Large deviations of random walks on random graphs

Abstract

We study the rare fluctuations or large deviations of time-integrated functionals or observables of an unbiased random walk evolving on Erdös-Rényi random graphs, and construct a modified, biased random walk that explains how these fluctuations arise in the long-time limit. Two observables are considered: the sum of the degrees visited by the random walk and the sum of their logarithm, related to the trajectory entropy. The modified random walk is used for both quantities to explain how sudden changes in degree fluctuations, similar to dynamical phase transitions, are related to localization transitions. For the second quantity, we also establish links between the large deviations of the trajectory entropy and the maximum entropy random walk.

Varied phenomenology of models displaying dynamical large-deviation singularities

“…However, large-deviation singularities do not necessarily indicate the existence of cooperative phenomena or distinct phases. For instance, singular features are seen in the large-deviation functions of finite systems in the reducible limit, when the connections between microstates are severed [23][24][25][26][27]. We show here that singularities can also appear in the limit of large system size of dynamical models, if the model's basic timescale (mixing time) diverges with system size.…”

“…The latter is suggestive of intermittency, although linear rate functions also arise in other types of process, such as relaxation to an absorbing state [5,35]. Moreover, simple switching models, which are by design intermittent, display, as the switching time increases, rate functions that broaden and vanish, becoming linear with zero gradient [23][24][25] (see, e.g., Fig. 5 of Ref.…”

Dynamical phase transition in drifted Brownian motion

“…The escape itself is analogous to Markov processes with absorbing states, which provide the simplest examples of rate functions having linear parts (see [64] and the appendix of [65]). The absorbing state in our case is represented by the complement of [−a, a] (i.e., the state "X t / ∈ [−a, a]"), which is eventually reached by dBM and serves as a trap for it, as this process is not recurrent, meaning that it has a zero probability to ever return close to x = 0 [66].…”

Generalized optimal paths and weight distributions revealed through the large deviations of random walks on networks

“…In this paper we propose a theoretical approach to unveil such generalized optimal paths and weight distributions by studying the statistics of trajectories using large-deviation methods [26][27][28]. Specifically, we analyze the large deviations of random walks on graphs [29,30]. This allows us to find paths that are optimal in the statistical sense outlined above, or weight distributions that make a network optimal for a given statistical characterization pertaining to the flow of information or physical entities.…”

“…In the first work, as far as we are aware, on large deviations of time-integrated observables of random walks on networks [29], localization and mode-switching dynamical phase transitions are revealed. More recently, in a contribution that has strongly influenced our methodology [30], such localization phenomena are explained by means of the generalized Doob transform, which is also used to shed new light on the relationship between the maximum entropy random walk and the standard random walk. Various other processes have been explored with related methodologies in publications dealing with, e.g., percolation transitions in single or multilayer networks subject to rare initial configurations, [37,38], paths leading to epidemic extinction [39,40], the connection between the rate of rare events and heterogeneity in population networks [41], or large-fluctuation-induced phase switch in majority-vote models [42].…”