Dynamically accelerated cover times

Abstract

Among observables characterizing the random exploration of a graph or lattice, the cover time, namely, the time to visit every site, continues to attract widespread interest. Much insight about cover times is gained by mapping to the (spaceless) coupon collector problem, which amounts to ignoring spatiotemporal correlations, and an early conjecture that the limiting cover time distribution of regular random walks on large lattices converges to the Gumbel distribution in d 3 was recently proved rigorously. Furthermore, a number of mathematical and numerical studies point to the robustness of the Gumbel universality to modifications of the spatial features of the random search processes (e.g., introducing persistence and/or intermittence, or changing the graph topology). Here we investigate the robustness of the Gumbel universality to dynamical modification of the temporal features of the search, specifically by allowing the random walker to "accelerate" or "decelerate" upon visiting a previously unexplored site. We generalize the mapping mentioned above by relating the statistics of cover times to the roughness of 1/ f α Gaussian signals, leading to the conjecture that the Gumbel distribution is but one of a family of cover time distributions, ranging from Gaussian for highly accelerated cover, to exponential for highly decelerated cover. While our conjecture is confirmed by systematic Monte Carlo simulations in dimensions d > 3, our results for acceleration in d = 3 challenge the current understanding of the role of correlations in the cover time problem.

The effects of socio-economic factors on the food security status of rural households in the Eastern Cape Province: Evidence from farming households

“…This study uses the Lower-Bound Poverty Line (LBPL) as it has emerged as the preferred threshold that is commonly used in South Africa's poverty reduction targets outlined in the Medium-Term Strategic Framework, National Development Plan, and Sustainable Development Goals. This study is in line with Maziya et al (2020), who made use of this model to estimate household food security.…”

“…The FGT poverty measures were calculated using the Distributive Analysis Stata Package (DASP) version 2.3 (Maziya et al, 2020;Araar and Duclos, 2013). Food security restriction is a non-negative parameter indicating the degree of sensitivity of the food security measure to inequality among the poor.…”

Analytical results for the distribution of cover times of random walks on random regular graphs

“…It was thus concluded that the Gumbel distribution is a universal distribution of cover times. Interestingly, it was recently shown that by accelerating the search process one can modify the distribution of cover times from the Gumbel distribution to narrower distributions such as the Gaussian distribution [119].…”

Unexpected advantages of exploitation for target searches in complex networks

“…By definition, the GMFPT allows us to estimate how long a walker will take to reach the target node from a randomly selected starting node. The mean cover time (MCT) C i (r) is defined here as the average time needed to visit all nodes at least once from a starting node i by the SERW with r. Some analytical results of the MCT of NRW or other random walks have been studied using lattices, Erdős-Rényi (ER) networks, Barabási-Albert (BA) networks, etc 38,[43][44][45][46] . For any connected network, it is well-known that the MCT of NRW C NRW i is bounded as 47,48 (…”

“…When r = 0, Eq. (B1) becomes almost equivalent to the probability of collecting a new coupon in the coupon collector problem 46 . The MCT C i is then represented by the sum of individual waiting times 1/p z between the zth and (z + 1)th new node:…”