Immunization of networks with community structure

In this study, an efficient method to immunize modular networks (i.e. networks with community structure) is proposed. The immunization of networks aims at fragmenting networks into small parts with a small number of removed nodes. Its applications include prevention of epidemic spreading, protection against intentional attacks on networks, and conservation of ecosystems. Although preferential immunization of hubs is efficient, good immunization strategies for modular networks have not been established. On the basis of an immunization strategy based on eigenvector centrality, we develop an analytical framework for immunizing modular networks. To this end, we quantify the contribution of each node to the connectivity in a coarse-grained network among modules. We verify the effectiveness of the proposed method by applying it to model and real networks with modular structure.


Introduction
The spread of epidemics can be considered to occur on networks that describe contacts between individuals [1][2][3].The size and dynamics of epidemics heavily depend on the structure of the contact network.In particular, in networks in which the number of contacts per individual (i.e., the degree) is heterogeneous, as represented by scale-free networks, epidemic spreading can occur on a large scale even at a small infection rate [4][5][6][7][8][9].
Given a limited dose of immunization, it is practically necessary to establish efficient immunization strategies against epidemics occurring on networks.If an appropriate ordering of immunization of nodes in a network is followed, the potential risk of large-scale epidemic spreading can be suppressed.We measure the efficiency of immunization by assessing the capability of the immunization strategy to fragment the network into small parts with a small number of sequentially removed nodes.This method of assessment has a wide applicability beyond prevention of epidemics.In ecology, it is important to identify the nodes in a food web whose removal causes the catastrophic disintegration of the food web, which would seriously damage the ecosystem [10,11].The possibility of efficient immunization of a network also implies that the network is vulnerable to intentional attacks such as those attributable to terrorism [12].
The standard solution to an immunization problem is to immunize hubs (i.e., nodes having large degrees) preferentially [8,12,13].The degree-based immunization strategy and its variants are very efficient for scale-free network models and some real data [8,[12][13][14][15][16].However, many real networks are more structured than merely having heterogeneous degree distributions.
Partly because of this factor, an immunization strategy based on a graph partition algorithm performs better than degree-based strategies [17].Immunization strategies involving the preferential removal of nodes with large betweenness centrality (betweenness-based strategies; see Sec. 3 for definition) also perform better than degree-based strategies in some networks [14,18].
Developing efficient immunization strategies for general complex networks is an unresolved question.
In this study, we focus on networks with modular structure.By definition, nodes in a network with modular structure are partitioned into multiple modules (also called communities) such that the number of links connecting the nodes in the same module is relatively large.The number of links connecting different modules is relatively small.Such networks abound in various fields [19][20][21][22].In simple cases in which modules are homogeneous and of equal size, epidemic dynamics [23][24][25] and immunization [23,26,27] have been mathematically analyzed in the limit of the infinite network size.However, in practical applications, relevant networks are finite, modules in a network are heterogeneous in various aspects [22], and nodes in a module play different roles [28][29][30].Metapopulation modeling is a promising approach to the understanding of epidemic dynamics in such modular networks [31,32].Establishing practical immunization strategies for general modular networks is an important issue.
We develop an immunization strategy for modular networks by extending an analytical framework proposed recently [33].It is our contention that it is important to consider the role of each node in the coarse-grained network among modules rather than in the original network so as to preferentially immunize nodes that bridge important modules.Some algorithms for community detection effectively solve the same problem [20,22].We believe that our method is much less computationally expensive than these methods and therefore is suitable for large modular networks.

