Fig. 1

Random walk models. Each of the six panels represents a random walk model. In each case, the ribbon above the blue time axis represents the dynamics at node level, that is, the waiting time X_w of the walker, possibly extended by a period where the walker is trapped. The ribbon under the time axis represents the dynamics at edge-level, with the associated random variables X_u and X_d for the up-times and down-times respectively. In the left column, the three classical models with only one timescale are represented, whereas in the right column, the walks have two (model 6) or three (models 4 and 5) competing timescales. For each model, the arrival time on the node is located on the time axis by the leftmost man icon (standing or sitting), and the time of the jump corresponds to the rightmost jumping-man icon. When the node-level ribbon (above the blue axis) and the edge-level ribbon (below the blue axis) have a synchronous start from the left, as in models 1, 2 and 5, the walk is fully active. Model 6 is also fully active, even though the reset of the edges happens after a term X_w. On the other hand, models 3 and 4 are passive at edge-level. This means that the edges dynamics is independent of the walker, in the sense that the arrival of the walker on a node does not mean the state of outgoing edges is reset