Rare events statistics of random walks on networks: localisation and other dynamical phase transitions

Abstract

Rare event statistics for random walks on complex networks are investigated using the large deviations formalism. Within this formalism, rare events are realized as typical events in a suitably deformed path-ensemble, and their statistics can be studied in terms of spectral properties of a deformed Markov transition matrix. We observe two different types of phase transition in such systems: (i) rare events which are singled out for sufficiently large values of the deformation parameter may correspond to localized modes of the deformed transition matrix; (ii) "mode-switching transitions" may occur as the deformation parameter is varied. Details depend on the nature of the observable for which the rare event statistics is studied, as well as on the underlying graph ensemble. In the present letter we report on the statistics of the average degree of the nodes visited along a random walk trajectory in Erdős-Rényi networks. Large deviations rate functions and localization properties are studied numerically. For observables of the type considered here, we also derive an analytical approximation for the Legendre transform of the large-deviations rate function, which is valid in the large connectivity limit. It is found to agree well with simulations.

Large deviations of random walks on random graphs

“…Our contribution is to clarify some of the hypotheses used in that work to approximate the large deviation functions characterising the cost fluctuations, and to derive the so-called driven process, which is a modified random walk that explains how specific cost fluctuations are created in time [22][23][24][25]. This is useful, as we will see, to understand how dynamical phase transitions arising in cost fluctuations are linked to localization transitions, as first reported in [18]. On a more practical level, the driven process can also be controlled to identify nodes with low or high degree, in addition to other graph properties, without knowing the detailed structure of the graph.…”

“…if we average the random walk in an annealed way over the whole ensemble of ER graphs. In our case, we consider random instances of the ER ensemble (quenched regime) and initialise the URW on the giant component, so as to avoid non-ergodic effects related to small disconnected components of the graph or single disconnected nodes [18]. As a result, the stationary average with P Poisson above must be computed instead with Q(k), as given by (17), since we are considering nodes that are necessarily connected (k ≥ 1), leading to…”

“…The results are shown for different graph sizes in the left column of Fig. 1 for α = 20 (top) and α = 4 (bottom), and are also compared with a random degree or mean-field (mf) approximation of Ψ(s) proposed in [18], having the form…”

“…In this paper, we follow a more dynamical approach initiated by De Bacco et al [18], whereby a single random walker hops on a random graph and accumulates each time it jumps a certain "cost" related, for example, to the degree of the node visited or other characteristics of that node. If the random walk is ergodic, then the mean cost (i.e., the total cost divided by the total number of jumps) will converge with probability 1 to the ergodic average of the cost as the number of jumps increases, meaning that the cost of most trajectories of the random walk is very close to that average.…”

Simple bounds on fluctuations and uncertainty relations for first-passage times of counting observables

“…An important observation is that the bounds on fluctuations of a counting observable and its FPTs are controlled by the average dynamical activity, in analogy to the role played by the entropy production in the case of currents [1,13]. We hope these results will add to the growing body of work applying large deviation ideas and methods to the study of dynamics in driven systems [29][30][31][32][33][34][35][36][37][38][39][40][41], glasses [25,[42][43][44][45][46][47], protein folding and signaling networks [48][49][50][51], open quantum systems [52][53][54][55][56][57][58][59][60][61][62][63][64], and many other problems in nonequilibrium [65][66][67][68][69][70][71][72].…”