Complexity

Volume 2018 (2018), Article ID 6386324, 14 pages

https://doi.org/10.1155/2018/6386324

## Analysis on Invulnerability of Wireless Sensor Network towards Cascading Failures Based on Coupled Map Lattice

Institute of Logistics Science and Engineering, Shanghai Maritime University, Shanghai, China

Correspondence should be addressed to Xiuwen Fu

Received 21 August 2017; Revised 11 December 2017; Accepted 28 December 2017; Published 28 January 2018

Academic Editor: Ilaria Giannoccaro

Copyright © 2018 Xiuwen Fu 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

Previous research of wireless sensor networks (WSNs) invulnerability mainly focuses on the static topology, while ignoring the cascading process of the network caused by the dynamic changes of load. Therefore, given the realistic features of WSNs, in this paper we research the invulnerability of WSNs with respect to cascading failures based on the coupled map lattice (CML). The invulnerability and the cascading process of four types of network topologies (i.e., random network, small-world network, homogenous scale-free network, and heterogeneous scale-free network) under various attack schemes (i.e., random attack, max-degree attack, and max-status attack) are investigated, respectively. The simulation results demonstrate that the rise of interference and coupling coefficient will increase the risks of cascading failures. Cascading threshold values and exist, where cascading failures will spread to the entire network when or . When facing a random attack or max-status attack, the network with higher heterogeneity tends to have a stronger invulnerability towards cascading failures. Conversely, when facing a max-degree attack, the network with higher uniformity tends to have a better performance. Besides that, we have also proved that the spreading speed of cascading failures is inversely proportional to the average path length of the network and the increase of average degree can improve the network invulnerability.

#### 1. Introduction

Wireless sensor networks (WSNs) are usually made up of hundreds, even thousands, of distributed sensor nodes organized in ad hoc paradigm to monitor the environment. Since they can be easily deployed and self-organized, WSNs can cover a wide range of applications domains [1, 2]. As in most of the scenarios WSNs are expected to operate in unattended environments, the sensor nodes always suffer from the risks of energy depletion, hardware malfunction, or deliberate attacks [3, 4]. Failures of sensor nodes would split originally connected network topology, would reduce the coverage of the network, and might even lead to a global network paralysis. Therefore, how to establish an invulnerable WSN has been a hot research issue in recent years.

Most of current research on invulnerability of WSNs mainly focuses on the connectivity and availability of the network topology after removing a certain number of nodes or links. Although some promising progress has been made in building an invulnerable network topology, all this work fails to take into consideration the impacts of load redistribution on topology invulnerability. In real WSNs, the changes in network topology would give rise to the redistribution of data flow in the network, thus leading to the dynamic changes of network load. The capacity of a sensor node tackling or transmitting data is always limited due to the constrained hardware cost. When the real-time data load is beyond its capacity, it is highly likely to turn into failure. When a node fails, those nodes which transmit data via this failure node have to choose new paths to transmit data, thus leading to the load redistribution in the network. This load-redistribution process might make some new nodes fail due to capacity spilled and these failure nodes would lead to a new round of cascading failures. Consequently, more and more failure nodes are removed from the topology and this process will be repeated until there is no new node turning into failure. Therefore, the cascading process is a common phenomenon in WSNs, which is also a crucial factor to influence the network invulnerability [5, 6]. Especially with the wider application of wireless multimedia sensor networks, the risks of overload in WSNs tend to be higher and the threats of cascading failures cannot be ignored anymore. But unfortunately, current research about cascading failures of WSNs is still rare.

