Entropy rate of random walks on complex networks under stochastic resetting

Optimizing search processes with stochastic resetting on the pseudofractal scale-free web

The pseudofractal scale-free web (PSFW) is a well-known model for a scale-free network with small-world characteristics. Understanding the dynamic properties of this network can provide valuable insights into dynamic processes occurring in general scale-free and small-world networks. In this study we investigate search processes using discrete-time random walks on the PSFW to reveal the impact of the resetting position on optimizing search efficiency, as measured by the mean first-passage time (MFPT). At each step the walker has two options: with a probability of 1-γ, it moves to one of the neighboring sites, and with a probability of γ, it resets to the predefined resetting position. We explore various choices for the resetting position, present rigorous results for the MFPT to a given node of the network, determine the optimal resetting probability γ^{*} where the MFPT reaches its minimum, and evaluate the ratio of the minimum for MFPT to the MFPT without resetting for each case. Results show that, in large PSFWs, both the degree of the resetting position and the distance between the target and the resetting position significantly affect the search efficiency. A higher degree of the resetting position leads to a slower convergence of the walker to the target, while a greater distance between the target and the resetting position also results in a slower convergence. Additionally, we observe that resetting to a vertex randomly selected from the stationary distribution can significantly expedite the process of the walker reaching the target. The findings presented in this study shed light on optimizing stochastic search processes on large networks, offering valuable insights into improving search efficiency in real-world applications, where the target node's location is unknown.

Random Walks on Networks with Centrality-Based Stochastic Resetting

We introduce a refined way to diffusely explore complex networks with stochastic resetting where the resetting site is derived from node centrality measures. This approach differs from previous ones, since it not only allows the random walker with a certain probability to jump from the current node to a deliberately chosen resetting node, rather it enables the walker to jump to the node that can reach all other nodes faster. Following this strategy, we consider the resetting site to be the geometric center, the node that minimizes the average travel time to all the other nodes. Using the established Markov chain theory, we calculate the Global Mean First Passage Time (GMFPT) to determine the search performance of the random walk with resetting for different resetting node candidates individually. Furthermore, we compare which nodes are better resetting node sites by comparing the GMFPT for each node. We study this approach for different topologies of generic and real-life networks. We show that, for directed networks extracted for real-life relationships, this centrality focused resetting can improve the search to a greater extent than for the generated undirected networks. This resetting to the center advocated here can minimize the average travel time to all other nodes in real networks as well. We also present a relationship between the longest shortest path (the diameter), the average node degree and the GMFPT when the starting node is the center. We show that, for undirected scale-free networks, stochastic resetting is effective only for networks that are extremely sparse with tree-like structures as they have larger diameters and smaller average node degrees. For directed networks, the resetting is beneficial even for networks that have loops. The numerical results are confirmed by analytic solutions. Our study demonstrates that the proposed random walk approach with resetting based on centrality measures reduces the memoryless search time for targets in the examined network topologies.