 Regular article
 Open Access
Centrality in modular networks
 Zakariya Ghalmane^{1, 3},
 Mohammed El Hassouni^{1},
 Chantal Cherifi^{2} and
 Hocine Cherifi^{3}Email authorView ORCID ID profile
 Received: 13 July 2018
 Accepted: 30 April 2019
 Published: 9 May 2019
Abstract
Identifying influential nodes in a network is a fundamental issue due to its wide applications, such as accelerating information diffusion or halting virus spreading. Many measures based on the network topology have emerged over the years to identify influential nodes such as Betweenness, Closeness, and Eigenvalue centrality. However, although most realworld networks are made of groups of tightly connected nodes which are sparsely connected with the rest of the network in a socalled modular structure, few measures exploit this property. Recent works have shown that it has a significant effect on the dynamics of networks. In a modular network, a node has two types of influence: a local influence (on the nodes of its community) through its intracommunity links and a global influence (on the nodes in other communities) through its intercommunity links. Depending on the strength of the community structure, these two components are more or less influential. Based on this idea, we propose to extend all the standard centrality measures defined for networks with no community structure to modular networks. The socalled “Modular centrality” is a twodimensional vector. Its first component quantifies the local influence of a node in its community while the second component quantifies its global influence on the other communities of the network. In order to illustrate the effectiveness of the Modular centrality extensions, comparison with their scalar counterparts is performed in an epidemic process setting. Simulation results using the SusceptibleInfectedRecovered (SIR) model on synthetic networks with controlled community structure allows getting a clear idea about the relation between the strength of the community structure and the major type of influence (global/local). Furthermore, experiments on realworld networks demonstrate the merit of this approach.
Keywords
 Influential nodes
 Centrality measures
 Community Structure
 SIR model
1 Introduction
Identifying the most influential nodes in a network has gained much attention among researchers in recent years due to its many applications. Indeed, these key nodes play a major role in controlling the epidemic outbreak [1], increasing the publicity on a new product [2], controlling the rumor spreading [3]. The most popular approach to uncover these central nodes is to quantify their influence using centrality measures. Various centrality measures have been proposed to quantify the influence of nodes based on their topological properties. Degree centrality, betweenness centrality, closeness centrality are among the most basic and the most widely used centrality measures.
The majority of realworld networks exhibit the modular organization of nodes, the socalled community structure [4–8]. Although there has been a tremendous effort regarding the definition of this property, there is no formal consensus on a definition that captures the gist of a community. It is intuitively apprehended as densely connected groups of nodes where individuals interact with each other more intensely than with those in the rest of the network. Therefore, communities are groups of nodes sharing some common properties and play similar roles in the interacting phenomenon within networks. Besides their various definitions, communities have been also found to show a number of interesting features such as the overlapping configuration of modules [9]. Some of the nodes can then be shared by multiple communities. Indeed, in some social networks, individuals can take part simultaneously in different groups, such as work colleagues, friends or family. In this work, we do not consider the overlaps between communities.
Previous works have shown that community structure has an important effect on the spreading process in networks [10–13]. However, classical centrality measures [14] do not take into account the influence of this major topological property on the spreading dynamics. In a modular network, we can distinguish two types of links [15, 16] that support the diffusion process: the links that connect nodes belonging to the same community (intracommunity links or strong ties) and the links that bridge the communities (intercommunity links or weak ties). The former exercise a local influence on the diffusion process (i.e., at the community level), while the latter have a global influence (at the network level). Therefore, we believe that these two types of links should be treated differently. Indeed, the intracommunity links contribute to the diffusion in localized densely connected areas of the networks, while the intercommunity links allow the propagation to remote areas of the network. Suppose that an epidemic starts in a community, as it is highly connected, the intracommunity links will tend to confine the epidemic inside the community, while the intercommunity links will tend to propagate it to the other communities. As their role is quite different, we propose to represent the centrality of modular networks by a twodimensional vector where the first component quantifies the intracommunity (or local) influence and the second component quantifies the intercommunity (or global) influence of each individual node in the network. To compute these components, we need to split the original network into a local and global network. The local network is obtained by removing all the intercommunity links from the original network. The global network is obtained by removing all the intracommunity links from the original network. Note that if the original network is made of a single connected component the global and local networks split into many connected components. Therefore, care must be taken to adapt the centrality definition to networks with multiple components. In the following, we restrict our attention to nonoverlapping community structure (i.e. a node belongs to a single community). Furthermore, we consider undirected and unweighted networks for the sake of simplicity, but results can be easily extended to more general situations.

