Wireless Communications and Mobile Computing

Volume 2019, Article ID 6528431, 15 pages

https://doi.org/10.1155/2019/6528431

## Influential Node Identification in Command and Control Networks Based on Integral *k*-Shell

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

Correspondence should be addressed to Yunming Wang; moc.621@82117891gnaw

Received 6 July 2019; Revised 27 September 2019; Accepted 9 October 2019; Published 30 October 2019

Academic Editor: Xianfu Lei

Copyright © 2019 Yunming 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

Influential nodes act as a hub for information transmission in a command and control network. The identification of influential nodes in a network of this nature is a significant and challenging task; however, it is necessary if the invulnerability of the network is to be increased. The existing *k*-shell method is problematic in that it features a coarse sorting granularity and does not consider the local centrality of nodes. Thus, the degree of accuracy with which the influential nodes can be identified is relatively low. This motivates us to propose a method based on an integral *k*-shell to identify the influential nodes in a command and control network. This new method takes both the global and local information of nodes into account, introduces the historical *k*-shell and a 2-order neighboring degree, and refines the *k*-shell decomposition process in a network. Simulation analysis is carried out from two perspectives: to determine the impact on network performance when influential nodes are removed and to obtain the correlation between the integral *k*-shell value and its propagation value. The simulation results show that the integral *k*-shell method, which employs an algorithm of lower complexity, accurately identifies the influence of those nodes with the same *k*-shell values. Furthermore, the method significantly improves the accuracy with which the influential nodes can be identified.

#### 1. Introduction

As the hinge for the command and control system to transfer information and fight order, command and control network (C2 network) is the key to win the war [1–3]. With the constant improvement of battlefield information, the organizational structure of the C2 network is becoming more and more complex, and the interaction of information is more frequent [4]. The C2 network shows the characteristics of heterogeneous nodes, multilayer structure, and so on. Command and control networks, which have the typical complex network characteristics, have scale-free characteristics. These networks exhibit the characteristics of “robustness and fragility” [5–7]; consequently, the problem associated with their vulnerability is a challenge that needs to be addressed. If the influential nodes in a network are attacked and damaged, the network structure is severely compromised, thus reducing the invulnerability of the command and control network and affecting the operational capability of the command and control system [8, 9]. Therefore, identifying the influential nodes of a command and control network is vital to ensure the reliable operation of a network and enhance the operational capability of the system, which has both important theoretical significance and practical value.

The methods used to identify the influential nodes in a complex network are mainly divided into two categories: system science analysis methods and social network analysis methods [10]. The basic concept of system science analysis methods is that “destructiveness is equal to importance”; that is, the influence of a node is equal to the degree of damage to the network after the deletion of that node. The main identification methods are the shortest path method, which identifies the influential nodes by evaluating the difference in the shortest path after the deletion of any one node; the spanning tree method, which identifies the influential nodes by evaluating the number of spanning trees after the deletion of any one node; and the node shrinkage method, which identifies the influential nodes from the contraction of the network after the deletion of any one node. These methods have prompted research to identify the influential nodes in complex networks. However, the computational complexity of these methods is generally high, and the network nodes need to be tentatively tested, which is difficult in practice [11]. The main concept of social network analysis is that “importance is equal to saliency,” with the influential nodes being identified by gathering statistics on the static characteristic indices of a network [12]. These indexes include the degree centrality [13], betweenness centrality [14, 15], approach degree [16], eigenvector [17], PageRank algorithm [18, 19], HITS algorithm [20, 21], structural hole [22, 23], and *k-*shell decomposition method [24].

