1 Introduction
The complex network has become a hot topic in recent research, because it is inextricably correlated with various research issues. For example, the CyberPhysical Systems (CPS) can be transformed into complex network to study the system operation Guan2018Internet, optimization Xu2012Performance; Guan2010Microgrid, and reliability Wei2018Measuring issues. The traffic network can also use complex networks to study traffic congestion XU2017AMM, path planning Wu2019TDPP, intelligent transportation yang2019network; YANG2019121259method, et al. Therefore, the study of the basic property of complex networks has become more important Rosenberg2017Minimal, like the fractal property Song2007How and selfsimilarity property Song2005Self 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 wentao2019similar; predicting the potential links in networks to find possible relationships in social software Lu2011Link
; exploring the game theory in networks to find the role of evolutionary game in human progress
wang2017onymity; wang2016statistical; Matja2017Statistical; measuring the vulnerability of networks to guide the reconstruction of networks wentao2018evaluating. In particular, only a part of nodes plays an important role to most network properties, i.e. a small number of individuals has a great influence on society L2016Vital. In network, this influence means the propagation, representation, and dynamics of nodes. Thus, finding the influential nodes in networks not only has significant theoretical significance but also practical significance. These nodes would have more important influence to the function and structure of networks Liao2017Ranking.Lots of centrality measures have been proposed to identify these nodes with huge influence in the complex network Iannelli2018Influencers, the number of vital nodes is very small, but the impact would be indeed much larger than the other nodes. The classical centrality measures contain Betweenness Centrality Newman2005A, Closeness Centrality Freeman1979Centrality, Degree Centrality Newman2003Newman, PageRank Brin1998anatomy, and lots of other measures wang2018amodified. In addition, part of the algorithm has been wildly used in various aspects of society, like ranking relevant website Zhang2019long, detecting threat and managing disaster Srinivas2019Community, designing searching algorithm Pelusi2018Gravitational; Pelusi2018Neural; affecting synchronization of interconnected network Feng2018Synchronization and so on jiang2018Correlation; Pelusi2019Redundancy; DNTIJAR2019; Jiang2019Znetwork. However, these existing centrality 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 filed to address this problem. For example, Mariani et al. Zhou2019Fast proposed a local centrality measure named social capital can fast identify influencer which is based on the local network structure properties. Andrade et al. de2019pmean proposed pmeans centrality based on the average of the geodesic distances, and obtained the greatest spreading capacity node in the network. Deng et al. feiidentifying2018 identified the vital nodes by inversesquare law in the complex network. Zhou et al. Li2019Identifying 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 filed, such as TOPSIS Zareie2018TOPSIS, evidence theory Liu2019Identifying; MO2019121538evidence, entropybased method Zareie2017Influential, nodes’ relationship SHEIKHAHMADI2017517online; Zhang2019Groups, optimal percolation theory Ferraro2018Finding; Morone2015Influence, and so on SHEIKHAHMADI2017517marketing.
Because fractal property is important for various fields Kunze2018Collage; Davide2018Fractal, it has been applied to compress image Kunze2017IFSM, maximize the expected return LaTorre2018Portfolio, optimize population size Davide2019optimal
, give metric between probability distributions
Mendivil2017Computing, and get solutions of a classical integral equation KUNZE2019SELF. The fractal property and selfsimilarity property in networks can not only show the network’s feature Rosenberg2017Maximal; Rosenberg2017Non, but also reveal the nodes’ properties wentao2018information. Recently, Pu et al. Pu2014Identifying modified local dimension in the network to identify the influential nodes. Then, Bian et al. Bian2018Chaos measured the information dimension of node to rank the influence of node which is a new research perspective. After that, Jiang et al. wentao2019nodes proposed the fuzzy local dimension to identify the influential nodes. Thus, the fractal and selfsimilarity properties have been proved to be significant for nodes’ importance identification.In this paper, a novel centrality measure is proposed based on multilocal dimension which is from the view of the fractal property.This proposed method considers the structure around the central node by the box. The size of box 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. Different chosen of would consider the information in different scale which can cause different representation of multilocal dimension. MLD would degenerate to local information dimension and variant of local dimension when and
respectively. Finally, the multilocal dimension of node can be obtained by the slope of linear regression. Thus, this proposed measure is a more general model to identify the vital nodes because the existence of coefficient
