Mobile Information Systems

Volume 2018, Article ID 6404136, 11 pages

https://doi.org/10.1155/2018/6404136

## Cascading Failure Model for Command and Control Networks Based on an *m*-Order Adjacency Matrix

^{1}School of Electrical and Information Engineering, Dalian Jiaotong University, Dalian 116028, Liaoning, China^{2}Communication and Network Laboratory, Dalian University, Dalian 116622, Liaoning, China^{3}College of Mechanical and Electronical Engineering, Lingnan Normal University, Zhanjiang 524048, China

Correspondence should be addressed to Yun-ming Wang; moc.621@82117891gnaw

Received 14 September 2018; Revised 12 November 2018; Accepted 21 November 2018; Published 18 December 2018

Academic Editor: Yuh-Shyan Chen

Copyright © 2018 Yun-ming Wang 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

Cascading failure models for command and control networks (C2 networks) continue to be a challenging and important research area. Current solutions share a common limitation because the solutions focus only on the importance of each node in isolation using one index rather than considering the contribution degree of neighboring nodes, which makes the initial load definition inaccurate and affects the cascading invulnerability of the network. To address this limitation, a new cascading failure model for C2 networks is proposed. The new model CFM_{AdjM}, which is based on an *m*-order adjacency matrix, proposes a method of initial load definition using the contribution degree of *m*-order neighboring nodes and defines the nonlinear load capacity model according to the nonlinear relationship between load and capacity. Finally, the influence of model parameters on the cascading failure of C2 networks is analyzed through simulation, and the results demonstrate that the new model effectively resists the cascading failure and enhances the survivability of the network by defining the initial load and adjusting the coefficient appropriately.

#### 1. Introduction

Command and control networks (C2 networks), as the hub of command and information transmission, are critical to victory in warfare. With continuous improvements in the degree and level of battlefield information, interactions within C2 networks have become more frequent. In addition, the organizational structure of C2 networks has become increasingly complex, exhibiting the characteristics of heterogeneous nodes, multiple links, and so on. In addition, C2 networks exhibit the characteristics of typical complex networks [1]. Because of their vital role, achieving invulnerability of C2 networks is essential. Each node in C2 networks has corresponding loads, and the network must adapt when one or more nodes are destroyed in combat [2, 3]. For the network to adapt to the loss of one or more nodes in combat, the loads of the destroyed nodes are reallocated to the neighboring nodes [4, 5]. However, the reallocation of loads may result in neighboring nodes exceeding their load capacity, causing the nodes to fail and new rounds of load redistribution. This can lead to a chain reaction, which eventually results in partial or complete network collapse. Such phenomena are referred to as network cascading failure [6, 7]. The role of cascading failure mechanisms can have a substantial impact on the entire network, even resulting in its collapse [8, 9]. In other words, cascading failure of C2 networks can be more destructive than common faults [10, 11]. Therefore, studying the cascading failure mechanism of C2 networks with complex network theory is of great significance to contain the probability of cascading failure and improve network invulnerability. This has become a hot research topic in the field of military research.

At present, many scholars at home and abroad study the cascading failure problem of complex networks. The main method still involves establishing a load capacity cascading failure model. In particular, the three key problems that must be addressed include the following: the means of determining the initial load of the nodes [12, 13], the ways of defining the maximum capacity of the nodes, and the means of reassigning the load of a failed node to other nodes in the network [14, 15].

In terms of initial load definition, there are two primary means: one is to define the initial load based on nodes [16, 17] and the other is to define the initial load according to the edge [18]. The former mostly uses node importance to define the initial load [19], which is computed based on the degree of the node, the node strength, the mean degree of the neighbor node, the betweenness, the random walk betweenness, and other conventional and improved indicators. The latter defines the initial load based on the contribution of the edge. In [20], an initial load definition method based on the degree of a node was proposed. This method considers the local information of the network but demonstrates limited accuracy. In [21], the influence of the initial load and the parameter distribution of the tolerance coefficient on the cascading failure were studied, and a cascading failure probability model was proposed based on the mean field theory. An initial load definition method based on the degree of a node and betweenness centrality was proposed in [22]. This method improves the robustness of the scale-free network against cascading failure. Based on the coupling effect of a network, a recent report [23] established a cascading failure model by adjusting the contribution of links to the load, and the more the links contribute to the load, the better the robustness of the network. Literatures [24, 25] define the initial load of an edge based on the betweenness of the endpoints of the edges and explain the cascading failure mechanism in the network by using the node capability coefficient and the betweenness-degree contribution value.

