Global mean first-passage times of random walks on complex networks

We present a general framework, applicable to a broad class of random walks on complex networks, which provides a rigorous lower bound for the mean first-passage time of a random walker to a target site averaged over its starting position, the so-called global mean first-passage time (GMFPT). This bound is simply expressed in terms of the equilibrium distribution at the target, and implies a minimal scaling of the GMFPT with the network size. We show that this minimal scaling, which can be arbitrarily slow for a proper choice of highly connected target, is realized under the simple condition that the random walk is transient at the target site, and independently of the small-world, scale free or fractal properties of the network. Last, we put forward that the GMFPT to a specific target is not a representative property of the network, since the target averaged GMFPT satisfies much more restrictive bounds, which forbid any sublinear scaling with the network size.

Probing microscopic origins of confined subdiffusion by first-passage observables

Subdiffusive motion of tracer particles in complex crowded environments, such as biological cells, has been shown to be widepsread. This deviation from brownian motion is usually characterized by a sublinear time dependence of the mean square displacement (MSD). However, subdiffusive behavior can stem from different microscopic scenarios, which can not be identified solely by the MSD data. In this paper we present a theoretical framework which permits to calculate analytically first-passage observables (mean first-passage times, splitting probabilities and occupation times distributions) in disordered media in any dimensions. This analysis is applied to two representative microscopic models of subdiffusion: continuous-time random walks with heavy tailed waiting times, and diffusion on fractals. Our results show that first-passage observables provide tools to unambiguously discriminate between the two possible microscopic scenarios of subdiffusion. Moreover we suggest experiments based on first-passage observables which could help in determining the origin of subdiffusion in complex media such as living cells, and discuss the implications of anomalous transport to reaction kinetics in cells.