Choose a standard centrality measure.

Compute the local network by removing all the intercommunity links from the modular network.

Compute the Local component of the Modular centrality using the standard centrality.

Compute the global network by removing all the intracommunity links from the modular network.

Compute the Global component of the Modular centrality using the standard centrality.
Experiments are conducted on modular synthetic networks in order to better understand the relative influence of the Local and Global component of the Modular centrality in the propagation process. Extensive comparisons with the standard centrality measures show that Modular centrality measures provide more accurate rankings. Simulations on realworld networks of diverse nature have also been performed. As their community structure is unknown, a community detection algorithm has been used. Results confirm that node rankings based on the Modular centrality are more accurate in terms of the epidemic size than those made by the standard centrality measures which have been designed for networks with no community structure.
The rest of the paper is organized as follows. Related modularbased measures are discussed in the next section. In Sect. 3, a general definition of the Modular centrality is given. In this framework, we present the extensions to modular networks of the most influential centrality measures (closeness, betweenness and eigenvector centrality). The experimental setting is described in Sect. 4. We report and analyze the results of the experiments performed on both synthetic and realworld networks in Sect. 5. Finally, the main conclusions are presented in Sect. 6.
2 Related works
Ranking the nodes according to their centrality constitutes the standard deterministic approach to uncover the most influential nodes in a network. These measures rely usually on various network topological properties. However, the community structure of the network is rarely taken into consideration. Few researchers have paid attention to this property encountered in many realworld networks [10–13, 17–23]. In this section, we give a brief overview of the main deterministic methods that motivates our proposition.
a. Community centrality
The magnitude of a node vector \(\vert x_{i} \vert \) specifies how central the node i is in its community in terms of the number of connections. Thus, the node i has a large positive contribution to the modularity when this measure is large. On the other hand, a higher value of \(\vert y_{i} \vert \) means that the node i has many connections to other nodes from foreign communities. Therefore, the Community centrality is defined to be equal to the vector magnitude \(\vert x_{i} \vert \). It measures the strength with which a given node i is assigned to its community. This measure was tested in the network of coauthorships between scientists. Results show that it is not very correlated with the degree centrality. Moreover, some nodes with high Community centrality measure have a relatively low degree. However, they have more connections with nodes of their communities. Thus, nodes with high Community centrality value play a central role in the spreading process in their local neighborhood.
b. Comm centrality
N. Gupta et al. [25] proposed a degreebased centrality measure for networks with nonoverlapping community structure. It is based on a nonlinear combination of the number of intracommunity links and intercommunity links. The goal is to select nodes that are both hubs in their community and bridges between the communities. This measure gives more importance to community bridges. Indeed, the number of intercommunity links is raised to the power of two. The comparison has been performed with deterministic and random immunization strategies using the SIR epidemic model and both synthetic and realworld networks. Nodes are immunized sequentially from each community in the decreasing order of their centrality value in their respective community. The number of nodes to be removed from a community are kept proportional to the community size. Results show that the Comm strategy is more effective or at least works as well as Degree and Betweenness centrality while using only information at the community level.
c. Number of neighboring communities centrality
In a previous work [26], we proposed to rank the nodes according to the number of neighboring communities that they reach in one hop. The reason for selecting these nodes is that they are more likely to have a big influence on nodes belonging to various communities. Simulation results on different synthetic and realworld networks show that it outperforms Degree, Betweenness and Comm centrality in term of the epidemic size in networks with a community structure of medium strength (i.e. when the average number of intracommunity links is of the same order than the number of intercommunity links).
