Anomalous biased diffusion in networks

TL;DR

This paper investigates how biased diffusion towards a target in networks affects the mean first passage time, revealing a threshold bias where the scaling behavior shifts from anomalous to logarithmic, with implications for network routing.

Contribution

It provides a theoretical and simulation-based analysis of biased diffusion in networks, identifying a threshold bias and deriving exact scaling laws for mean first passage time.

Findings

Existence of a threshold probability $p_{th}$ for regime transition.

For $p<p_{th}$, MFPT scales as $N^eta$, with $eta$ depending on $p$.

For $p>p_{th}$, MFPT scales logarithmically with network size.

Abstract

We study diffusion with a bias towards a target node in networks. This problem is relevant to efficient routing strategies in emerging communication networks like optical networks. Bias is represented by a probability $p$ of the packet/particle to travel at every hop towards a site which is along the shortest path to the target node. We investigate the scaling of the mean first passage time (MFPT) with the size of the network. We find by using theoretical analysis and computer simulations that for Random Regular (RR) and Erd\H{o}s-R\'{e}nyi (ER) networks, there exists a threshold probability, $p_{th}$, such that for $p<p_{th}$ the MFPT scales anomalously as $N^\alpha$, where $N$ is the number of nodes, and $\alpha$ depends on $p$. For $p>p_{th}$ the MFPT scales logarithmically with $N$. The threshold value $p_{th}$ of the bias parameter for which the regime transition occurs is found to depend only on the mean degree of the nodes. An exact solution for every value of $p$ is given for the scaling of the MFPT in RR networks. The regime transition is also observed for the second moment of the probability distribution function, the standard deviation.