Vital spreaders identification in complex networks with multi-local dimension☆
Introduction
The complex network has become a useful approach in recent research, because it is inextricably correlated with various research issues. For example, the Cyber–Physical Systems (CPS) can be transformed into complex networks to study the system operation [1], optimization [2], [3], and reliability [4], [5] issues. The traffic network can also use complex networks to study traffic congestion [6], path planning [7], intelligent transportation [8], et al. Therefore, the study of the basic property of complex networks has become more important [9], [10], like the fractal property [11] and self-similarity property [12] of complex networks. These properties have been used in various fields in the network. Currently, lots of relevant studies have been carried out to study the significant properties of the network, like measuring the similarity between nodes to find the same user in different apps [13], [14]; predicting the potential links in networks to find possible relationships in social software [15], [16]; exploring the game theory in networks to find the role of evolutionary game in human progress [17], [18], [19], [20]; measuring the vulnerability of networks to guide the reconstruction of networks [21], [22], [23]. In particular, only a part of nodes plays an important role in most network properties, i.e. a small number of individuals has a great influence on society [24], [25], [26]. In network, the influence ability of each node means the speed of propagation caused by this node. Thus, finding the influential nodes in networks not only has significant theoretical significance but also practical significance. These nodes would have a more important influence on the function and structure of networks [27], [28].
Lots of centrality methods have been proposed to identify these nodes with huge influence in the complex network [29], the number of vital nodes is very small, but the impact would be indeed much larger than the other nodes. The classical centrality methods contain Betweenness Centrality [30], Closeness Centrality [31], Degree Centrality [32], PageRank [33], and lots of other methods [34]. In addition, some algorithms have been wildly used in various aspects of society, like ranking relevant website [35], detecting threat and managing disaster [36], [37], designing searching algorithm [38], [39], affecting synchronization of interconnected network [40], [41] and so on [42], [43], [44], [45]. However, these existing centralities have their own limitations. For instance, Betweenness Centrality has a high computational complexity, and lots of nodes’ value would be 0 which cannot identify their importance; Closeness Centrality cannot be applied in the network with disconnected components; Degree Centrality considers the neighbor nodes’ influence but ignores the influence all over the network.
Recently, some novel centralities have been proposed in this field to address this problem. For example, Tang et al. [46] presented a probabilistic greedy-based local search strategy to enhance the exploitation operation of discrete bat algorithm which can maximize the spread of influence. Shi et al. [47] considered influence maximization from an online–offline interactive setting and proposed the location-driven influence maximization problem. Deng et al. [48] identified the vital nodes by inverse-square law in the complex network. Zhou et al. [49] modified the gravity model to detect the influential nodes in the complex network which achieve a good performance. There still are lots of methods used in this field, such as TOPSIS [50], evidence theory [51], [52], entropy-based method [53], nodes’ relationship [54], [55], evidential network [56], [57], optimal percolation theory [58], [59], and so on [60], [61], [62], [63].
The fractal property and self-similarity property in networks can not only show the network’s feature [64], [65], but also reveal the nodes’ properties [66]. Recently, Pu et al. [67] modified the local dimension in the network to identify the influential nodes. Then, Bian et al. [68] measured the information dimension of nodes to rank the influence of node which is a new research perspective. After that, Jiang et al. [69] proposed the fuzzy local dimension to identify the influential nodes. Thus, the fractal and self-similarity properties have been proved to be significant for nodes’ importance identification. Although the local dimension can describe the local structure properties around the central nodes, it is inadequate to use a single local dimension to describe the fractal property of nodes in the complex network. To describe the spatial heterogeneity of fractal objects systematically, the multi-local dimension is introduced in this paper. By adjusting the weighting coefficient , MLD can measure the fractal property around the central node from different scales, and there are different representations about the MLD. In particular, MLD can degenerate to the local information dimension [70] and variant of local dimension [67] when and respectively. The different scales of MLD can give sufficient consideration to the fractal property of local structure.
In this paper, a novel centrality method is proposed based on the multi-local dimension which is from the view of the fractal property. This proposed method considers the structure around the central node by the box. The box radius would increase from 1 to the maximum value of the shortest distance from the central node. The information in each box is represented by the number of nodes in the box. Then, a weighting coefficient is used to deal with the information. The different chosen values of would consider the information in different scale which can cause different representations of the multi-local dimension. Finally, the multi-local dimension of nodes can be obtained by the slope of linear regression. Thus, this proposed method is a more general method to identify the vital nodes because of the existence of coefficient . Some real-world and theoretical complex networks have been used in this paper, the effectiveness and reasonableness of this proposed method are demonstrated in comparison with some existing centrality methods. Observing from the experiment results, the superiority of this proposed method and the relationship between this proposed method and other comparison methods can be obtained.
The organization of the rest of this paper is as follows. This proposed multi-local dimension is defined in Section 2 to identify the vital spreaders in the network. Meanwhile, some real-world complex networks and existing comparison methods are performed to illustrate the reasonableness and effectiveness of the proposed method in Section 3. The conclusion is conducted in Section 4.
Section snippets
The proposed vital spreaders identification method
In this section, a novel method is proposed based on the multi-local dimension (MLD) to identify the influential spreaders in the complex network. This proposed method can consider the information in boxes with different scale . When has different values, different expressions of MLD would be given to identify influential nodes. In addition, this proposed method would degenerate to local information dimension and variant of local dimension when and respectively. The flow chart of MLD
Experimental study
In this section, six different scale real-world complex networks, three theoretical complex networks, and three comparison methods are used in this section to show the reasonableness and effectiveness of this proposed method. Four kinds of experiments are utilized in this section, including giving top-10 nodes lists, obtaining the individuation of each nodes’ rank results, measuring the infectious ability of initial nodes, describing the relationship between different methods and infectious
Conclusion
In this paper, a novel method is proposed to identify the influential nodes based on the multi-local dimension in the complex networks. Different from previous methods, this proposed method is a more general method, because it can degenerate to local information dimension and variant of local dimension with the different chosen weighting coefficient . In addition, this proposed method is a negative correlated with existing methods which means the influential nodes would have small values of
CRediT authorship contribution statement
Tao Wen: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Resources, Software, Visualization, Writing - original draft. Danilo Pelusi: Validation, Writing - review & editing. Yong Deng: Funding acquisition, Project administration, Supervision, Validation, Writing - review & editing.
Acknowledgments
The authors greatly appreciate the reviewers’ suggestions and editor’s encouragement. This work is partially supported by the National Natural Science Foundation of China (grant nos. 61973332, 61573290, and 61503237).
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No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.knosys.2020.105717.