d. Community HubBridge centrality
We also proposed the Community HubBridge centrality measure in [26]. It is based on the combination of the number of intracommunity links weighted by the size of the community and the intercommunity links weighted by the number of neighboring communities. This measure tends to select preferentially nodes that can be considered as hubs inside large communities and bridges having many connections with various neighboring communities. According to experimental results, on both synthetic and realworld networks, this centrality measure is particularly suited to networks with strong community structure (i.e., when there are few intercommunity links as compared to the number intracommunity links). In this situation, it can identify effectively the most influential spreaders as compared to Degree, Betweenness, Comm and the Number of Neighboring Communities centrality measures. A variation of this centrality measure called the Weighted Community HubBridge centrality has also been introduced. It is weighted such that, in networks with welldefined community structure, more importance is given to bridges (intercommunity links), while in networks with weak community structure the hubs in the communities dominate. The goal is to target the bridges or the hubs according to the community structure strength. This measure has proved its efficiency as compared to the alternatives particularly in networks with weak community structure.
e. Kshell with community centrality
 (i)
After the removal of nodes with intracommunity links, the Kshell decomposition of the remaining nodes is computed. It is associated with an index of \(k^{W}_{\mathrm{core}}\).
 (ii)
After the removal of nodes with intercommunity links, the Kshell of the remaining nodes is computed. It is associated with an index of \(k^{S}_{\mathrm{core}}\).
 (iii)
A new measure is then calculated and assigned to each node based on the linear combination of both \(k^{W}_{\mathrm{core}}\) and \(k^{S}_{\mathrm{core}}\) in order to find nodes that are at the same time bridges and hubs located in the core of the network.
f. Global centrality
In [28], M. Kitromilidis et al. propose to redefine the standard centrality measures in order to characterize the influence of Western artist. Based on the idea that influential artists have connections beyond their artistic movement, they propose to define the centrality of modular networks by considering only the intercommunity links. In other words, an influential artist must be related to multiple communities, rather than being strongly embedded in its own community. Considering a painter collaboration network where edges between nodes represent biographical connections between artists, they compared Betweenness and Closeness centrality measures with their classical version. Results show that the correlation values between the standard and modified centrality measures are quite high. However, the modified centrality measures allow to highlight influential nodes who might have been missed as they do not necessarily rank high in the standard measures.
All these works suggest that it is of prime interest to disentangle the local influence from the global influence in order to characterize a node centrality in modular networks. Indeed, these complementary types of influence may carry very different meanings and be more or less important in different situations. This is the reason why we propose to exploit the mesoscopic granularity level in order to extend the definition of the centrality measures that are agnostic about the community structure to modular networks. We propose to represent the centrality measures in modular networks as a twodimensional vector made of its Local and Global component. If needed, these two components can be merged in a scalar value, but the combination can be made in multiple ways according to complementary available information about the network nature and topological properties.
3 Modular centrality
Our main objective is to take into account the community structure in order to identify influential nodes. Indeed, in modular networks, a node has two types of influence: a local influence which is linked to its community features and a global influence related to its interactions with the other communities. Under this assumption, we provide a general definition of centrality in modular networks. We design a generic algorithm for computing the centrality of a node under this general definition. The Modular centrality extension can be naturally inferred from the various existing definitions of centrality designed for networks without community structure. To illustrate this process, we give the modular extensions of the most influential centrality measures (Betweenness, Closeness, and Eigenvector).
