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

Abstract

Numerous problems of both theoretical and practical interest are related to finding shortest (or otherwise optimal) paths in networks, frequently in the presence of some obstacles or constraints. A somewhat related class of problems focuses on finding optimal distributions of weights which, for a given connection topology, maximize some kind of flow or minimize a given cost function. We show that both sets of problems can be approached through an analysis of the large-deviation functions of random walks. Specifically, a study of ensembles of trajectories allows us to find optimal paths, or design optimal weighted networks, by means of an auxiliary stochastic process (the generalized Doob transform). The paths are not limited to shortest paths, and the weights must not necessarily optimize a given function. Paths and weights can in fact be tailored to a given statistics of a time-integrated observable, which may be an activity or current, or local functions marking the passing of the random walker through a given node or link. We illustrate this idea with an exploration of optimal paths in the presence of obstacles, and networks that optimize flows under constraints on local observables.

Dynamical phase transition to localized states in the two-dimensional random walk conditioned on partial currents

“…Beyond a certain point, the band practically disappears, and the (large) current is almost completely sustained by vortices localized at the edges, as shown for s = −3. Such cyclic, localized behavior, which has recently been observed in random walks on general graphs [51] and the zero-range process on a diamond lattice [52], presents some similarities to the vortices of the two-dimensional simple exclusion process [45].…”

Dynamical phase transition to localized states in the two-dimensional random walk conditioned on partial currents

“…Beyond a certain point the band practically disappears, and the (large) current is almost completely sustained by vortices localized at the edges, as shown for s = −3. Such cyclic, localized behavior, which has recently been observed in random walks on general graphs [48] and the zero-range process on a diamond lattice [49], presents some similarities to the vortices of the two-dimensional simple exclusion process [45].…”

Graph-combinatorial approach for large deviations of Markov chains

“…Just to give an example, the observable in (1) can represent the number of transitions over [1, n] in a particular subset of the state space [1], obtained by fixing f = 1 ∆ , with ∆ the characteristic function of the subset. Furthermore, in certain contexts, C n can also express heat [2], two-point correlation functions, activities [3][4][5][6], particle and energy currents [7], efficiency [8][9][10][11][12], entropy production [6,13,14], and many others.…”