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 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.
翻译:在这项工作中,我们提出了一个新颖的方法,用于计算图表中随机行走的平均第一通时间(MFPTs),该方法基于马可夫州模型(称为当地平衡(LE))的维度减少技术。我们展示了对于包括树木在内的广大类图表而言,LE粗毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛的图(MFPTs ) 。我们证明,对于每个组仅通过原始图的单一节点连接到拉特贝的单维宽边线线形图(MFPTs)来说,这一关系是精确的。对于一个广为人所知的基本边缘列米毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛毛的图(MFMFMFMTPTs)是精确的。对于不跌落到这一类的图表来说,一般的ELLELELEL提供有用的近毛毛毛毛毛毛毛毛毛毛毛细的图, 和粗略的图结构在原始图上,如果我们的直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直观的直径直径直径直径直径直径直径直径直径直径直径直的直的直的直的直的直的直的直的直的直的直的直的直的直的直的直的图,如果我们的直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直的直的直的直径直径直径直径直径直径直径直径直径直的直的直的直的直的直的直的直的直的直直直直直直的直的直直直直的直的直的直的直径直的直的直的直的直的直的直的直的直的直