3.1 Definitions
3.1.1 Local component of the Modular centrality
Let’s consider a network denoted as \(G(V,E)\), where \(V=\{v_{1},v_{2},\ldots,v _{n}\}\) and \(E=\{(v_{i},v_{j}) \setminus v _{i},v_{j} \in V\}\) denotes respectively the set of vertices and edges. Its nonoverlapping community structure \(\mathcal{C}\) is a partition into a set of communities \(\mathcal{C}=\{C_{1},\ldots,C_{k},\ldots,C_{m}\}\) where \(C_{k}\) is the kth community and m is the number of communities. The local network \(G_{l}\) is formed by the union of all the disjoint modules of the network \(G_{l} = \bigcup_{k=1}^{m} C_{k}\). These components are obtained by removing all the intercommunity links between modules from the original network G. Each module represents a community \(C_{k}\) denoted as \(C_{k}(V_{k},E_{k})\). Where \(V_{k}=\{ v _{i}^{k} \setminus v_{i} \in V \}\) and \(E_{k}=\{(v_{i}^{k_{1}},v_{j} ^{k_{2}}) \setminus v_{i},v_{j} \in V\text{ and }k_{1}=k_{2} \}\), while \(v_{i}^{k}\) refers to any node \(v_{i}\) belonging to the community \(C_{k}\).
For a selected centrality measure β, we define \(\beta _{L}(v _{i}^{k})\) as the Local centrality of the node \(v_{i} \in V_{k}\). It is computed separately in each module \(C_{k}\) of the local graph \(G_{l}\).
3.1.2 Global component of the Modular centrality
Let’s consider the network \(G(V,E)\), the global network \(G_{g}\) is formed by the union of all the connected components of the graph that are obtained after removing all the intracommunity links from the original network \(G(V,E)\). Let’s suppose that \(\mathcal{S}=\{S_{1},\ldots,S _{q},\ldots,S_{p}\}\) is the set of the revealed connected components and \(p= \vert \mathcal{C'} \vert \) is the size of the set \(\mathcal{S}\), the global network is defined by \(G_{g} = \bigcup_{q=1}^{p} S_{q}\). Each component \(S_{q}\) is denoted as \(S_{q}(V_{q},E_{q})\). Where \(V_{q}=\{ v_{i}^{q} \setminus v_{i} \in V \}\) and \(E_{q}=\{(v_{i}^{q_{1}},v_{j}^{q_{2}}) \setminus v_{i},v_{j} \in V\text{ and }q_{1}=q_{2} \}\), while \(v_{i}^{q}\) refers to any node \(v_{i}\) belonging to the component \(S_{q}\). In this network, there may be some isolated nodes (i.e., nodes that are not linked directly to another community). These nodes are removed from \(G_{g}\) in order to obtain a trimmed network formed only by nodes linked to different communities by one hop. Consequently, the set of nodes of \(G_{g}\) is defined then by \(V_{g}=\{ v_{i} \in V \setminus \vert \mathcal{N}_{v_{i}}^{1} \vert \neq 0\}\). Where \(\mathcal{N}_{v_{i}}^{n}\) is the neighborhood set of nodes reachable in n hops. It is defined by \(\mathcal{N}_{v_{i}}^{n}=\{v_{j} \in V \setminus v_{i} \neq v_{j}\text{ and } d_{G}(v_{i},v_{j})\leq n\}\), \(d_{G}\) is the geodesic distance.
For a selected centrality measure β, we define \(\beta _{G}(v _{i}^{q})\) as the Global centrality of the node \(v_{i} \in V_{q}\). It is computed over each connected component \(S_{q}\) included in the global graph \(G_{g}\). Remember that the Global centrality measure of the removed isolated nodes is set to 0.
3.1.3 Modular centrality
3.2 Algorithm
 Step 1.
Choose a standard centrality measure β.
 Step 2.
Remove all the intercommunity links from the original network G to obtain the set of communities \(\mathcal{C}\) forming the local network \(G_{l}\).
 Step 3.
Compute the Local measure \(\beta _{L}\) for each node in its own community.
 Step 4.
Remove all the intracommunity links from the original network to reveal the set of connected components \(\mathcal{S}\) formed by the intercommunity links.
 Step 5.
Form the global network \(G_{g}\) based on the union of all the connected component. Isolated nodes are removed from this network and their Global centrality value is set to 0.
 Step 6.
Compute the Global measure \(\beta _{G}\) of the nodes linking the communities based on each component of the global network.
 Step 7.
Add \(\beta _{L}\) and \(\beta _{G}\) to the Modular centrality vector \(B_{M}\).
3.3 Modular extensions of standard centrality measures
In order to illustrate the process allowing to extend a given centrality defined for a network without community structure to a modular network, we give as examples the modular definitions of the Betweenness, Closeness and Eigenvector centrality.
3.3.1 Modular Betweeness centrality
3.3.2 Modular Closeness centrality
3.3.3 Modular Eigenvector centrality
3.4 Toy example
Standard Degree centrality, Global and Local Component of the Modular Degree Centrality of the nodes in the toy example
Node ID  4  11  18  22  1  2  3  16  5  7  8 