The degree centrality method identifies the influential nodes by counting the number of neighboring nodes. A higher number of neighboring nodes imply that a node is more significant and therefore has a greater influence. The identification of influential nodes based on their degree centrality is simple and intuitive and has low computational complexity. However, this method considers only the local information of a node such that the identification accuracy is low. Chen et al. [25, 26] proposed a new method for identifying influential nodes based on local information. Their ultimate goal was to improve the recognition precision of the degree centrality of influential nodes. This method was based on the 4-order neighboring information of nodes and had relatively high recognition accuracy and lower computational complexity. These two methods for influential node identification considered only the number of neighboring nodes and disregarded the location of the node within the network structure. Nodes that are located in the core of real networks may have larger influence but a smaller degree. The betweenness centrality [14] method identifies influential nodes by calculating the number of paths through a node. A higher number of paths correspond to a greater degree of influence. It assumes that information is transmitted along only the shortest path between any pair of nodes. The PageRank algorithm [18, 19] is based on a cumulative nomination and is mainly used for web-page ranking. When web page A has a link pointing to web page B, web page B accrues a certain score. The PageRank algorithm provides a ranking for web-page matching according to the information provided by users, and the algorithm complexity is low. When an isolated node or community exists on a web page, the ranking of the PageRank algorithm is not unique. The Hypertext-Induced Topic Search (HITS) algorithm [20, 21] determines the most relevant web pages for a search by using an iterative approach. This algorithm offers wide applicability and low computational complexity. Kitsak et al. [24] defined the importance of nodes according to their location within a network and proposed a *k-*shell decomposition method. Although this method has low computational complexity and has been widely accepted, the method is not suitable for application to tree, star, or scale-free networks. This is because the identification results are too coarse, making the influential node difficult to distinguish within a shell. Zeng and Zhang [27] proposed a method based on mixing degree decomposition (MDD), which considers the number of remaining and deleted neighboring nodes to improve the visibility of the node influence. However, the scientific determination of the adjustable parameter was not explained. As a means of processing a large number of nodes with the same *k-*shell value, which the *k-*shell decomposition method is unable to distinguish, Liu et al. [28] considered the shortest distance between a target node and the node set with the highest *k-*core value. They proposed a method using no parameters, which they named the distance to network-core (DNC) method. This method ranks the influence of the nodes in terms of the distance from a node to the network core, which is defined as the node set with the highest *k-*shell value. The *k-*shell decomposition method could identify the most influential spreaders of a network and also assign some nodes with the same value regardless of their role in the spreading process. In the same way, Ren et al. [29] aimed to overcome the difficulty of identifying influential nodes with the same *k-*shell value. Their approach was to combine the information on the neighboring nodes to determine the influential nodes with a minimum *k-*shell value, without changing the *k-*shell decomposition method. Hou et al. [30] proposed a new method, named the degree-betweenness-*k-*shell (DBK) method, to analyze the influence of the important nodes by using the Euler formula to combine the indexes of degree, betweenness, and *k-*shell. This method is characterized by high computational complexity. Finally, the social network analysis method depends on the network topology, without considering the heterogeneity and hierarchy of the node. There is a certain sidedness to the exploration of influential nodes when relying on a single index such that it is not possible to fully reflect the characteristics of the networks. Subsequently, Helbing and Podobnik et al. [31, 32] highlighted that suitable system design and management can prevent undesirable cascade effects and promote favorable kinds of self-organization in the system. Perc et al. [33, 34] reviewed models that describe information cascades in complex networks, with an emphasis on the role and consequences of node centrality. Mahdi also showed that some centrality measures, such as the degree and betweenness, are positively correlated with the spreading influence.

Research into influential node identification in a command and control network led to the introduction of the concept of a combat ring [35–37]. A combat network model for a weapons and equipment system based on a combat ring is established. The natural connectivity index is proposed as a measure of the survivability of a combat network. The measurement model has high sensitivity and low computational complexity. A network robustness measure based on a maximal connected subgraph was proposed. The measure enabled the robustness of networks to be evaluated for all kinds of attacks, including random attacks, degree rank attacks, and betweenness rank attacks [38]. The weighted algebraic connectivity was employed to analyze the robustness of the network structure when faced with an uncertain disturbance [39]. This work concluded that a high-density node posse would induce more connections and greater usage, leading to increased unreliability. The entropy of the network structure was used to measure the order and stability of the supply chain system, which could be used to analyze the influence of the network scale and node connection probability on the stability of the system structure [40]. Note that research into the invulnerability of C2 networks is still in the early stages. The invulnerability of a combat network was investigated based on the connectivity of the nodes and edges [41]. The average path length was employed to simulate the relationship between the network structure and the efficiency of the network operations on the combat unit. This study concluded that the network average distance was an effective index for measuring network*-*centric warfare, where the essential concept was to employ traditional measures in complex networks. However, no special attention was paid to the applicability of a C2 network. From this perspective, the Perron–Frobenius eigenvalue (PFE) of the adjacency matrix is often used to measure the network performance of an information age combat model (IACM), as introduced by Cares [42]. Although this idea has been proposed, the specific application of an IACM has not been adequately investigated. This means that the PFE and other problems have also not been validated, neither by theoretical derivation nor by experimental verification. Subsequent research [43, 44] was carried out based on Cares’s IACM theoretical model, whereby a simulation was also conducted based on NetLogo. This study first validated the PFE as an evaluation index for measuring the operational effectiveness of a network. The simulation experiments incurred various limitations. For example, decision nodes were not connected to the network, the network scale was too small, and the difference in the capability of a node itself was not adequately considered. The theory of “structural holes” to identify the influential nodes in complex networks was proposed [22]. This method used local information to calculate the constraints of structural holes with certain limitations. The hierarchical structure of a command and control network was considered, the concept of hierarchical flow betweenness was defined, and the constraint coefficient of the structural holes was calculated [23]. Although the algorithm was computationally efficient, it had poor versatility.

