anatomy of a Large-Scale Hypertextual Web Search Engine , 1998, Comput. Networks.

[9]  Mason A. Porter,et al.  Random walks and diffusion on networks , 2016, ArXiv.

[10]  R. Bapat.  On the first passage time of a simple random walk on a tree , 2011 .

[11]  J. Meyer.  The Role of the Group Generalized Inverse in the Theory of Finite Markov Chains , 1975 .

[12]  A. Hellander,et al.  A multiscale compartment-based model of stochastic gene regulatory networks using hitting-time analysis , 2021, The Journal of chemical physics.

[13]  L. Asz.  Random Walks on Graphs: a Survey , 2022 .

[14]  Vincenzo Nicosia,et al.  First-passage times to quantify and compare structural correlations and heterogeneity in complex systems , 2020, Communications Physics.

[15]  H. Stanley,et al.  Optimizing the success of random searches , 1999, Nature.

[16]  O. Bénichou,et al.  From first-passage times of random walks in confinement to geometry-controlled kinetics , 2014 .

[17]  Vittorio Loreto,et al.  Ring structures and mean first passage time in networks. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  D. Wales,et al.  Optimal dimensionality reduction of Markov chains using graph transformation. , 2020, The Journal of chemical physics.

[19]  J. Stoyanov.  A Guide to First‐passage Processes , 2003 .

[20]  Havlin,et al.  Mean first-passage time on loopless aggregates. , 1989, Physical review. A, General physics.

[21]  G. Hummer,et al.  Optimal Dimensionality Reduction of Multistate Kinetic and Markov-State Models , 2014, The journal of physical chemistry. B.

[22]  Heiko Rieger,et al.  Random walks on complex networks. , 2004, Physical review letters.

[23]  Jim Pitman,et al.  Tree formulas, mean first passage times and Kemeny’s constant of a Markov chain , 2016, Bernoulli.

[24]  Nicolas E. Humphries,et al.  Environmental context explains Lévy and Brownian movement patterns of marine predators , 2010, Nature.

[25]  O. Martin,et al.  Return probabilities and hitting times of random walks on sparse Erdös-Rényi graphs. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  R. Kuehn,et al.  A random walk perspective on hide-and-seek games , 2018, Journal of Physics A: Mathematical and Theoretical.