Discrete-time random walks and Lévy flights on arbitrary networks: when resetting becomes advantageous?

Abstract

The spectral theory of random walks on networks of arbitrary topology can be readily extended to study random walks and Lévy flights subject to resetting on these structures. When a discrete-time process is stochastically brought back from time to time to its starting node, the mean search time needed to reach another node of the network may be significantly decreased. In other cases, however, resetting is detrimental to search. Using the eigenvalues and eigenvectors of the transition matrix defining the process without resetting, we derive a general criterion for finite networks that establishes when there exists a non-zero resetting probability that minimizes the mean first passage time (MFPT) at a target node. Right at optimality, the coefficient of variation of the first passage time is not unity, unlike in continuous time processes with instantaneous resetting, but above 1 and depends on the minimal MFPT. The approach is general and applicable to the study of different discrete-time ergodic Markov processes such as Lévy flights, where the long-range dynamics is introduced in terms of the fractional Laplacian of the graph. We apply these results to the study of optimal transport on rings and Cayley trees.

A measure of dissimilarity between diffusive processes on networks

“…The matrices W and Π(r; γ) are stochastic matrices: knowing their eigenvalues and eigenvectors allows the calculation of the occupation probability at any time, including the stationary distribution at t → ∞, as well as the mean first passage time to any node. The eigenvalues and eigenvectors of Π(r; γ) are related to those of W, which is recovered in the limit γ = 0 (see a detailed discussion in [23,24,48,49]).…”

“…, c N−1 ordered in such a way that c 0 describes the diagonal elements and C ij = c (i−j)mod N . In cases where the network is a regular structure formed by cycles [13], different types of random walks can be defined with a circulant matrix L; in particular, random walks with long-range displacements [10,26], biased transport on directed rings [27], among many others [10,24]. The elementary circulating matrix E is defined with all its null elements except c 1 = 1.…”

“…Our analysis in figure 5 for D(t) reveal how the average differences between two processes evolve; it also allows the identification of the times in which its maximum values occur and shows us how D(t) reaches a stationary value for large times. In particular, the limit D(t → ∞) can be obtained analytically introducing into equation (24) the stationary distributions given by equation (45). On the other hand, | D| max gives the maximum average dissimilarity.…”

“…In section 5, we examine processes for which the network is the same but considering modifications in the definition of the random walker. We explore degree-biased random walks [5,22] and dynamics with stochastic reset to the initial node [23,24]. This last case is explored analytically to measure the effect of reset when compared with the case without restart in different types of structures including deterministic and random networks.…”

Preface: stochastic resetting—theory and applications

“…Using a renewal approach, these authors find expressions for the mean first passage time to an absorbing state, as well as the probabilities to be absorbed at any specific absorbing state, and their corresponding (conditional) mean first passage times. Riascos et al examined when resetting of discrete-time random walks and Lévy flights is advantageous [50]. Using the eigenvalues and eigenvectors of the transition matrix defining the process without resetting, they derived a criterion that establishes when resetting in discrete time can minimizes the mean first passage time to a target site.…”