Elsevier

Knowledge-Based Systems

Volume 198, 21 June 2020, 105893
Knowledge-Based Systems

Identifying critical nodes in complex networks via graph convolutional networks

https://doi.org/10.1016/j.knosys.2020.105893Get rights and content

Abstract

Critical nodes of complex networks play a crucial role in effective information spreading. There are many methods have been proposed to identify critical nodes in complex networks, ranging from centralities of nodes to diffusion-based processes. Most of them try to find what kind of structure will make the node more influential. In this paper, inspired by the concept of graph convolutional networks(GCNs), we convert the critical node identification problem in complex networks into a regression problem. Considering adjacency matrices of networks and convolutional neural networks(CNNs), a simply yet effectively method named RCNN is presented to identify critical nodes with the best spreading ability. In this approach, we can generate feature matrix for each node and use a convolutional neural network to train and predict the influence of nodes. Experimental results on nine synthetic and fifteen real networks show that under Susceptible–Infected–Recovered (SIR) model, RCNN outperforms the traditional benchmark methods on identifying critical nodes under spreading dynamic.

Introduction

Nowadays, many real-world systems can be described by complex networks, the nodes represent the elements and edges stand for the relationships among different elements. It is well-known that many mechanisms such as cascading, spreading, and synchronizing are highly affected by a few critical nodes [1]. Therefore, identifying critical nodes in complex networks has caused great concern in the research community [2], [3], [4], [5]. And the spreading ability of nodes has been applied to solve many problems, such as controlling propagation of information and rumors in social networks, ordering reputation of scientists [6] and identifying leaders in social networks [7].

Identifying critical nodes based on the structure and function of networks is a hot research topic on static networks and plays vital role in controlling the spread of epidemic or negative information [8], [9]. Up to now, many methods have been proposed to measure the importance of nodes in static networks. Among of these, degree centrality [10], k-shell [11], H-index [12], [13] and semi-local centrality [14] select influential nodes by node’s own influences. Based on paths in networks, eccentricity centralities [15], betweenness [16] and closeness [17] have been proposed. And HITs [18], LeaderRank [19] and PageRank [20] are based on eigenvector. Zhang et al. have presented VoteRank [21] to identify a set of spreaders with the best spreading ability. Nodes vote in their neighbors in each turn, and the voting ability of neighbors of elected spreader will be decreased permanently. Li et al. proposed a classified neighbors algorithm [22] by assigning different weights for each class of neighbors and summing up the contributions of neighbors to quantify the nodal spreading capability and further to differentiate the influence of various nodes. Fei et al. [23] thought the mutual attraction between different nodes has been defined in complex network, which is inversely proportional to the square of the distance between two nodes. And proposed a method based on the inverse-square law to identify influential nodes. In order to adjust the decreasing factor adaptively according to the degree of global network, Guo et al. [24] proposed a new method named EnRenew based on node’s entropy to select a set of influential nodes. Lv et al. [25]. proposed an average shortest path centrality to identify critical nodes, in which the relative change of the average shortest path of the whole network is taken into account.

Although there are many methods to measure the influence of nodes in complex networks, most of them can be regarded as a mission to find what kind of structure will make the node more influential. With the development of machine learning, some researchers have proposed the concept of GCNs [26], [27], which aims at learning low-dimensional latent representation of graphs. These representations can be used as features for a wide range of tasks on graphs such as classification, clustering, link prediction and visualization. Inspired by this concept of GCNs, we convert the critical node identification problem in complex networks into a regression problem. Many traditional influential nodes mining algorithms usually directly use the adjacency matrix, which may contain both the intrinsic structure and the redundant information. In order to reduce the noise or redundant information and preserve the intrinsic structure information. By using adjacency matrices of networks and GCNs, we propose a simply yet effectively iterative method named RCNN to rank nodes. The performance of the proposed method is compared with degree centrality, betweenness centrality, k-shell and H-index in SIR model [28], [29] on nine synthetic and fifteen real networks. The results show that the proposed method in this paper can effectively identify critical nodes which have greater impacts on information spreading in complex networks.

The rest of paper is organized as follows: The related works are presented in Section 2 and the background is presented in Section 3. The proposed method to identify critical nodes based on graph convolutional networks is defined in Section 4. Experiment results and the time complexity are analyzed and discussed in Section 5. Conclusions and future interested research topics are given in Section 6.

Section snippets

Related works

In recent years, with the wide application of machine learning, GCNs have been used in community detection [30], [31], link prediction [32], [33] and etiology of diseases [34]. GCNs are semi-supervised methods for graphs and usually can be trained with task-specific loss through back-propagation like standard CNNs. However, due to the lack of grid structures, standard convolution of image data or text data cannot be directly applied to graphics. So the difficulty of GCNs lies in how to extract

Background

In this section, some background information about networks, convolutional neural networks and graph convolutional networks is given.

Method

We aim to bring convolutional neural networks to bear on the problem of identifying critical nodes in complex networks. We propose a framework for learning representations for arbitrary networks. Similar to convolutional neural network for images, we construct locally connected neighborhoods from the input networks. These neighborhoods serve as the receptive fields of a convolutional architecture, allowing the framework to learn effective network representations. The details of RCNN are

Experiments and discussions

The performance of the RCNN is evaluated by SIR spreading model, and compared with well-known existing metrics such as degree centrality, betweenness centrality and k-shell in nine synthetic and fifteen real networks.

Conclusion

How to identify critical nodes in complex networks is an interesting and important topic in many applications. Most of previous researches concentrate on finding what kind of structure will make the node more influential. Inspired by the concept of GCNs, by using adjacency matrices of networks and CNNs, RCNN is proposed in this paper. According to the experimental results on nine synthetic and fifteen real networks, RCNN performs much better than other four benchmark methods in identifying the

CRediT authorship contribution statement

En-Yu Yu: Investigation, Data curation, Methodology, Validation, Visualization, Writing - original draft. Yue-Ping Wang: Writing - review & editing. Yan Fu: Writing - review & editing. Duan-Bing Chen: Supervision, Funding acquisition, Writing - review & editing. Mei Xie: Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work is jointly supported by the National Natural Science Foundation of China with Grant Nos. 61673085 and 61433014, Science Strength Promotion Programme of UESTC, China (Grant No. Y03111023901014006) and National Key R&D Program of China (Grant No. 2017YFC1601005).

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