β  7  7  5  5  4  4  4  4  3  3  3 
\(\beta _{L} \)  3  7  3  3  4  2  4  2  3  2  3 
\(\beta _{G} \)  4  0  2  2  0  2  0  2  0  1  0 
Node ID  9  10  14  15  17  19  21  6  12  20  13 

β  3  3  3  3  3  3  3  2  2  2  1 
\(\beta _{L} \)  2  3  2  3  3  3  3  1  2  2  1 
\(\beta _{G} \)  1  0  1  0  0  0  0  1  0  0  0 
To sum up, it can be noticed from this example that when we consider the Degree centrality, the community hubs are the most influential spreaders locally due to their ability to reach a high number of nodes in their own communities. The bridges which are linked to various communities are the most influential spreaders globally as they allow to reach a high number of communities all over the network.
3.5 Modular centrality ranking strategies
In order to rank the nodes according to their centrality, it is necessary to derive a scalar value from the Modular centrality vector. To do so, we can proceed in many different ways. In order to highlight the essential features of centrality in modular networks, we choose to consider three strategies. The first two are straightforward. Indeed, a simple way to combine the components of the Modular centrality is to use the modulus and the argument of this vector. The third strategy uses more information about the community structure in order to see if this can be beneficial.
\(k^{\mathrm{inter}}(v_{i}^{k})\) is the number of intercommunity links of node \(v_{i}^{k}\) and \(k(v_{i})\) is the degree of node \(v_{i}^{k}\).

A community \(C_{k}\), where the intracommunity links predominate is densely connected and therefore it has a very welldefined community structure. If an epidemic starts in such a cohesive community, it has more chance to stay confined than to propagate through the few links that allows to reach the other communities of the network. In this case, priority must be given to local immunization. Consequently, more weight is given to the Local component of the Modular centrality \(\beta _{L}\) to target the most influential nodes in the community since it is well separated from the other communities of the network.

A community \(C_{k}\) where the intercommunity links predominate has a noncohesive community structure. It is more likely that an epidemic starting in this community diffuses to the other communities through the many links that it shares with the other communities. Consequently, more weight is given to the Global component of the Modular centrality measure \(\beta _{G}\) in order to target nodes that can propagate the epidemic more easily all over the network due to the loose community structure of \(C_{k}\).
4 Experimental setting
In this section, we give some information about the synthetic and realworld dataset used in the empirical evaluation of the centrality measures. The SIR simulation process is recalled, together with the measure of performance used in the experiments.
4.1 Dataset
4.1.1 Synthetic networks
LFR network parameters
Number of nodes  4000 
Average degree  7 
Maximum degree  80 
Exponent for the degree distribution  2.8 
Exponent for the community size distribution  2 
Mixing parameter  0.1, 0.4, 0.7 
Community size range  [15 200] 
4.1.2 Realworld networks

Social networks: Four Samples of the Facebook Network are used. The egoFacebook network collected from survey participants using the Facebook app. [33] and the Facebook friendship network at 3 US universities (Caltech, Princeton, Georgetown) collected by Traud et al. [34]. Nodes represent individuals (survey participant or members of the University), and edges represent online friendship links between two individuals. In the University network, in order to obtain data that are relevant for the spread of epidemic infections, only the relationship of individuals who live in the same dormitory or study the same major are considered.

Communication network: The EmailEucore^{1} network has been generated using email data from a large European research institution. The dataset contains only communication between institution members. Each node corresponds to an email address and an edge is established between two nodes u and v, if at least one email has been exchanged between address u and address v.

Technological network: PowerGrid^{2} is a network containing information about the topology of the Western States Power Grid of the United States. An edge represents a power supply line. A node is either a generator, a transformer or a substation.

Collaboration network: GRQC^{a} (General Relativity and Quantum Cosmology) collaboration network has been collected from the eprint arXiv and covers scientific collaborations between authors of papers submitted to the General Relativity and Quantum Cosmology category. If an author i coauthored a paper with author j, the graph contains an edge from i to j. If the paper is coauthored by k authors this generates a completely connected (sub)graph on k nodes.
Description of the structural properties of the realworld networks. N is the total numbers of nodes, E is the number of edges. \(\langle k\rangle \), \(k_{\max }\) are respectively the average and the max degree. C is the average clustering coefficient. \(\alpha _{\mathrm{th}}\) is the epidemic threshold of the network
Network  N  E  〈k〉  \(k_{\max }\)  C  \(\alpha _{\mathrm{th}}\) 