. Some realworld complex networks have been used in this paper, the effectiveness and reasonableness of this proposed method is demonstrated in comparison with some existing centrality measures. 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 multilocal dimension is defined in Section 2 to identify the vital spreaders in the network. Meanwhile, some realworld 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.
2 The proposed vital spreaders identification model
In this section, a novel measure is proposed based on multilocal 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 is shown in Fig. 1.
2.1 The structure of complex network
In a given complex network , is the set of nodes and is the set of edges in the network, and is the number of nodes and edges respectively in the network. Firstly, the adjacency matrix of is given to describe the topological structure of the complex network. The element in the adjacency matrix shows the connection edge between node and node . represents there is an edge between node and , and is the opposite. Then, the shortest distance between any two nodes can be obtained by the adjacency matrix (the known information) through Dijkstra algorithm Dijkstra1959, and the definition of the shortest distance between node and node can be shown below,
(1) 
where is one element of which can show network’s connection, are the IDs of different nodes. The shortest distance matrix can be constructed by the know shortest distance between any two nodes, and it is denoted as . The element represents the shortest distance between node and , and the shortest distance matrix would be a symmetric matrix. The maximum value of the shortest distance from node is denoted as and defined as follows,
(2) 
where would vary from the chosen of central node , which can show the scale of locality of central node .
2.2 The local dimension of complex network
After getting the relevant basic characteristics of complex networks, the local dimension which is the basis of this proposed method is introduced in this section. The local dimension (LD) is modified from the fractal dimension and firstly proposed to accurately measure the local property of each node, i.e. the change of dimensionality among vertices in the network Silva2013Local. Then, Pu et al. Pu2014Identifying modified local dimension to identify the vital nodes in the complex network. In this method, the volume scaling property has been considered in different topological scale. In general, the number of nodes within a given radius (including ) for any node follows a power law which is shown as follows,
(3) 
where is the local dimension of node . Thus, the local dimension of any node can be obtained by the slope of double logarithmic curves which is detailed shown below,
(4) 
where is the symbol of derivative. Due to the discrete property Ben2004Complex of complex network, Eq. (4) can be rewritten as follows,
(5) 
where is the radius of the box, represents the number of nodes whose shortest distance from central node equal to , and represents the number of nodes whose shortest distance from central node is less than or equal to . The radius whose central node is node would increase from 1 to , and the local dimension of node would be the slope of double logarithmic curves.
2.3 This proposed multilocal dimension of complex network
Take node as the central node as an example in this section. In this proposed method, there is a box covering the network with node as the central node. The size of the box would increase from 1 to , and the entire network would be covered by this box when . These nodes in the network with different distance from central node is shown in Fig. 2. The information in this box is related with the number of nodes in this box, and it is defined as follows,
(6) 
where is the number of nodes covered by this box, i.e. the shortest distance from these nodes to the central node is less than the size of the box , and is the number of nodes in the network. For this given measure , the partition consideration of the box is defined as follows,
(7) 
where is the real number () which can be changed. In addition, plays a weighting coefficient for Eq. (7). In addition, when , the partition consideration . Thus, the partition consideration of the box would have following property: .
Then, the multilocal dimension of node is defined as follows,
(8) 
because the denominator would equal to 0 when , MLD would have different expression in this situation. When , the numerator would be defined as follows,
(9) 
where has been given in (7), and is the size of the box. When , would be defined as follows,
(10) 
where the partition consideration follows the expression of Shannon entropy.
In this proposed method, the scale of locality for each nodes is different, which is decided by the maximum value of the shortest distance from central node . The box size would change from 1 to
for each node. The numerical estimation of MLD would be obtained by the linear regression of
against for , and when , MLD would be obtained by the linear regression of against . Similar to multifractal dimension, the multilocal dimension can degenerate to other dimensions with different values of , and it is detailed shown below.
When , MLD would degenerate to local information dimension Wen2019local.

