Random walks in small-world exponential treelike networks

TL;DR

This paper analyzes random walks on small-world exponential treelike networks, deriving exact scaling laws for trapping and first-passage times, and compares these results with other tree structures to understand information transmission efficiency.

Contribution

It provides exact analytical results for mean trapping time, mean sending time, and global mean first-passage time in small-world exponential trees, linking structure to random walk dynamics.

Findings

Mean trapping time scales linearly with network size N.

Mean sending time and global MFPT scale as N log N.

Shortest path length influences FPT scaling in trees.

Abstract

In this paper, we investigate random walks in a family of small-world trees having an exponential degree distribution. First, we address a trapping problem, that is, a particular case of random walks with an immobile trap located at the initial node. We obtain the exact mean trapping time defined as the average of first-passage time (FPT) from all nodes to the trap, which scales linearly with the network order $N$ in large networks. Then, we determine analytically the mean sending time, which is the mean of the FPTs from the initial node to all other nodes, and show that it grows with $N$ in the order of $N \ln N$. After that, we compute the precise global mean first-passage time among all pairs of nodes and find that it also varies in the order of $N \ln N$ in the large limit of $N$. After obtaining the relevant quantities, we compare them with each other and related our results to the efficiency for information transmission by regarding the walker as an information messenger. Finally, we compare our results with those previously reported for other trees with different structural properties (e.g., degree distribution), such as the standard fractal trees and the scale-free small-world trees, and show that the shortest path between a pair of nodes in a tree is responsible for the scaling of FPT between the two nodes.