Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 101059, 8 pages

http://dx.doi.org/10.1155/2015/101059

## Analysis of Road Traffic Network Cascade Failures with Coupled Map Lattice Method

^{1}Beijing Key Laboratory for Cooperative Vehicle Infrastructure Systems and Safety Control, School of Transportation Science and Engineering, Beihang University, Beijing 100191, China^{2}Jiangsu Province Collaborative Innovation Center of Modern Urban Traffic Technologies, SiPaiLou No. 2, Nanjing 210096, China

Received 26 March 2015; Revised 18 June 2015; Accepted 21 June 2015

Academic Editor: Yuanchang Xie

Copyright © 2015 Yanan Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In recent years, there is growing literature concerning the cascading failure of network characteristics. The object of this paper is to investigate the cascade failures on road traffic network, considering the aeolotropism of road traffic network topology and road congestion dissipation in traffic flow. An improved coupled map lattice (CML) model is proposed. Furthermore, in order to match the congestion dissipation, a recovery mechanism is put forward in this paper. With a real urban road traffic network in Beijing, the cascading failures are tested using different attack strategies, coupling strengths, external perturbations, and attacked road segment numbers. The impacts of different aspects on road traffic network are evaluated based on the simulation results. The findings confirmed the important roles that these characteristics played in the cascading failure propagation and dissipation on road traffic network. We hope these findings are helpful to find out the optimal road network topology and avoid cascading failure on road network.

#### 1. Introduction

In many large-scale networks, the failure of a node or edge would make the other nodes fail and lead to a chain reaction due to the coupling relationships among nodes. This phenomenon is known as network cascading failure. Cascading failure problems may take place on many natural or artificial networks, such as the Internet [1, 2], power grids [3–6], and traffic networks [7–11]. The effects from large destruction may be caused by cascading failures on the entire networks. Many cities have suffered from serious traffic paralysis that brought great inconvenience to people’s normal life (e.g., Beijing urban traffic was shut down completely due to the rainstorm on July 21, 2012). Therefore, it is essential to understand the cascading failure on traffic network to prevent or reduce the influences of large-scale failure.

Many scholars have studied the impacts of network topology [7, 12], network connectivity [13], different attack strategies [4, 14, 15], and network robustness [16–18] on cascading failure. To describe the cascading failure, coupled map lattice (CML) model has been widely applied in previous literatures. For example, using the basic CML method, Xu and Wang [12] studied the cascading failures in different network topologies. Based on the proposed edge-based CML method, Di et al. [19] investigated the cascading failure on random networks and scale-free networks. Though most studies paid attention to the artificial network on cascading failure, the research that applies CML model to investigate the natural road traffic network on cascading failure is limited.

Because the properties of natural road traffic network are different from artificial networks, the particular road traffic network properties need to be concerned when we use CML model. One of the particular properties is aeolotropism. Due to the fact that there are one-way and two-way streets in the city, the road traffic network is supposed to be described as directed graphs. Another particular property is restorability, which means that road congestions can dissipate over a certain range. The road traffic network consists of intersections and road segments. Vehicles travel on the network and form the distributed traffic flow. If traffic congestions occur in one or some road segments, congestions can be gradually dissipated after a period of time due to the redistribution of traffic flow. These two particular properties may lead to unique cascading failures rules in road traffic network.

Considering the above particular properties, the original CML model will be improved for analyzing cascading failures of road traffic network. The improved CML model is expected to express the aeolotropism of road traffic network topology, which will be proposed in the following section. Besides, in order to match the pattern of road congestion dissipation, a recovery mechanism has been put forward in the next part. For the purpose of deliberating cascading failures roundly, an empirical network in Beijing is tested to investigate the impacts of different attack strategies, coupling strengths, external perturbations, and attacked road segment numbers on road traffic network.

The remainder of this paper is organized as follows. The next section will introduce the improved CML model and the recovery mechanism. Then, the simulations based on the empirical network are conducted. Finally, the highlights of this paper are concluded.

#### 2. Road Traffic Network Cascading Failures Model Based on CML

The original CML model is formulated as follows [12]:where is the state of the th node at the th time step. is defined as the coupled strength. is the sum of all nodes. represents the degree of the th node. Adjacency matrix is used to represent the topology of the network. If there is an edge between node and node , then ; otherwise, . Chaotic Logistic map with is used to denote dynamic behaviors of nodes. is closer to 4; the value of is more evenly distributed throughout the region of 0 to 1. Therefore, it is always set to be . The absolute value notation is used in (1) for ensuring nonnegative saturation state of each node.

To describe the aeolotropism of road traffic network, an improved CML model is proposed to investigate the cascading failures on road traffic network. In the original CML model, means the state of th node at the th time step, while, for road traffic network, it is expressed by road saturation:

In (2), means the road saturation of the th road segment at the th time step. is the value of adjacency matrix of the road traffic network. If there is an edge from node to node , then ; otherwise, . and delegate the coupled strengths of the start point and endpoints, respectively. is the sum of all nodes’ out-degree, and is that of in-degree. and , respectively, represent the in-degree and out-degree of the th node which means the number of downstream segments and upstream segments for the road traffic network.

Cascading failure on road traffic network may be triggered by some internal and external factors (e.g., traffic congestion or crash) that lead to the failure of one or more roads. To describe this situation, an external perturbation is added to the node at the th time as follows:

If when , the node is in a normal state; if , the node is defined to be failed at the th time step. For the situation when the node fails at the th time step, with is defined in previous studies [12, 19]. However, with regard to the road traffic network, the failure state could not continue all the time due to the fact that traffic congestion will gradually dissipate with the redistribution of traffic flow. A recovery mechanism to fit with road traffic characteristics is proposed in this study as follows.

If an external perturbation is added to the node at the th time step, means that road segment has been in a blocked state at th time step. If the upstream vehicles cannot enter, coupled strength of upstream segments and road segment is set to be 0 (i.e., ). The saturation state of the th node from th step to th step can be represented by (4). If after steps, the saturation state of the th node returns to normal, represented by (2). The recovery mechanism of the road traffic network on cascading failure is shown in Figure 1.