egoFacebook  4039  88,234  43.69  1045  0.605  0.009 
Caltech  620  7255  43.31  248  0.443  0.012 
Princeton  5112  28,684  88.93  628  0.298  0.006 
Georgetown  7423  162,982  90.42  1235  0.268  0.006 
EmailEucore  986  25,552  33.24  347  0.399  0.013 
Powergrid  4941  6594  2.66  19  0.107  0.092 
CRQC  4158  13,428  5.53  81  0.529  0.059 
4.2 SIR simulations
4.3 Evaluation criteria
5 Experimental results
Extensive experiments have been performed in order to evaluate the effectiveness of the most popular Modular centrality extensions (Degree, Betweenness, Closeness and Eigenvector centrality) as compared to their standard definition. First, the Local and Global component of the various Modular centrality measures is compared to their standard counterpart. Next, the three ranking methods based on the combination of the components of the Modular centrality are also evaluated. These experiments are conducted on both synthetic and realworld networks.
5.1 Synthetic networks
5.1.1 Evaluation of the local and the global component of the Modular centrality
a. Welldefined community structure
In networks with welldefined community structure, the Local component of the Modular centrality always outperforms the standard measures for all the centrality measures as it is shown in the left panels of the Fig. 2 (when \(\mu =0.1\)). The gain is around 20% as compared to the standard measure for Closeness, Degree and Eigenvector centrality. The smallest gain is for Betweenness centrality with an average value of 10%. On the contrary, the Global component of the Modular centrality is always less performing than the standard measures. These results clearly demonstrate that it is more efficient to immunize the influential nodes inside the communities when there are few intercommunity links in the networks. Indeed, as there are few intercommunity edges, the infection may die out before reaching other communities. So, the local influence of nodes is more important than global influence in networks with strong community structure.
b. Community structure with medium cohesiveness
The middle panels of Fig. 2 show the performance of the various ranking methods in networks with community structure of medium strength (\(\mu =0.4\)). In this case, both the Global and Local components of the Modular centrality are always more efficient than the standard centrality. The gain in performances of the Global component of the Modular centrality is always greater than for the Local component. Indeed, the Global component outperforms the standard measure with a Gain around 12% for Betweenness, Closeness and Eigenvector centrality. The largest gain is for the Degree centrality with an average value of 17%. The Local component of the Modular centrality performs better than the standard measure with a gain around 5% for Betweenness, Closeness and Eigenvector centrality and around 12% for Degree centrality. These results send a clear message: In networks with medium community structure strength, the global influence is more important than the local influence. Indeed, with a greater number of intercommunity links, there are more options to spread the epidemics to the other communities of the network.
c. Loose community structure
The right panel of Fig. 2 reports the comparison between the Modular centrality and the traditional centrality measures in networks with noncohesive community structure (\(\mu =0.7\)). It appears that the relative difference of the outbreak size between the Global component of the Modular centrality and the standard centrality is always positive while it is always negative for the Local component of the Modular centrality. And this is true for all the centrality measures under test. In fact, there is a gain of around 5% using the Global component of the Modular centrality, while the Local component performs worse than the traditional measure with an average of 5% for the Degree, Betweenness, Closeness and Eigenvector centrality measures. Consequently, we can conclude that in networks with a loose community structure the global influence is dominant, even if the difference with the standard measure is not as important than for networks with a medium community structure. Indeed, in this situation (\(\mu =0.7\)), the intercommunity edges constitute the majority of edges in the network (around 70% of links lie between the communities). In fact, as the community structure is not well defined, minor differences are observed with a network that has no community structure.
5.1.2 Evaluation of the ranking methods of the Modular centrality
Figure 2 reports also the relative difference of the outbreak size Δr as a function of the fraction of the initial spreaders \(f_{0}\) for the three ranking methods (Modulus and Tangent of the argument of the Modular centrality, Weighted Modular measure) and for the various centrality measure and community structure strength under study. The first observation that can be made from these results is that combining the components of the Modular centrality is always more efficient than using either a single component or the conventional centrality. This remark holds for all the centrality measures studied and whatever the community structure strength. Additionally, the ranking of the three combination methods in terms of efficiency is always the same. The Weighted Modular measure ranks first. It is followed by the Modulus and then the Tangent of the argument of the Modular centrality. We believe that it is due to the fact that the Weighted Modular measure uses more information about the topology of the community structure than its alternatives. Indeed, the weights introduced in this measure allow tuning locally the relative importance of the Local and Global component for each community in the network. Thus, the Weighted Modular measure can adapt to the structure of each community in the network. As a result, it is more efficient than the other proposed ranking methods. One of the main benefits of this result is to highlight the fact that significant gains can result from improving the way the Local and Global component are combined and that there may be still room for improvement in this direction. In other words, even more effective measures can be obtained if relevant additional information about the community structure is used. Furthermore, it is noticed that the ranking strategies show their best performance in networks with a welldefined community structure. For instance, The Modulus of the Modular centrality outperforms the standard measure with a gain, on average, of 40% in networks with strong community structure, 25% in networks with community structure of medium strength and 20% in networks with unclear community structure for all the centrality measures. For the Weighted Modular measure, the gain is around 42% in networks with strong community structure, 29% in networks with community structure of medium strength and 25% in networks with unclear community structure for all the centrality measures. The gain, therefore, decreases as the community structure becomes more and more loose. The reason behind that behavior is that the Local centrality is typical of networks with a community structure while the Global centrality is also a feature of networks with no community structure. As the mixing proportion increases, the differences with networks without community structure become less and less important. Indeed, the global network size increases until it tends to represent the major part of the original network. In the limiting case, it is a network with no community structure and the Modular centrality reduces to its Global component which is identical to the classical centrality measures.
5.2 Realworld networks
The estimated mixing parameter μ and modularity Q of the realworld networks
Network  egoFacebook  Powergrid  caGrQc  Princeton 

