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Bundled networks, obtained by attaching a copy of a fiber structure to each node on the base structure, serve as important realistic models for the geometry and dynamics of nontranslationally invariant systems in condensed matter physics. Here, we analyze the first-passage properties, including the mean first-passage time, the mean-trapping time, the global-mean first-passage time (GFPT), and the stationary distribution, of a biased random walk within such networks, in which a random walker moves to a neighbor on base with probability γ and to a neighbor on fiber with probability 1 − γ when the walker at a node on base. We reveal the primary properties of both the base and fiber structure, which govern the first-passage characteristics of the bundled network. Explicit expressions between these quantities in the bundled networks and the related quantities in the component structures are presented. GFPT serves as a crucial indicator for evaluating network transport efficiency. Unexpectedly, bases and fibers with similar scaling of GFPT can construct bundled networks exhibiting different scaling behaviors of GFPT. Therefore, bundled networks can be tailored to accommodate specific dynamic property requirements by choosing a suitable base and fiber structure. These findings contribute to advancing the design and optimization of network structures. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The Watts–Strogatz networks are important models that interpolate between regular lattices and random graphs, and Barabási–Albert networks are famous models that explain the origin of the scale-free networks. Here, we consider the first encounters between two particles (e.g., prey A and predator B) embedded in the Watts–Strogatz networks and the Barabási–Albert networks. We address numerically the mean first-encounter time (MFET) while the two particles are moving and the mean first-passage time (MFPT) while the prey is fixed, aiming at uncovering the impact of the prey's motion on the encounter time, and the conditions where the motion of the prey would accelerate (or slow) the encounter between the two particles. Different initial conditions are considered. In the case where the two particles start independently from sites that are selected randomly from the stationary distribution, on the Barabási–Albert networks, the MFET is far less than the MFPT, and the impact of prey's motion on the encounter time is enormous, whereas, on the Watts–Strogatz networks (including Erdős-Rényi random networks), the MFET is about 0.5–1 times the MFPT, and the impact of prey's motion on the encounter time is relatively small. We also consider the case where prey A starts from a fixed site and the predator starts from a randomly drawn site and present the conditions where the motion of the prey would accelerate (or slow) the encounter between the two particles. The relation between the MFET (or MFPT) and the average path length is also discussed. [ABSTRACT FROM AUTHOR]