Fig. 5

Steady states as a ranking application. Models 5 and 6 on the bottom panels (C) and (D) behave in the limit for the fast walker (μ→∞) as a Markovian passive edge-centric random walk, where the steady state is uniform across the symmetric network. In the small- μ limit, the steady state probability is proportional to the degree, like in the Markovian active node-centric walk. In model 4 of panel (a), this behaviour does not hold true. Indeed, when μ→∞, the steady-state residence probability is high when degree is low. This inversion with respect to the degree-based ranking is captured by the inset of panel (a). On this inset, the evolution of τ, which denotes Kendall’s Tau coefficient, is represented in function of μ. When Kendall’s Tau is one, the rankings of panels (a) and (c) or (d) are the same; when it is -1, the two rankings are reversed. There is no link between the two when it is 0. Panel (b) displays the result of Monte-Carlo simulation of 10⁴ trajectories of a walker subject to the motion rules of model 4, in the range of values of μ corresponding to the reversed ranking. These simulations do account for the presence of cycles, and confirm quantitatively the ranking predicted by the analytical formula. We have selected η=1=λ for all plots. The strongly connected symmetric network is Erdös-Rényi with 30 nodes and connection probability \(\frac {1}{5}\)