Immunization based on dynamical importance of nodes
Consider an undirected and unweighted contact network with N nodes.Even though our results can be easily extended to the case of weighted networks, this study is confined to the immunization of unweighted networks for simplicity.An immunization strategy is an ordering of all the nodes in a network according to which the nodes are removed.The fraction of the removed nodes is set equal to 1 − p (0 ≤ p ≤ 1); the fraction of the remaining nodes is equal to p.The fraction of nodes contained in the largest connected component (LCC) is denoted by S.
In a good immunization strategy, S is small with a small number of removed nodes, i.e., with a large p.
Restrepo and colleagues proposed an immunization strategy based on the so-called dynamical importance of nodes [33].Although the dynamical importance is defined for directed networks [34], the exposition of their results in this section is concerned with the undirected version.The adjacency matrix is denoted by A; A ij = 1 when node i and node j are adjacent, and A ij = 0 otherwise.Because A is a symmetric matrix, all the eigenvalues of A are real.The largest eigenvalue of A and the corresponding eigenvector, which is called the Perron vector, are denoted by λ and u, respectively.In networks with low clustering (i.e., small density of triangles), the LCC is large (i.e., S = O(1)) if and only if λ exceeds unity [33].The Perron vector u is the mode that survives after multiplying A repeatedly to an almost arbitrary initial N-dimensional vector.Intuitively, the multiplication of A implies the spread of epidemics to the nearest neighbors.A large λ implies the efficient expansion of the LCC.
When p < 1, we generate the effective adjacency matrix from the network composed of only the remaining nodes and the links among these nodes.We apply the threshold condition to the effective adjacency matrix to determine whether the LCC is large.In this way, we can estimate the critical value of p with regard to the percolation transition.
The dynamical importance of node k, denoted by I k , is defined by the decrement of λ owing to the removal of node k.The linearized eigenequation after removing node k is expressed as where (∆A) ij = −A ij (δ ik + δ jk ) and δ is Kronecker's delta.Because the kth element of u, denoted by u k , necessarily becomes zero owing to the removal of node k, the appropriate perturbation is given by ∆u = ∆ ′ u − u k êk , where êk is the unit vector for the kth component and ∆ ′ u is an N-dimensional small vector.By inserting these expressions and Au = λu into Eq.( 1), we obtain the following equation to first order: Therefore, I k for undirected networks is equal to the square of the eigenvector centrality [35].
In the immunization strategy developed by Restrepo et al. [33], which we label as the Res strategy, we first remove the node with the largest I k .Then, we recalculate the dynamical importance of each node in the updated network to determine the second node to be removed.
We repeat this procedure.This method works efficiently in various networks [33].

Localized epidemics in modular networks
The threshold condition λ > 1 is ineffective for modular networks.To demonstrate this, consider an ad hoc modular network composed of N M homogeneous modules of equal size N/N M .A node is connected to each of the N/N M − 1 nodes in the same module with probability 1 and to each of the N − (N/N M ) nodes in the other modules with probability ǫ.A small value of ǫ implies modular structure of the network [20,22].We can approximate the adjacency matrix A by the following block-circulant matrix composed of N M × N M blocks, each of which is an (N/N M ) × (N/N M ) matrix.Let E be the (N/N M ) × (N/N M ) unit matrix, and J be the (N/N M ) × (N/N M ) matrix whose all elements are unity.The N M diagonal blocks of A are equal to J − E. If we approximate the probability that a link exists between two nodes in different modules by the weight of the link, which is not crucial for the following arguments, the N M (N M − 1) off-diagonal blocks of A are equal to ǫJ.An example network in the case of N = 8 and N M = 2 is shown in Fig. 1.
The N M leading eigenvalues of the approximated adjacency matrix are represented by where ρ is an N M th generic root of unity.Although Eq. (3) more simply indicates the existence of an (N M − 1)-fold degenerate eigenvalue −1 + (1 − ǫ)N/N M and a nondegenerate eigenvalue /N M , we use Eq.(3) for theoretical developments below.For further analysis, we fix a specific ρ.The corresponding eigenvectors are given by where ⊤ denotes the transpose.The other N − N M eigenmodes have degenerated eigenvalues −1 and are irrelevant to the percolation transition.
When ǫ is small, λ 1 , . .., λ N M are almost the same.In the limit ǫ → 0, we obtain In this limit, a proper linear summation of u i (1 ≤ i ≤ N M ) yields a localized mode represented by where a block of ones appears from the (((i − 1)N/N M ) + 1)th element to the (iN/N M )th element.Each u ′ i represents a mode that is localized in a module.According to the criterion explained in Sec.2.1, the LCC is large when any of the values λ 1 , . .., λ N M exceeds unity.When there are more than two nodes in each module (i.e., N/N M > 2), the LCC is large even in the limit ǫ → 0, because λ = N/N M − 1.However, when ǫ → 0, the LCC does not extend beyond a single module, i.e., S ≤ N/N M .When there are many modules (i.e., large N M ), the result for ǫ → 0 implies that the actual S is small.When ǫ > 0 is small, a similar relation holds true.In this case, the largest eigenvalue is not degenerated.However, for a moderate p, the LCC tends to contain a majority of nodes in a single module and does not extend beyond the module.Such a LCC is regarded to be large by the Res strategy, whereas it is actually small when N M is large.
In summary, the Res strategy applied to modular networks may be inefficient, because it does not distinguish between local and global epidemics.The same is the case for degree-based immunization strategies in which hubs are preferentially immunized.If a considerable number of hubs contribute to intramodular but not to intermodular connectivity, alternative strategies may work better.Even though we have dealt with networks with modules of equal size, the discussions above can also be applied to modular networks in which the size of modules is heterogeneous.

