Mathematical Problems in Engineering

Volume 2018, Article ID 8268436, 11 pages

https://doi.org/10.1155/2018/8268436

## Identifying Important Nodes in Complex Networks Based on Multiattribute Evaluation

^{1}College of Computer Science and Technology, Harbin Engineering University, Harbin, China^{2}Library, Heilongjiang University of Chinese Medicine, Harbin, China^{3}College of Computer Science and Information Engineering, Harbin Normal University, Harbin, China

Correspondence should be addressed to Jianpei Zhang; nc.ude.uebrh@iepnaijgnahz

Received 13 February 2018; Accepted 16 April 2018; Published 31 May 2018

Academic Editor: Yang Tang

Copyright © 2018 Hui Xu 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

Assessing and measuring the importance of nodes in a complex network are of great theoretical and practical significance to improve the robustness of the actual system and to design an efficient system structure. The classical local centrality measures of important nodes only take the number of node neighbors into consideration but ignore the topological relations and interactions among neighbors. Due to the complexity of the algorithm itself, the global centrality measure cannot be applied to the analysis of large-scale complex network. The k-shell decomposition method considers the core node located in the center of the network as the most important node, but it only considers the residual degree and neglects the interaction and topological structure between the node and its neighbors. In order to identify the important nodes efficiently and accurately in the network, this paper proposes a local centrality measurement method based on the topological structure and interaction characteristics of the nodes and their neighbors. On the basis of the k-shell decomposition method, the method we proposed introduces two properties of structure hole and degree centrality, which synthetically considers the nodes and their neighbors’ network location information, topological structure, scale characteristics, and the interaction between different nuclear layers of them. In this paper, selective attacks on four real networks are, respectively, carried out. We make comparative analyses of the averagely descending ratio of network efficiency between our approach and other seven indices. The experimental results show that our approach is valid and feasible.

#### 1. Introduction

In recent years, the research of node importance ranking has attracted more and more attention, not only because of its important theoretical significance, but also because of its extensive practical application value [1, 2]. In complex networks, the most important nodes can help us effectively to prevent network attacks [3], to obstruct the spread of computer viruses on the networks [4], to prevent the epidemic of infectious diseases in the population [5], to inhibit the spread of gossip in society [6], and to guide the dissemination of information in social networks [7, 8]. The commonly used centrality measures include degree centrality [9], closeness centrality [10], betweenness centrality [11], and Katz centrality [12], but these indices are highly dependent on the topology of the network.

Kitsak et al. [13] found that a node with high betweenness or degree value in social network research is not necessarily the most important node. The k-shell decomposition method is proposed and decomposes a network into hierarchically ordered shells by recursively pruning the nodes with degree lower than or equal to k. The k-shell decomposition method can determine the location of nodes in the network, and the core layer is considered as a highly important node set [14]. Due to the low computational complexity of the k-shell decomposition method, it is widely used to excavate and analyze important nodes in biological networks, scientific cooperation networks, friend networks, communication networks, and so on. However, some limitations are found from this approach [1]. The k-shell decomposition method only considers the influence of the residual degree in the network decomposition, but the ranking results are too coarse-grained, which makes much difference on the node. This method is not suitable for tree diagrams, regular networks, and BA networks [15]. The k-shell decomposition method has been extended and improved by many scholars. Zeng et al. [16] evaluated the residual degree and the exhausted degree simultaneously and proposed the mixed degree decomposition. Liu et al. [17] considered the k-shell information of the target nodes and the distance from the maximum k-shell nodes of the network comprehensively, which overcome the defect that the nodes importance can not be accurately measured due to the existence of a large number of nodes with the same coreness value. Garas et al. [18] performed k-shell decomposition method on weighted networks. Liu et al. [19, 20] found that the core after the k-shell decomposition was not the true core because there existed small groups in the network which were too close to each other. Based on the definition of entropy in the theory of information entropy, they proposed the connection entropy to measure the diversity of network shell connections.

Identification of important nodes in complex networks has important theoretical significance for the structure, propagation, and synchronization of complex networks. It has a very practical value for understanding the communication and control of information, disease, and rumor, marketing the promotion of new products. In order to identify the important nodes in the complex network efficiently and accurately, this paper combines the local environment, the location, and the influence of the node to the network function and describes the importance of the complexity of the complex network nodes. In this paper, the main contributions include the following: (1) the outward links diversity assessment index is proposed and the defect, caused by the same coreness value of a large number of nodes in network after the k-shell decomposition, which can not accurately measure the importance of nodes is solved. The index not only considers the position of nodes in the network, but also takes the different nuclear layer of interaction between neighbor nodes into consideration. (2) An index of important metrics is put forward, based on the multiattribute evaluation and node deletion, which not only considers the node and its neighborhood topology structure and the interaction characteristics between nodes, but also is able to dig out the important node in the core position and can identify the key node in the structure of hole position as well. (3) After the network’s deliberated attack simulation experiment was carried out in the real data set, importance of the nodes can be calculated and quantified by the decreasing ratio of the network efficiency before and after the network attack. The experiment result shows that the method proposed in the paper has better performance in identifying important nodes in complex networks and is quite suitable for large-scale quantitative analysis of important nodes.

The remainder of this paper is structured as follows. In Section 2, we will propose and describe our method. Section 3 briefly reviews seven typical centrality indices for subsequent comparative analysis. According to the calculation of the monotonicity index and the decreasing ratio of the network efficiency, it is verified that the method proposed in this paper is better than other seven indices. Section 4 summarizes the full text and looks forward to future research directions.

#### 2. Method

Consider an undirected network with nodes and edges. Given the adjacency matrix of the network G, the degree of node i can be expressed as

Then the sum of the adjacency degrees of node i is defined as

denotes the neighbor set of node i. Degree centrality only considered the neighborhood information of nodes but ignored the topological relations between neighbors and the location of nodes in the network. Therefore degree centrality can not reflect the interaction between neighbor nodes in the calculation, and the calculation result is not accurate enough.

The k-shell decomposition method can determine the position of nodes in the network, but it only considers the influence of the residual degree when it is decomposed, which causes the ranking results to be too coarse-grained and to make the nodes less distinguishable. When an intentional attack simulation is conducted on the network, the node in the innermost kernel after the k-shell decomposition is deleted and it will be easily replaced by other nodes, if its local structure is too close and its outbound links are too small. That is to say, the node rarely interacts with other nodes and deleting it can not cause system paralysis. So the node importance is reduced. As shown in Figure 1, node B, which is in the innermost layer, does not have any outward links. Obviously, when deliberately attacking node B, other nodes in the same layer can replace it and cannot cause the system to be paralyzed. In order to overcome the defect when the coreness values of a large number of nodes in the network after the k-shell decomposition are the same, not measuring the importance of the nodes accurately, we propose the outward links diversity assessment index, which is expressed by .