We study diffusion with a bias toward 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 or particle to travel at every hop toward a site that 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 Erdős-Rényi networks, there exists a threshold probability, $p_{\mathrm{th}}$, such that for $p<p_{\mathrm{th}}$ the MFPT scales anomalously as $N^{\alpha }$, where $N$ is the number of nodes, and $\alpha$ depends on $p$. For $p>p_{\mathrm{th}}$, the MFPT scales logarithmically with $N$. The threshold value $p_{\mathrm{th}}$ 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. For the case of scale-free (SF) networks, we present analytical bounds and simulations results showing that the MFPT scales at most as $\ln N$ to a positive power for any finite bias, which means that in SF networks even a very small bias is considerably more efficient in comparison to unbiased walk.

• Received 12 March 2013

DOI:https://doi.org/10.1103/PhysRevE.88.012817