Many methods exist for identifying the influential nodes in complex networks; however, their application to a command and control network is still in its infancy, given its complex network characteristics such as the hierarchical structure, heterogeneous nodes, local-area collaboration, and large network scale. The huge number of nodes and changeable network topology in a command and control network make it difficult to apply existing methods to identify influential nodes in networks with high real-time requirements. In addition, a command and control network comprises a large number of leaf nodes. The degree of these nodes is 1 and the betweenness is 0. The use of existing methods to identify influential nodes is problematic in that the computational complexity and accuracy cannot be taken into account. Based on this, this paper proposes a method to identify influential nodes based on the integral *k-*shell. This method inherits the advantage of the *k-*shell decomposition method, i.e., its low computational complexity. The proposed method overcomes the problem that arises when a large number of nodes have the same *k-*shell value, i.e., that it is impossible to identify the influential nodes. Therefore, it greatly improves the recognition accuracy of the *k-*shell algorithm.

#### 2. Integral *k*-Shell Method

The *k*-shell decomposition method is a classical influential node identification algorithm for application to complex networks. The principle of the algorithm is to shell the network layer and divide the nodes into different layers around the core. As the influence of the nodes in the central core increases, their importance also increases [45–47]. Although the complexity of the *k*-shell algorithm is low, the recognition accuracy is overly coarse-grained such that the influence of those nodes with the same *k*-shell value cannot be distinguished. To overcome the shortcomings of the *k*-shell decomposition method, an integral *k*-shell (IKS) decomposition method is proposed. This section first briefly introduces the principle of the *k*-shell decomposition method and then describes the IKS method in detail.

##### 2.1. *k*-Shell Decomposition Method

Within a network, the *k-*shell decomposition method recursively shells those nodes with a degree that is less than or equal to *k*. The method considers the location of the nodes in the network and the aggregation characteristics of the other nodes, thus overcoming the limitations of the degree centrality method. Assuming that there are no isolated nodes with a degree of 0 in the network, those nodes with a degree of 1 are the least influential nodes in the network from the perspective of the degree index. Therefore, those nodes with a degree of 1 and their edges are deleted from the network first. After this deletion, new nodes with a degree of 1 appear in the network. These new nodes and their edges are then deleted, and the process is repeated until there are no new nodes with a degree of 1 within the network. At this point, all the deleted nodes form the first level, that is, the 1*-*shell, and the *k-*shell value of these nodes is 1. The degree of each of the nodes remaining in the network is at least 2. By repeating the above deletion operation, it is possible to acquire a second layer for which the *k-*shell value is equal to 2, that is, the 2*-*shell. This process is continued until all the nodes in the network are assigned *k-*shell values. A diagram of the *k-*shell decomposition is shown in Figure 1. The nodes in the different circles belong to different cores, namely, the 1-shell, 2-shell, and 3-shell, from the outside to the inside. The *k-*shell decomposition method has a low time complexity of only *o*(*m*). This is extremely advantageous for large-scale network analysis; however, because a large number of nodes are assigned the same *k-*shell value, their influence cannot be identified, resulting in approximate partitioning results.