Modeling Multistability and Hysteresis in Urban Congestion Spreading

Abstract

Growing evidence suggests that the macroscopic functional states of urban road networks exhibit multistability and hysteresis, but microscopic mechanisms underlying these phenomena remain elusive. Here, we demonstrate that in real-world road networks, the recovery process of congested roads is not spontaneous, as assumed in existing models, but is hindered by connected congested roads, and such hindered recovery can lead to the emergence of multistability and hysteresis in urban traffic dynamics. By analyzing real-world urban traffic data, we observed that congestion propagation between individual roads is well described by a simple contagion process like an epidemic, but the recovery rate of a congested road decreases drastically by the congestion of the adjacent roads unlike an epidemic. Based on this microscopic observation, we proposed a simple model of congestion propagation and dissipation, and found that our model shows a discontinuous phase transition between macroscopic functional states of road networks when the recovery hindrance is strong enough through a mean-field approach and numerical simulations. Our findings shed light on an overlooked role of recovery processes in the collective dynamics of failures in networked systems.

Figures (3)

• Figure 1: Measuring transition rates between traffic states of individual roads using real-world traffic velocity data. (a) Schematics of interactions between road segments separated by intersections in an urban road network. The direction of arrows denotes the direction of vehicles from each road segments. The road $i$ is a upstream and downstream of roads indicated by red and orange arrows, respectively. (b) The representation of the road-to-road network described in (a). A node (i.e., a road segment) is represented by an empty circle. The representative node $i$ has degree $k_i = 6$ (3 downstream and 3 upstream). (c) Measured average transition rates of individual roads in Seoul with respect to the number of adjacent congested road segments. Blue and orange dashed lines represent the propagation and recovery rates of local congestion, respectively. The grey boxes in the background represent the total number of events that a road has a certain number of adjacent congestion in the entire dataset.
• Figure 2: Representing a mean-field equation and phase diagram of our model. (a) Representation of the equation of a deviation of the congestion probability $\Delta z \,(=P(C|z)-z)$ with a given congestion probability $z$. Each dotted line denotes the conditional deviation of congestion probability based on the number of adjacent congested roads $\theta$ (Blue: 0, Orange: 1, Green: 2). The Blue solid line represents a total deviation of congestion probability, which is the convolution of conditional deviations and the distribution function of $\theta$. Right panel is zooming out for the blue solid line around $\Delta z = 0$ in left panel. (b) Phase diagram of the mean-field analysis for $n=2$ with $\beta_0 = 0$. Green and orange area represent global free-flow and congestion states regardless of the initial state, respectively. The purple area denotes the hysteresis region which means the global congestion density depends on the initial state.
• Figure 3: Numerical results of our modified SIS model. (a) Numerical simulation results on a square grid. We have calculated the time average of the global congestion density $z$, varying the contagion ratio $R=\beta/\mu_0$ from 0 to 0.5 (forward, dotted line and circle markers) or from 0.5 to 0 (backward, dashed line and triangle markers) with $10^5$ relaxation and simulation steps. Each color represents a different recovery reduction ratio $\xi$ and the process of varying $R$ (the other parameters $\beta_0$ and $\mu_0$ are set to $10^{-6}$ and 0.5, respectively). The colored solid lines represent the corresponding mean-field solutions of $n=3$. The black dash-dotted line denotes the epidemic threshold of the original SIS model which can be read as $\beta n/\mu_0 = 1$. (b) Autocorrelation function of empirical data of Seoul (left) and numerical simulations which are the spontaneous recovery model (middle) and the hindered recovery model (right). Each color represents the degree of averaged nodes.