  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.