When , MLD would degenerate to variant of local dimension Pu2014Identifying.
Both of these two measures have been applied to identify the influential spreaders in the complex network. Thus, this proposed method MLD is a more general method.
2.4 Vital spreaders identification
When the multilocal dimension is obtained, the importance of spreaders can be ranked by the value of multilocal dimension. Different with previous methods, the spreader would be more important with smaller MLD. The details can be shown in Section 3.
3 Experimental study
In this section, four different scale realworld 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 top10 nodes lists, obtaining the individuation of each nodes’ rank results, measuring the infectious ability of initial nodes, describing the relationship between different measures and infectious ability obtained by SI model.
3.1 Data
There are four different scale realworld complex networks used in this section to show the effectiveness and reasonableness of this proposed method, and they are:

The Zacharys Karate network: This network demonstrates the relationship between many individuals in one USA university karate club;

The Jazz musicians network: This network shows the collaborations between different jazz musicians;

The USA airline network: This network represents the airlines between the big city airports in the USA;

The Political blogs network: This network demonstrates the blogs’ connection in two camps in the USA.
These network can download from http://vlado.fmf.unilj.si/pub/networks/data/. The detailed structural information of these four networks are shown in Table 1. and is the number of nodes and edges in the network respectively. and is the average value and maximum value of degree of node in the entire network. and represents the average value and maximum value of the shortest distance in the network.
Network  

Karate  34  78  4.5882  17  2.4082  5 
Jazz  198  5484  27.6970  100  2.2350  6 
USAir  332  2126  12.8072  139  2.7381  6 
Political blogs  1222  19021  27.3552  351  2.7375  8 
3.2 Existing centrality measures
Before the experiment begins, let’s introduce some existing centrality measures to identify the influential nodes as comparison methods in this section. Because MLD would degenerate to local information dimension and variant of local dimension, these two measures would not be used as comparison measures in this section.
Definition 3.1.
Betweenness centrality (BC) Newman2005A. The betweenness centrality of node is expressed as , and it is defined as follows,
(11) 
where means the shortest path between node and node which go through node , and means the shortest path between node and node . Node and node would traverse all nodes in the network. Thus, BC highlights the intermediary role of selected node.
Definition 3.2.
Closeness centrality (CC) Freeman1979Centrality. The closeness centrality of node is expressed as , and it is defined as follows,
(12) 
where is the shortest distance between node and node which belongs to the shortest distance matrix , and node would traverse all nodes in the network. Thus, CC highlights that the selected node can quickly reach any node in the network.
Definition 3.3.
Degree centrality (DC) Newman2003Newman. The degree centrality of node is expressed as , and it is defined as follows,
(13) 
where is the element in adjacency matrix , and node would traverse all nodes in the network. When there is an edge between node and node , would equal to 1, and represents the opposite situation. In fact, the degree centrality of node represents the number of edges connected with node . Thus, DC highlights the number of neighbor nodes around selected node in the network.
3.3 Experiment I: Top10 nodes
In this experiment, the top10 nodes lists are obtained by different measures to show the difference and correlation between these methods, and these lists are shown in Table 2. Because these methods consider different parts of information in the network, their rank lists may be different with the others. When two methods’ top10 nodes lists are similar, their consideration information would be similar. In addition, the same nodes between MLD and other methods can bring more credibility to this proposed method. These nodes which only appear in MLD result would have a significant improvement to the propagation process.

Observing the result in Karate network from Table 2, the most similar lists to MLD is BC,and there are eight same nodes between BC and MLD. The number of same nodes between CC, DC, and MLD is five and six nodes respectively, which is relatively low compared to the results between BC and MLD. The result means that the most similar method to MLD is BC, which is different from the later experiments.

In Jazz network, the result between BC and MLD is the most dissimilar, and there are only three same nodes between these two measures which is the lowest same number of all results. Compared CC with this proposed method, there are 7 same top10 nodes. In addition, the top10 nodes lists are almost the same using MLD and DC, and it is 9 same nodes in the top10 nodes lists.

Similar to Jazz network’s result, the number of the same top10 nodes between DC and MLD in USAir network is the highest in three comparison methods, and it is 8 same nods. There is six same nodes between this proposed method and CC in this top10 nodes lists. The number of same top10 nodes between BC and MLD is also the lowest in three comparison methods, and there is only four same nodes between two measures, which means there are difference between BC and MLD. The most influential node identified by three comparison methods and MLD is the same, and it is node 118, which means the accuracy of this proposed method.