Coupled map lattice (CML) is a dynamical system that models the behavior of nonlinear systems. As far as the network simulation is concerned, CML considers the network system as a time-discrete and space-discrete system. In the CML-modeling network system, by observing the interaction among nodes and the self-status changes of nodes, the dynamic behavior of the network can be well studied [7]. CML has been widely applied in the domain of complex networks due to its easy-modeling and high-computation efficiency advantages. The first CML model was proposed in [8] for the studies of spatiotemporal chaos. After that, many CML models have been developed for different applications. Leung et al. [9] developed a radial-based CML model for signal detection. Zhang and Wang [10] proposed a mixed linear-nonlinear CML model for image encryption. Konishi et al. [11] presented a car-following CML model for suppression of traffic congestion. Kohar et al. [12] used a quadratic CML method to research the role of network topology in noise reduction. For cascading failures, its inner essence is that a single node’s failure is possible to spread to the entire network due to the coupled relationship with others. Considering the coupled feature of cascading failures, CML is a convincing theoretical tool to research it. Due to this reason, Wang and Xu [13] researched the cascading process of globally coupled networks based on CML. Cui et al. [14] studied the cascading failures of small-world networks. Di et al. [15] investigated the tolerance of edge cascades with CML method. Xu and Wang [16] researched the cascading process of scale-free networks.

But as WSN is a physical network featured by limited transmission radius, which is quite different from general complex network, its cascading process would demonstrate evident differences compared with other networks. Therefore, in this paper our goal is to investigate the invulnerability of various WSN topologies towards cascading failures with CML method. The contribution of this paper covers four aspects:

(1) We develop a cascading model of WSNs based on CML and propose four network topology construction methods (i.e., random network, small-world network, homogeneous scale-free network, and heterogeneous scale-free network) considering the limited transmission radius feature of WSNs.

(2) We analyze the degree distribution of the network topology generated by our methods theoretically. We prove that the generated scale-free network topologies are featured by pow-law degree distribution and the degree distributions of generated random network and small-world network are characterized by Poisson distribution.

(3) We design three attack schemes (i.e., random attack, max-degree attack, and max-status attack) as the trigger conditions for cascading process of WSNs. We investigate the cascading invulnerability under three attack schemes.

(4) Simulation results demonstrate that scale-free networks have stronger invulnerability when facing a random attack. Random network and small-world network perform better when facing a max-degree attack. Max-status attack can trigger cascading failures with less interference. The spreading speed of cascading failures is inversely proportional to the average path length of the network and increasing average degree can improve the network invulnerability.

The remainder of this paper is organized as follows. Section 2 describes the related work. Section 3 provides the cascading model of WSNs based on CML. Then, we give the topology construction methods of WSNs in Section 4 and give a theoretical analysis on their network characteristics from the perspective of degree distribution in Section 5. In Section 6, we propose the attack schemes and investigate the cascading failures invulnerability of WSNs under different attack schemes. Finally, we summarize our work and draw conclusions in Section 7.

#### 2. Related Works

Current research about how to build an invulnerable WSN topology can be classified into three types: scale-free network, small-world network, and* k*-connectivity network. In scale-free networks, a few numbers of central nodes possess most of connections in the network, making the network invulnerable to random failures. In this area, Zhu et al. [17] proposed two scale-free evolution models EAEM and EBEM. In EAEM, the newly joined node prefers to connect the existing nodes with higher degree. In EBEM, the newly joined node is more likely to build connections with the existing nodes with higher degree and more remaining energy. Simulation results proved that both models are able to generate scale-free WSN topologies, but EBEM model is more energy-efficient. Luo et al. [18] proposed a scale-free model by introducing a link adding/deleting action. In Luo et al.’s model, besides adding new nodes into the network at each time round, the links between poor-energy nodes are likely to be removed and a new link might be built between a pair of rich-energy nodes. Since, in real WSNs, the failures of wireless links are more likely to occur than nodes failures, the scale-free topology generated by Luo et al.’s model is closer to the real scenario. The small-world network theory has also proved to be an effective tool to improve the network invulnerability. Helmy [19] firstly proved that, by introducing wired links as shortcuts into WSNs, the network can maintain relatively low average path length and high cluster coefficient. In our previous work [20], we found that when the number of shortcuts reaches 20% of the total number of nodes in the network, the fault tolerance of the network can be improved by 50%. Compared with scale-free network and small-world network,* k*-connectivity topology is the most common method in improving network invulnerability. The basic idea behind* k*-connectivity topology is to ensure each node in the network maintaining at least paths towards other nodes. In this way, even if paths were cut off, the node can still deliver messages to other nodes successfully. Joshi and Younis [21] found that when the network size is large enough,* k*-connectivity network tends to be similar to random network, both of which degree distributions follow Poisson distribution. In WSNs, we can adopt two methods to achieve* k*-connectivity. One is to introduce relay nodes into the network. Compared with the common nodes, relay nodes are equipped with more powerful batteries and transmission modules. Han et al. [22] proposed a relay node placement scheme PFRP. In this scheme, common nodes choose nearest relay nodes as their cluster heads and the backbone network that is composed of relay nodes are designed for* k*-connectivity. Another one is to adjust transmission power of sensor nodes to achieve* k*-connectivity. Since in WSNs the transmission radius of sensor nodes is always limited, network connectivity can only reach or 3 in most cases. Lin et al. [23] firstly simplified the transmission power adjusting issue as the transmission range assignment issue and proved that this issue in two-dimensional space is NP-hard.

