Emergence and co-existence of periodic and unstructured motion in future-avoiding random walks

Abstract

Self-avoiding random walks on graphs can be seen as walkers interacting with their own past history. This letter considers a complementary class of dynamics: Mutual future avoiding random walks (MFARWs), where stochastically driven walkers are avoiding each others planned future trajectories. Such systems arise naturally in conceptual models of shared mobility. We show that periodic behavior emerges spontaneously in such MFARWs, and that periodic and unstructured behavior coexist, providing a first example of Chimera style behavior of non-oscillatory paths on networks. Further, we analytically describe and predict the onset of structure. We find that the phase transition from unstructured to periodic behavior is driven by a novel mechanism of self-amplifying coupling to the periodic components of the stochastic drivers of the system. In the context of shared mobility applications, these Chimera states imply a regime of naturally stable co-existence between flexible and line-based public transport.

Figures (5)

• Figure 1: Networks used in this paper: a) star graph with 10 nodes, b) line graph with 10 nodes, c) ring graph with 10 nodes, d) Cayley tree with 40 nodes, e) grid graph with 16 nodes, f) Wheel graph with 5 nodes. Driving from one node to a neighbouring node takes one time unit.
• Figure 2: TSP tours predict link usage better than random shortest path walks. a) Example for shortest path walk on 4x4 grid, visiting the nodes (4,2,9,11,6,13,15). b) The edge betweenness or link usage for a perfect taxi service predicts the link usage observed in the simulations with $R^2=0.81$ for the NDH and $R^2=0.82$ for the MFD dispatcher for a single vehicle and a load of $x=100$. Each point refers to one edge in one of the networks from Fig. \ref{['fig:networks']}. c) One of the shortest tours visiting each point in the 4x4 grid at least once. d) The mix of all TSP tours including rotations and mirrored versions predicts link usages with $R^2=0.96$ or $R^2=0.92$ respectively.
• Figure 3: Recurrence plots reveal periodic and unstructured routes. All results are from tests with $B=100$, $x=10$ and a vehicle capacity of 20. a) Accepted request OD-pairs of periodic route (index 0) from MFD simulation. b) Corresponding recurrence plot of route a). c) Accepted request OD-pairs of unstructured route (index 20) from MFD simulation. d) Corresponding recurrence plot of route c). e) Accepted request OD-pairs of periodic route (index 0) from NDH simulation. f) Corresponding recurrence plot of route e). g) Accepted request OD-pairs of unstructured route (index 20) from NDH simulation. h) Corresponding recurrence plot of route g).
• Figure 4: Persistent periodic patterns emerge in some cases. Fraction of route periodicity for each vehicle of the fleet evaluated on all 6 networks for both dispatcher algorithms across loads from $x=0.1$ to $x=20$. Across all simulations $B=100$ and $N_{req}=100000$. Persistent periodic routes emerge for cycle, line and wheel network in both dispatchers and for the grid only in the NDH dispatcher, while routes always remain unstructured in the Cayley tree and star network.
• Figure 5: Chimera states emerge once periodicity is statistically inevitable. a) Transition to partially periodic states steepens with fleet size for NDH dispatcher for different fleet sizes $B$ (blue to red dots) and the theoretical approximation for $B=500$ (solid black line). b) Transition to partially periodic states steepens with fleet size for MFD dispatcher.