In terms of load capacity models, the first one involves statistical distribution based on node attributes [26, 27]. The second is based on the classic ML model defined by Motter and Lai [28], in which the capacity is directly linear to the initial load. This model is widely used. The third is based on the KM model proposed by Kim et al. [29], in which there is a nonlinear relationship between capacity and initial load. By studying four practical networks, the KM model reflects the nonlinear relationship between load and capacity. In other words, smaller capacity nodes have larger idle capacity. This conclusion has drawn the attention of many scholars. Subsequent research [30] introduced a stochastic method of achieving an optimal heterogeneous allocation of node capacity and compares the load capacity allocation performance of *N* order and *N* − 1 order by using node deliberate attacks and random failure. In addition, a new model of cascading failure capacity was proposed in [31], which allocates additional capacity to nodes with larger load and larger degree to ensure network robustness.

Considering the hierarchical structure characteristics of C2 networks and the influence of neighboring nodes, a cascading failure model of a C2 network based on an *m*-order neighborhood matrix is established, hereafter referred to as CFM_{AdjM}. A method of initial load definition based on *m*-order neighboring node contribution is first proposed, considering both the importance of nodes and the contribution degree of the *m*-order neighboring nodes. Secondly, a nonlinear load capacity model is established. Finally, the effectiveness and feasibility of this cascading failure model for C2 networks proposed in this paper is verified through simulation.

#### 2. Establishing a Cascading Failure Model for Command and Control Networks

##### 2.1. The Method of Defining Initial Load Based on the Contribution Degree of the *m*-Order Neighboring Nodes

The definition of the initial load has an important influence on the cascading failure of C2 networks, and the rationality of the initial load definition determines the ability of the network to resist cascading failure [32, 33]. Furthermore, the definition of initial load largely depends on the importance of nodes. This implies that the evaluation of node importance is directly related to determining the merits and failures of a cascading failure model [34]. Most of the existing cascading failure models of C2 networks directly use the single index such as degree of the nodes [4, 11] or betweenness [35] to quantify the initial load of nodes. The complexity of a C2 network structure makes it difficult for a single index to accurately measure the importance of the nodes. The importance of nodes is affected by the network structure and exhibits a strong correlation with its neighboring nodes.

According to the theory of spatial autocorrelation, the closer the distance between node pairs in complex networks, the greater the contribution to each other’s importance, conversely, the lesser the contribution to each other’s importance. In addition, the importance of contribution values degrades exponentially with the increase in distance. The C2 network is a type of complex network with scale-free characteristics, and the degree follows the power law distribution, which implies that fewer nodes have a larger degree and a larger number of nodes have relatively smaller degree. Therefore, this paper considers the important contribution of neighboring nodes in evaluating the importance of nodes in C2 networks and proposes an initial load definition method based on the contribution degree of the *m*-order neighboring nodes. Betweenness is a global variable that reflects the role and influence of nodes in the entire network. According to the concept of betweenness and the operation command process of OODA [2], we first provide the definition of a sensing node, a command node, and a fire node in C2 networks. On this basis, the concepts of combat link and combat link betweenness are proposed.

*Definition 1. *Sensor nodes refer to combat units with capabilities such as early warning, detection, reconnaissance, and surveillance, including early warning radars, reconnaissance radars, among others.

*Definition 2. *Command nodes refer to combat units with capabilities such as intelligence fusion, command and decision-making, and information coordination and distribution, including command organization, intelligence processing agency, among others.

*Definition 3. *Fire nodes refer to combat units with capabilities such as interception, attack, and damage, including all types of air defense weapons.

*Definition 4. *Combat link refers to one or more links of detection-command-fire formed by the operational information flow from sensor nodes to fire nodes via the decision fusion of command nodes.

A schematic of a combat link is shown in Figure 1. When the sensor nodes discover the enemy target, they send the information to the command nodes, and then, the command nodes forward the target information to the peer nodes or report/issue the information to higher/lower nodes. After cooperative decision-making of a number of command nodes, the combat command is issued to the fire strike nodes, and finally, the fire strike nodes attack the incoming enemy target. In this process, information transmission will pass through several combat links.