μ  0.03  0.034  0.095  0.354 
Q  0.834  0.934  0.86  0.753 
Network  EmailEucore  Caltech  Georgetown 

μ  0.42  0.448  0.522 
Q  0.569  0.788  0.662 
5.2.1 Evaluation of the local and global component of the Modular centrality
a. Welldefined community structure
In these networks, communities are densely connected and there are few links lying between the communities. Therefore, in most cases, contagious areas are found in the core of the communities and the spread of the epidemic may stop before even reaching the community perimeter. Thus, there is a low probability that a bridge (intercommunity link) propagates the epidemic to the other communities. This is the reason why the Local component of the Modular centrality performs always better than the Global component. Furthermore, we can also notice on Fig. 3 that when the mixing parameter value increases (i.e., the community structure gets weaker), the Local component of the Modular centrality gets less efficient while the Global component performs better. This is due to the fact that the Global component increases with the number of intercommunity links.
b. Community structure with medium cohesiveness
c. Loose community structure
5.2.2 Evaluation of the ranking methods of the Modular centrality
Figures 3 to 5 report also the relative difference of the epidemic outbreak size between the Modular centrality ranking methods and the standard centrality measures. The results are clear evidence of the efficiency of the Modular centrality. Whatever the ranking strategy of the Modular centrality adopted, it outperforms in all the situations the Local and Global component of the Modular centrality and the classical centrality. The improvements in terms of performance compared to classical centrality are quite significant. For instance, with a fraction of initial spreaders equal to 8%, the modulus of the Betweenness Modular centrality allows a gain of 45% on the egoFacebook network, 28% on Princeton and 24% on Georgetown. As the ranking strategies use both the local and the global information of each node, they are more efficient than measures relying on either local or global information taken separately. Furthermore, the Weighted Modular measure is usually the most efficient measure in most cases. It uses the fraction of intercommunity links as additional information to target the most influential spreaders in each community. It can give more or less weight to the Local and the Global component according to the individual community structure strength. This explains its superiority over the other ranking measures. To summarize, these experiments reveal that combining the components of the Modular centrality, allows designing efficient ranking methods. In addition, using more relevant information about the community structure at the community level allows designing even more efficient ranking methods. Moreover, the ranking measures exhibit their best results in networks with strong community structure.
5.2.3 Comparisons with the alternative measures
Overall, the Betweennessbased centralities (Local, Global, Weighted Modular centrality, Standard Betweenness) outperforms all the other alternatives. Note that, even the standard Betweenness centrality is more efficient than the other alternatives. This result is independent of the community structure strength. In networks with a welldefined community structure (i.e., Powergrid network), one can see on Fig. 6(a) that the Eigenvectorbased centralities are just below, followed by the Degreebased centralities. The lowest performance is obtained by the Closenessbased centrality measures. The Comm and Number of Neighboring Communities exhibit a lower efficiency as compared to the three versions (Standard, Local and Weighted Modular measures) of the four previous centralities. Their performance is, however, as good as the Global measures with slightly higher performance for the NNC measure. These two methods tend to target bridge nodes that have a high global influence in the network. This is the reason why they perform at the same level as global measures. Furthermore, the Community HubBridge measure has globally the same performance as the Weighted Modular Betweenness. As this centrality incorporates both local and global influence of nodes it performs better than most of the other measures. This corroborates the fact that both dimensions must be taken into account in order to design a centrality measure in modular networks.
In networks with a loose community structure (i.e., Georgetown network), one can notice from Fig. 6(b) that the ranks of the centrality measures in terms of efficiency are different. The Betweennessbased measures still rank first, the Degreebased centralities are the second bestperforming measures followed by the Closeness and the Eigenvectorbased centrality measures. In this type of networks, the Comm and the Number of Neighboring Communities perform better than the Local and the standard centrality measures. Their performance is as good as the Global measures since they highlight also nodes with high global influence. These nodes can play a major role in the spreading process in networks with noncohesive community structure. This is due to the large amount of intercommunity links. Hence the higher performance of these two communitybased measures. Additionally, the curves of the Community HubBridge centrality are usually at the top of all the figures. Thus, it is as expected more efficient than the standard and the Modular centrality components of all the tested centralities. It has an overall similar performance that most Weighted Modular measures (Degree, Closeness, Eigenvector), except the Weighted Modular Betweenness which performs better. Both of them targets efficiently nodes with high local and global influence in the network.
5.2.4 Influence of the community detection algorithms
The estimated mixing parameter μ and the modularity Q in Powergrid and Georgetown networks
Network  Metric  Detection algorithm  