Module-based immunization strategy
We develop an immunization strategy that can be applied to modular networks.By definition, intermodular links are rare compared to intramodular links in a modular network.If intermodular links are preferentially removed during immunization, the modular structure will be preserved throughout the immunization procedure.Therefore, if the LCC at a certain value of p contains a considerable number of modules that are connected at this value of p, many nodes in each of such modules are likely to belong to the LCC.On this basis, for simplicity, we assume that all the nodes in each module belong to the LCC or none of them belongs to the LCC.To establish an efficient immunization strategy for modular networks, we apply the Res strategy to the coarse-grained network representing the connectivity among modules.
Given a partition of nodes into N M modules, we define an N M ×N M coarse-grained adjacency matrix Ã as where M I denotes the Ith module.The matrix Ã is weighted, and ÃIJ is equal to the number of links between M I and M J .It should be noted that ÃII is set to 0 to respect the assumption that all the nodes in a module are simultaneously included in or excluded from the LCC.Otherwise, a localized mode such as (1 0 . . .0) ⊤ may become the Perron vector of Ã, owing to which epidemics restricted to a single module cannot be ruled out.
The Perron vector ũ = (ũ 1 . . .ũN M ) ⊤ of Ã is determined by Ã ũ = λ ũ, where λ is the largest eigenvalue of Ã. ũi represents the importance of the ith module in terms of the eigenvector centrality.
We calculate the shift in λ, denoted by ∆ λ, owing to the removal of a single node k.We denote the index of the module that node k belongs to by K.The removal of node k elicits a change in the coarse-grained adjacency matrix by where d kI is the number of intermodular links that exist between node k and module M I , i.e., It should be noted that ∆ ÃKK = 0.
To calculate ∆ λ, it is necessary to evaluate the amount of perturbation in ũ owing to the node removal.Generally, ũK is perturbed by an amount larger than ũI (I = K) because only the elements of Ã in the Kth row or those in the Kth column can decrease after node k is removed.However, as opposed to the formulation of the Res strategy (Sec.2.1), the removal of node k does not result in ũK = 0, unless node k is the only node contained in M K .Although this situation occurs after some nodes have been removed, it is not very common except near the percolation threshold.Therefore, we assume that the node removal changes the Perron where ∆ ′ ũ is a small vector.We determine x as follows.The Kth linear equations for the Perron vector before and after the removal of node k are represented by and respectively.By combining these equations and neglecting small-order terms ∆ ′ ũI ≪ ũI (1 ≤ I ≤ N M ) and ∆ λ ≪ λ, we obtain If node k is the last node in M K that is removed at a certain value of p, Eq. ( 12) becomes x = ũK .This relation is consistent with the fact that ũK vanishes after the removal of node k.
When there are many nodes in module K, Eqs. ( 9) and ( 12) imply x = ũK Therefore, the contribution of the node removal to ∆ λ is attributed to two factors: the importance of the module that node k belongs to (i.e., 2ũ K − x ≈ 2ũ K ) and the connectivity of node k to other important modules (i.e., I =K d kI ũI ).As the other extreme to the case described above, we consider the situation in which node k is the only node that constitutes module K.By substituting d kI = ÃKI (1 ≤ I ≤ N M ) in Eqs. ( 9), (12), and (13), we have ∆ λ = ũ2 K / I ũ2 I ; the Res strategy is reproduced.In other words, the Mod strategy is equivalent to the Res strategy when all the nodes form isolated modules.
To apply the Mod strategy to real data, we first partition the network into N M modules.
Then, we calculate ũI (1 ≤ I ≤ N M ) by the power method.This operation is fast unless the spectral gap of Ã is too small and N M is too large.The power method produces λ as a byproduct; this value is used in Eq. (12).Then, we remove the node that realizes the maximum (2ũ K − x) I =K d kI ũI .Next, we repeat this procedure.To save computation time, we do not apply a module detection algorithm in each step.On the basis of the modular structure determined for the original network, we recalculate ũI and remove the nodes one at a time.If all the modules are isolated, we sequentially remove the nodes in the descending order of d kK .
We recalculate d kK of all the remaining nodes after the removal of each node.This part of the Mod strategy is heuristic and can be replaced by other immunization strategies.

