  The co-evolution between network structure and functional performance is a
fundamental and challenging problem whose complexity emerges from the intrinsic
interdependent nature of structure and function. Within this context, we
investigate the interplay between the efficiency of network navigation (i.e.,
path lengths) and network structure (i.e., edge weights). We propose a simple
and tractable model based on iterative biased random walks where edge weights
increase over time as function of the traversed path length. Under mild
assumptions, we prove that biased random walks will eventually only traverse
shortest paths in their journey towards the destination. We further
characterize the transient regime proving that the probability to traverse
non-shortest paths decays according to a power-law. We also highlight various
properties in this dynamic, such as the trade-off between exploration and
convergence, and preservation of initial network plasticity. We believe the
proposed model and results can be of interest to various domains where biased
random walks and decentralized navigation have been applied.