Louvain  Infomap  
Powergrid  μ  0.034  0.038 
Q  0.92  0.93  
Georgetown  μ  0.522  0.491 
Q  0.521  0.601 
In networks with a loose community structure (e.g., Georgetown network), the standard measure is always better performing than the Local measure. The Global measure, however, performs better than the Standard one with an average gain of 17% and 16% for the Degree and Betweenness centrality measures respectively. Whereas the average gain is around 11% in the case of Louvain algorithm for both centrality measures. On the other hand, the overall gain of the Weighted Modular measure is around 29% and 25% for Degree and Betweenness measures when using Infomap, while it is around 20% and 19% for both centrality measures respectively when Louvain algorithm is employed. In this network, the Infomap algorithm has a relatively smaller mixing parameter and higher modularity. Infomap is then more accurate as compared to Louvain algorithm. That explains why the performance of the Modular centrality components enhances in networks with a noncohesive community structure when the Infomap detection algorithm is used.
Globally, the results of this set of experiments show that variations of the uncovered community structure impact the performance of the centrality measures. The efficiency of the measures increases with the modularity of the community structure.
6 Conclusion
In this paper, we propose a general definition of centrality measures in networks with nonoverlapping community structure. It is based on the fact that the intracommunity and intercommunity links should be considered differently. Indeed, the intracommunity edges contribute to the diffusion in localized densely connected areas of the network, while the intercommunity links allow the global propagation to the various communities of the network. Therefore, we propose to represent the centrality of modular networks by a twodimensional vector, where the first component measures the local influence of a node in its community and the second component quantifies its global influence on the other communities. Based on this assumption, centrality measures defined for networks with nocommunity structure can be easily extended to modular networks. Considering the most influential centrality measures as typical examples, we defined their modular extension. Experiments based on an epidemic spreading scenario using both synthetic and realworld networks have been conducted in order to better understand the influence of the two components of the Modular centrality. First of all, results on synthetic and realworld networks are quite consistent. It appears that the Local component is more effective in networks with a strong community structure while the Global component takes the lead as the community structure gets weaker. Comparison with the classic centrality always turns to the advantage of the Modular centrality. More precisely, in networks with strong community structure, the Local component of the Modular centrality outperforms the Global component and the standard centrality, while in networks with medium or weak community structure the Global component performs better than its alternatives. Moreover, it is also observed that combining both components of the Modular centrality in order to rank the nodes according to their influence is always more efficient than to use a single component. Furthermore, a further gain can be obtained if the ranking strategy incorporates more information about community structure strength. We perform also a set of experiments using the Infomap detection algorithm to uncover communities. Results show that the performance of the Modular centrality variants exhibit the same behavior in networks with a welldefined community structure. Their performance, however, is different in networks with a loose community structure. In this case, slightly better results are obtained with the Infomap algorithm.
Declarations
Acknowledgements
Not applicable.
Availability of data and materials
The datasets used in this article are all publicly available and cited in the references.
Funding
Not applicable.
Authors’ contributions
All the authors contributed to designing the proposed method. ZG implemented the model and all the analyses. All authors participated in the formulation and writing of this paper. All authors approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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