Large deviations of random walks on random graphs

Journal of Physics A: Mathematical and Theoretical, 2016

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.

Physical review letters, 2007

We study random walks on large random graphs that are biased towards a randomly chosen but fixed target node. We show that a critical bias strength bc exists such that most walks find the target within a finite time when b > bc. For b < bc, a finite fraction of walks drifts off to infinity before hitting the target. The phase transition at b = bc is a critical point in the sense that quantities like the return probability P (t) show power laws, but finite size behavior is complex and does not obey the usual finite size scaling ansatz. By extending rigorous results for biased walks on Galton-Watson trees, we give the exact analytical value for bc and verify it by large scale simulations.

Journal of Physics A: Mathematical and General, 2005

Random walks on graphs are widely used in all sciences to describe a great variety of phenomena where dynamical random processes are affected by topology. In recent years, relevant mathematical results have been obtained in this field, and new ideas have been introduced, which can be fruitfully extended to different areas and disciplines. Here we aim at giving a brief but comprehensive perspective of these progresses, with a particular emphasis on physical aspects. Contents 1 Introduction 2 Mathematical description of graphs 3 The random walk problem 4 The generating functions 5 Random walks on finite graphs 6 Infinite graphs 7 Random walks on infinite graphs 8 Recurrence and transience: the type problem 9 The local spectral dimension 10 Averages on infinite graphs 11 The type problem on the average 1 12 The average spectral dimension 21 13 A survey of analytical results on specific networks 23 13.1 Renormalization techniques. .

Performance Evaluation Review, 2012

In this paper we study the behavior of a continuous time random walk (CTRW) on a time varying dynamic graph. We establish conditions under which the CTRW is a stationary and ergodic process. In general, the stationary distribution of the walker depends on the walker rate and is difficult to characterize. However, we characterize the stationary distribution in the following cases: i) the walker rate is significantly larger or smaller than the rate in which the graph changes (time-scale separation), ii) the walker rate is proportional to the degree of the node that it resides on (coupled dynamics), and iii) the degrees of vertices belonging to the same connected component are identical (structural constraints). We provide numerical results of examples that illustrate our theoretical findings and other peculiarities, as well as two applications.

Physical Chemistry Chemical Physics, 2018

Percolation networks have been widely used in the description of porous media but are now found to be relevant to understand the motion of particles in cellular membranes or the nucleus of biological cells. We here study the influence of the cluster size distribution on diffusion measurements in percolation networks.

DOKTORSAVHANDLINGAR-CHALMERS TEKNISKA …, 2001

2015

Continuous-time random walks (CTRWs) on discrete state spaces, ranging from regular lattices to complex networks, are ubiquitous across physics, chemistry, and biology. Models with coarse-grained states, for example those employed in studies of molecular kinetics, and models with spatial disorder can give rise to memory and non-exponential distributions of waiting times and first-passage statistics. However, existing methods for analyzing CTRWs on complex energy landscapes do not address these effects. We therefore use statistical mechanics of the nonequilibrium path ensemble to characterize first-passage CTRWs on networks with arbitrary connectivity, energy landscape, and waiting time distributions. Our approach is valuable for calculating higher moments (beyond the mean) of path length, time, and action, as well as statistics of any conservative or non-conservative force along a path. For homogeneous networks we derive exact relations between length and time moments, quantifying t...

Bulletin of the Brazilian …, 2006

We study two versions of random walks systems on complete graphs. In the first one, the random walks have geometrically distributed lifetimes so we define and identify a non-trivial critical parameter related to the proportion of visited vertices before the process dies out. In the second version, the lifetimes depend on the past of the process in a non-Markovian setup. For that version, we present results obtained from computational analysis, simulations and a mean field approximation. These three approaches match.

Journal of Statistical Physics, 1995

Annales De L Institut Henri Poincare-probabilites Et Statistiques, 1994

We consider a random walk Xt, t E Z+ and a dynamical random Geld ~ (x ) , x E (t E Z+) in mutual interaction with each other. The interaction is small, and the model is a perturbation of an unperturbed model in which walk and field evolve independently, the walk according to i.i.d. finite range jumps, and the field independently at each site x E llv, according to an ergodic Markov chain. Our main result in Part I concerns the asymptotics of temporal correlations of the random field, as seen in a fixed frame of reference. We prove that it has a &quot;long time tail&quot; falling off as an inverse power of t. In Part II we obtain results on temporal correlation in a frame of reference moving with the walk. (*) Partially supported by C.N.R. (G.N.F.M.) and M.U.R.S.T. research funds. A.M.S. Classification : 60 J 15, 60 J 10. Annales de l’lnstitut Henri Poincaré Probabilités et Statistiques 0246-0203 Vol. 30/94/04/$ 4.00/@ Gauthier-Villars 520 C. BOLDRIGHINI, R. A. MINLOS AND A. PELLEGRIN...