A thorough analysis and overview of invulnerability of WSNs can be found in [24]. Through analyzing existing solutions, one can conclude that existing topology construction methods mainly focus on the improvement of fault tolerance in a static point of view, but fail to consider the dynamic impacts of cascading process caused by load redistribution. Therefore, in order to understand the cascading process of WSNs and find out which topology structure tends to be more vulnerable against cascading failures, in what follows we investigate the cascading invulnerability of different network topologies under three attack schemes based on CML.

#### 3. Cascading Model of WSNs Based on CML

Considering that the links between sensor nodes are bidirectional in most of WSNs, we use undirected graph to represent the topology of WSNs, where is the collection of sensor nodes and is the collection of links.

Based on the CML model proposed by Wang and Xu [13], we give a CML-based cascading model for WSNs:where means the status of node at time and is the total number of nodes in the network. If , node is at the normal status, which means its real-time load is within its capacity. On the contrary, if , node is in failure status which means its real-time load has already been beyond its capacity. In this case, for any moment , and the edges of node would be also removed from the network. In this model, link status among nodes is indicated by the adjacent matrix . If node connects with node at time* t*, . If no connection exists between nodes and* j*, . is the degree of node at time* t*, which is equal to the sum of each element in row of . In WSNs, represents the number of adjacent nodes that node has. is the coupled coefficient, representing the coupled level between a pair of adjacent nodes. indicates that adjacent nodes cannot influence each other. With the increase of , the mutual influence tends to be more evident. Nonlinear function means the dynamic behavior of a sensor node in WSNs. Here we choose the logistics function . This function is often adopted in the network in which nodes can be easily affected by adjacent nodes [25]. As far as WSN is concerned, on the one hand, the capacity of sensor nodes is usually limited due to low-cost reason, making them sensitive to load change. On the other hand, sensor nodes in WSNs need to receive messages from last-hop nodes and relay them to next-hop nodes, meaning that sensor nodes are required to maintain frequent load-exchange with their adjacent nodes. Therefore, using logistics function to represent the dynamic behavior of WSNs is a reasonable choice. For logistics function, when .

Aiming to monitor the large area, WSNs deliver the environmental data to the base station via multihop relay. Therefore, the load of a sensor node consists of two parts: sensing load and relay load. For sensor node , the sensing load is the load generated by its own sensing tasks, which is only related to its self-status function . The relay load is the load generated by relaying the data from its neighbors; thus it is only related to the status functions of its neighbors . Coupled coefficient is to adjust the proportion between the sensing load and the relay load. By combining the sensing load and relay load, we can get the total load in (1). To state this more clearly, here we present an example on a simplified network topology shown in Figure 1.