  We present a new approach of topology biased random walks for undirected
networks. We focus on a one parameter family of biases and by using a formal
analogy with perturbation theory in quantum mechanics we investigate the
features of biased random walks. This analogy is extended through the use of
parametric equations of motion (PEM) to study the features of random walks {\em
vs.} parameter values. Furthermore, we show an analysis of the spectral gap
maximum associated to the value of the second eigenvalue of the transition
matrix related to the relaxation rate to the stationary state. Applications of
these studies allow {\em ad hoc} algorithms for the exploration of complex
networks and their communities.