Observing the Political blogs network’s result from Table 2, all comparison methods have many same nodes in top10 lists. CC and DC both have nine same top10 nodes with MLD, and this only one different node is the ninth and tenth node respectively. The lowest number of same nodes is between BC and MLD, and it is seven, which is bigger than the results in other networks. The top2 nodes are the same in CC, DC, and MLD. From the result in this network, it can be found the similarity between this proposed method and other comparison methods is high.
In conclusion, observing from the number of the same top10 nodes, this proposed method has close performance with DC, and it is far from BC. The effectiveness and superiority would be demonstrated in the following sections. Because this proposed method MLD can degenerate to local information dimension and variant of local dimension, these two measures would not be contained in the following experiments.
Rank  Karate Network  Jazz Network  

BC  CC  DC  MLD  BC  CC  DC  MLD  
1  1  1  34  34  136  136  136  60 
2  3  3  1  1  60  60  60  136 
3  34  34  33  33  153  168  132  132 
4  33  32  3  24  5  70  168  83 
5  32  33  2  3  149  83  70  168 
6  6  14  32  2  189  132  108  99 
7  2  9  4  30  167  194  99  108 
8  28  20  24  6  96  122  158  158 
9  24  2  14  7  115  174  83  194 
10  9  4  9  28  83  158  7  7 
Rank  USAir Network  Political blogs Network  
BC  CC  DC  MLD  BC  CC  DC  MLD  
1  118  118  118  118  12  28  12  12 
2  8  261  261  261  304  12  28  28 
3  261  67  255  152  94  16  304  304 
4  47  255  182  230  28  14  14  14 
5  201  201  152  255  145  36  16  16 
6  67  182  230  182  6  67  94  94 
7  313  47  166  112  16  94  6  6 
8  13  248  67  147  300  35  67  67 
9  182  166  112  166  163  145  35  35 
10  152  112  201  293  35  304  145  36 
3.4 Experiment II: Individuation
Then, different methods’ capability to identify influential nodes are explored in this section. The importance of these nodes with same score (frequency) cannot be distinguished correctly, but it is a common situation in this field. Thus, a more useful method should be found to give nodes as individual values as possible. If one method can give all these nodes with unique score, this method can give a reasonable importance rank lists to avoid ambiguous rank results. So the individual of method can be considered an effectiveness indicator to show the quantity of different methods. The higher the individual of one method is, the more effective this method is.
Definition 3.4.
The individuation of one method is defined as follows,
(14) 
where is the number of nodes with unique score, is the number of nodes in the entire network. is the individuation of one method.
The frequency of nodes in each rank obtained by different measures is shown in Fig. 3. In these four network, it can be found that MLD has the least number of nodes in the same rank, and there are more ranks in this proposed method. In contrast, other three comparison methods have more nodes with same rank. In these four networks, DC has the least ranks which means there are lots of nodes with the same ranks. The frequency of nodes in most of the top ranks is relatively low in BC, but the last few ranks have a very high frequency (almost half of nodes), which means BC cannot identify these nodes with low . CC can give a relatively reasonable ranks, because most of the frequency in each ranks is low and there are relatively more ranks. However, compared with CC, MLD is more effective to identify the influential nodes. That is because the frequency of nodes in each rank is the least in these four methods, and there are the most ranks (almost one node have one unique ranks) in these networks.
Network  

Karate  0.4705  0.5882  0.3235  0.7058 
Jazz  0.6565  0.6414  0.3131  0.9494 
USAir  0.2771  0.5813  0.1746  0.7620 
Political blogs  0.5114  0.6743  0.1178  0.9525 
The individuation of different methods in realworld complex networks are shown in Table 3, where the highest is bold. It can be found that MLD have the highest individuation in these four methods, and DC has the lowest individuation . These results mean that this proposed method is an effective method to identify the influential nodes in the complex network.
3.5 Experiment III: SI model
In this section, SusceptibleInfected (SI) model L2016Vital is applied to show the effectiveness and reasonableness of this proposed method. The details of SI model is introduced below.

For the entire network, all nodes are classified into two states, and they are
susceptible state and infected state. 
At the beginning, the top10 nodes obtained by centrality measure (shown in Table 2) are set as infected state, and the other nods are set as susceptible state.

When the infection process begins, these susceptible node can be affected by their neighbor nodes with a given probability (spreading ability) in each time . In addition, the total number of susceptible nodes and infectious nodes equals to in any time .

