  Random walks represent an important tool for probing the structural and
dynamical properties of networks and modeling transport and diffusion processes
on networks. However, when individuals' movement becomes dictated by more
complicated factors, e.g., scenarios that involve complex decision making, the
linear paradigm of classical random walks lack the ability to capture
dynamically rich behaviors. One modification that addresses this issue is to
allow transition probabilities to depend on the current system state, resulting
in a nonlinear random walk. While the resulting nonlinearity has been shown to
give rise to an array of more complex dynamics, the patterns that emerge, in
particular on regular network topologies, remain unexplored and poorly
understood. Here we study nonlinear random walks on regular networks. We
present a number of stability results for the uniform state where random
walkers are uniformly distributed throughout the network, characterizing the
spectral properties of its Jacobian which we use to characterize its
bifurcations. These spectral properties may also be used to understand the
patterns that emerge beyond bifurcations, which consist of oscillating short
wave-length patterns and localized structures for negative and positive bias,
respectively. We also uncover a subcriticality in the bifurcation for positive
bias, leading to a hysteresis loop and multistability.
