  We propose local-biased random walks on general networks where a Markovian
walker can choose between different types of biases in each node to define
transitions to its neighbors depending on their degrees. For this ergodic
dynamics, we explore the capacity of the random walker to visit all the nodes
characterized by a global mean first passage time. This quantity is calculated
using eigenvalues and eigenvectors of the transition matrix that defines the
dynamics. In the first part, we illustrate how our framework leads to optimal
transport for small-size graphs through the analysis of all the possible bias
configurations. In the second part, we explore optimal bias in each node by
using simulated annealing. This heuristic algorithm allows obtaining
approximate solutions of the optimal bias in different types of networks. The
results show how the local bias can optimize the exploration of the network in
comparison with the unbiased random walk. The methods implemented in this
research are general and open the doors to a broad spectrum of tools applicable
to different random walk strategies and dynamical processes on networks.
