  Random walks are the simplest way to explore or search a graph, and have
revealed a very useful tool to investigate and characterize the structural
properties of complex networks from the real world, e.g. they have been used to
identify the modules of a given network, its most central nodes and paths, or
to determine the typical times to reach a target. Although various types of
random walks whose motion is node biased have been proposed, which are still
amenable to analytical solution, most if not all of them rely on the assumption
of linearity and independence of the walkers. We introduce a novel class of
nonlinear stochastic processes describing a system of interacting random
walkers moving over networks with finite node capacities. The transition
probabilities are modulated by nonlinear functions of the available space at
the destination node, with a bias parameter that allows to tune the tendency of
the walkers to avoid nodes occupied by other walkers. Firstly, we derive the
master equation governing the dynamics of the system, and we determine an
analytical expression for the occupation probability of the walkers at
equilibrium in the most general case, and under different level of network
congestions. Then, we study different type of synthetic and real-world
networks, presenting numerical and analytical results for the entropy rate, a
proxy for the network exploration capacities of the walkers.We find that, for
each level of the nonlinear bias, there is an optimal crowding that maximises
the entropy rate in a given network topology. The analysis suggests that a
large fraction of real-world networks are organised in such a way as to favour
exploration under congested conditions. Our work provides a general and
versatile framework to model nonlinear stochastic processes whose transition
probabilities vary in time depending on the current state of the system.
