Random Walks on Networks with Centrality-Based Stochastic Resetting

Fractal and first-passage properties of a class of self-similar networks

A class of self-similar networks, obtained by recursively replacing each edge of the current network with a well-designed structure (generator) and known as edge-iteration networks, has garnered considerable attention owing to its role in presenting rich network models to mimic real objects with self-similar structures. The generator dominates the structural and dynamic properties of edge-iteration networks. However, the general relationships between these networks' structural and dynamic properties and their generators remain unclear. We study the fractal and first-passage properties, such as the fractal dimension, walk dimension, resistance exponent, spectral dimension, and global mean first-passage time, which is the mean time for a walker, starting from a randomly selected node and reaching the fixed target node for the first time. We disclose the properties of the generators that dominate the fractal and first-passage properties of general edge-iteration networks. A clear relationship between the fractal and first-passage properties of the edge-iteration networks and the related properties of the generators are presented. The upper and lower bounds of these quantities are also discussed. Thus, networks can be customized to meet the requirements of fractal and dynamic properties by selecting an appropriate generator and tuning their structural parameters. The results obtained here shed light on the design and optimization of network structures.

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.