Once the susceptible node is infected into infectious node, it cannot return to the susceptible state, i.e. it is the irreversible process.

The number of infectious nodes would continue to increase over time until all nodes are infected.
These initial nodes with higher infection ability would infect the entire network as early as possible, so the number of infectious node can be a effective indicator to show the infection ability of initial nodes, i.e. the importance of initial nodes. More infectious nodes in each time is, higher infectious ability these initial nodes are, more important these initial nodes are. Because CC considers the nodes’ distance from the selected node, which is similar to this proposed method, CC is selected as the comparison method in this section. In these networks, all results would average the results of 30 SI experiments with , and the results are shown in Fig. 4.
Observing from Fig. 4, the number of infectious nodes continue to increase over all time . In Karate network shown in Fig. 4(a), the infection ability of initial nodes obtained by MLD is clearly superior to CC, and it can be seen that obtained by MLD is larger than obtained by CC from the whole process. In Jazz network shown in Fig. 4(b), the performance of MLD is better than CC which can be seen from early and middle propagation process in SI model. In USAir network shown in Fig. 4(c), the infection ability of initial nodes obtained by MLD is superior than these nodes obtained by CC, and it can be seen from the middle and late term of SI model, obtained by MLD is larger than obtained by CC in this term. In Political blogs network shown in Fig. 4(d), MLD is slightly better than CC, because they are almost same in the propagation process. But the number of infectious nodes obtained by MLD is bigger than the number obtained by CC between 5 to 15 time. In conclusion, this proposed method have a superiority performance in most of experiments, and some times the performance of MLD is close to the comparison method.
3.6 Experiment IV: The relationship between different methods
Because BC has lots of nodes with same value which would cause unusual relationship between BC and this proposed method, the comparison methods are chosen as CC and DC in this section. The relationship between the values obtained by different centrality measures and the infectious ability obtained by SI model are shown in Fig. 5 (CC VS MLD) and Fig. 6 (DC VS MLD). One point in the relationship graph represents one node in the network, the value of axis means the value obtained by different measures, and the color of point shows the infectious ability of this node obtained by SI model, i.e. the number of infected nodes () in 10 steps. The infectious ability of node is obtained by averaging 50 independent experiments results when . The positive correlation means the nodes would have large value obtained by comparison method and MLD, and negative correlation is the opposite. Observing from Fig. 5, CC and MLD is negative correlation, and their relationship is linear which can give similar rank results between these two measures. In addition, the values obtained by CC is relatively small than other methods (small order of magnitude) which cannot clearly show the difference in nodes’ importance. Observing from Fig. 6, the correlation between DC and MLD is also negative, which means the node with large MLD would have small DC. What’s more, it can be found that there are lots of nodes with small degree centrality, which is because of the scalefree property of the complex network. Thus, there would be lots of nodes with small DC that cannot correctly identify importance. However, MLD can overcome this shortcoming, because the MLD of node would be more scattered which can give each node with unique value and obtain a relatively reasonable rank lists. Overall speaking, this proposed method would be different withe existing methods, which is negative correlated with exiting methods. In addition, this proposed method can give a more reasonable rank list because it can identify the importance of nodes with close value obtained by existing methods.
4 Conclusion
In this paper, a novel method is proposed to identify the influential nodes based on multilocal dimension in the complex networks. Different with previous methods, this proposed method is a more general model, because it can degenerate to local information dimension and variant of local dimension with the different chosen of weighting coefficient . In addition, this proposed method is negative correlated with existing methods which means the influential nodes would have small value of MLD and large value of existing centrality measures. Comparing with exiting centrality methods, this proposed method can effectively identify the influential nodes in the network and give a reasonable rank to these nodes, which can overcome the limitations of previous methods.
However, this proposed method can still be improved to meet the high requirements in this field. For instance, there are still some nodes with same value of MLD, and the ranking of these nodes is relative top, which can mislead to form the correct node importance rank. Thus, in further research, the consideration factors of this method can be changed, which can demonstrate the property of the network more specifically.
Acknowledgment
The authors greatly appreciate the reviews’ suggestions and the editor’s encouragement. The work is partially supported by National Natural Science Foundation of China (Grant Nos. 61973332, 61573290, 61503237).
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