  We study the dynamics of nonlinear random walks on complex networks. We
investigate the role and effect of directed network topologies on long-term
dynamics. While a period-doubling bifurcation to alternating patterns occurs at
a critical bias parameter value, we find that some directed structures give
rise to a different kind of bifurcation that gives rise to quasi-periodic
dynamics. This does not occur for all directed network structure, but only when
the network structure is sufficiently directed. We find that the onset of
quasi-periodic dynamics is the result of a Neimark-Sacker bifurcation, where a
pair of complex-conjugate eigenvalues of the system Jacobian passes through the
unit circle, destabilizing the stationary distribution with high-dimensional
rotations. We investigate the nature of these bifurcations, study the onset of
quasi-periodic dynamics as network structure is tuned to be more directed, and
present an analytically tractable case of a four-neighbor ring.
