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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.