Mathematical Problems in Engineering

Volume 2015, Article ID 761818, 8 pages

http://dx.doi.org/10.1155/2015/761818

## Risk and Resilience Analysis of Complex Network Systems Considering Cascading Failure and Recovery Strategy Based on Coupled Map Lattices

School of Reliability and Systems Engineering, Beihang University, Beijing 100191, China

Received 3 July 2015; Accepted 13 September 2015

Academic Editor: Mark Leeson

Copyright © 2015 Fuchun Ren 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

Risk and resilience are important and challenging issues in complex network systems since a single failure may trigger a whole collapse of the systems due to cascading effect. New theories, models, and methods are urgently demanded to deal with this challenge. In this paper, a coupled map lattices (CML) based approach is adopted to analyze the risk of cascading process in Watts-Strogatz (WS) small-world network and Barabási and Albert (BA) scale-free network, respectively. Then, to achieve an effective and robust system and provide guidance in countering the cascading failure, a modified CML model with recovery strategy factor is proposed. Numerical simulations are put forward based on small-world CML and scale-free CML. The simulation results reveal that appropriate recovery strategies would significantly improve the resilience of networks.

#### 1. Introduction

In modern society, many real-world systems, such as Internet, transportation network, and power grid system [1–3], can be described by complex networks. They have made a great contribution to social development. However, despite the benefits to people’s life, risk or disadvantage of complex network system is also serious since a local failure may trigger a breakdown of the whole system due to cascading effect; many real-world examples have been witnessed around the world, for example, the power grid of North America in 2003 [4], in which a fault of three extrahigh voltage transmissions led to a cascading effect in power system affecting about 50 million people and caused an economic loss of 4 billion to 10 billion, the 2003 SARS plague [5], and the 2008 global financial crisis [6].

The traumatic experiences bring an urgent need of the study of risk and safety of complex network system which has been a hot topic for scientific researchers in recent years [7–10]. Amongst these researches, cascading failure of complex networks has been one of the hottest topics. Numbers of important aspects of cascading failure in complex networks have been discussed by scholars recently. Many of them provide a good research on the assessment and modeling of cascading effect [11, 12].

From the perspective of risk management, safety and resilience engineering, hazard, and risk should be identified and controlled timely and effectively. However, with the increasing of scale and complexity of modern complex network systems, the responsibility to manage an effective and safe system has been heavier than ever before. New theories, models, and methods are urgently demanded to protect the systems from an unacceptable risk condition.

Coupled map lattices (CML) are a dynamical system with discrete time, discrete space, and continuous state variables [13] which are initially used to study spatiotemporal chaos and then widely applied in various fields including but not limited to biology, mathematics, and engineering. In CML, the dynamical elements are situated at discrete spatial points while the time is discrete and the state is continuous. Each spatial element is coupled to its neighbors. Recently, CML has been widely investigated to model the dynamical behaviors and cascading failure in complex systems [14–18]. To counter the impact of cascading failure and provide an effective prevention of large-scale breakdown, strategies such as redistribution and restoration have been discussed [19–22]; however there is few considering recovery strategy of failed node in the process of cascading failure. In other words, risk recognition and analysis are just a prime step; actions can be taken to achieve an acceptable resilience of the whole system [23].

The remainder of this paper is organized as follows. Section 2 proposes the risk and resilience analysis approach based on CML and a modified CML model with recovery strategy factor is discussed. In Sections 3 and 4, the small-world CML and scale-free CML are adopted, respectively, to analyze the cascading failure and the effect of recovery strategy. Finally, Section 5 provides a conclusion of this paper.

#### 2. Risk and Resilience Analysis Based on Coupled Map Lattices

##### 2.1. Risk Analysis of Cascading Failure Based on Coupled Map Lattices

Coupled map lattices (CML) are a dynamical system with discrete time, discrete space, and continuous state variables; typically, CML of nodes can be described as follows:where is the state variable of the th node at the th time step. The connection information of the nodes is given by an adjacency matrix . If there is an edge between node and node , ; otherwise, . denotes the degree of node . represents the coupling strength. The function defines the local dynamics which is chosen in this work as the chaotic logistic map , where if . Absolute value notation in (1) is used to guarantee that the state of each node is always nonnegative. The th node is regarded to be in a normal state at the th time step if . In contrast, if at the th time step, the th node is regarded to be failed and .

However, to simulate and analyze the cascading failure based on the CML, an external perturbation is added to a specific node at the th time step as in

Then, at the th time step; other nodes would be impacted according to (1) afterwards and a cascading failure would be triggered.

Topological structure (static) and restoration strategy (dynamic) of the complex network would be fundamental to cascading effect, spread threshold, and scope [11, 21, 23]. In CML, it is the coupling strength and external perturbation that get an important impact [16]. However, coupling strength is something like topological factor and external perturbation is something like external attack that is not controllable once happened. Even though restoration strategy in weighted networks is discussed, there are few discussions about the recovery strategy in CML; that is, the failed units would be treated as failed evermore in the cascading process. So a factor of recovery strategy against cascading failure should be added in the research of cascading failure in CML.

##### 2.2. A Modified CML Model with Recovery Strategy Factor

As illustrated above, if the th node got failed at the th time step, that is, , then the state variable of the th node is identically equal to 0 after the th time step; that is, . But in actual world, recovery strategies would be implemented at proper time after the malfunction, as illustrated in Figure 1; when the th node got failed at the th time step it would be recovered to normal after time step while the states of its neighbors, that is, th and th nodes, have been changed to failed at the th time step. The timeliness and effectiveness of recovery would make a critical influence on the cascading process. In cascading, we take the reduction of failure rate and recovery number of failed nodes as the two metric parameters of the effectiveness of recovery.