Results
We compare the efficiency of the Mod strategy on various networks with those of other immunization strategies.
To detect modules in networks, we apply either the greedy algorithm proposed by Clauset and colleagues [36,37] that approximately maximizes the modularity of a network, the fast heuristic algorithm to the same end proposed by Blondel and colleagues [38], or the algorithm based on random walks proposed by Rosvall and Bergstrom [39].For all the examined data sets, Blondel's and Rosvall's algorithms identify the smallest and the largest number of modules among the three algorithms, respectively (Tab.1).We call the Mod strategy combined with the community detection algorithms of Clauset, Blondel, and Rosvall as the Mod-C, Mod-B, and Mod-R strategies, respectively.
We compare the efficiency of the Mod strategy with the following immunization strategies.
• Degree-based (D) strategy: We remove the nodes in decreasing order of their degree in the original network.If there exists more than one node with the same degree, we select one of them with equal probability.
• Recalculated degree-based (RD) strategy: We sequentially remove the nodes with the largest degree.This strategy differs from the D strategy in that we recalculate the degrees of all the remaining nodes after removing each node.
• Betweenness-based (B) strategy: We remove the nodes in decreasing order of the betweenness centrality.The betweenness centrality of a node is the normalized number of shortest paths between node pairs that pass through the node [2,19,40].
• Recalculated betweenness-based (RB) strategy: We sequentially remove the nodes with the largest betweenness centrality.We recalculate the betweenness centralities of all the remaining nodes after removing each node.
If, in any strategy, there are multiple nodes that realize the maximum value of the relevant quantity, we select one of these nodes with equal probability.
Because the B strategy performs poorly compared to other strategies in all the networks described in the following sections, we do not show the numerical results of this strategy.
Although the D strategy performs worse than the RD strategy (and many other strategies) in most cases, we present the results obtained from the D strategy because it is a typical strategy [8,12,13].While efficiencies of the D, RD, B, and RB strategies are were compared in a previous study for some networks [14], we examine these strategies with regard to modular networks.

Results for model networks
Our methods do not improve upon the previous methods for networks without modular structure.To verify this, we generate a scale-free network with N = 5000 and the degree distribution p(k) ∝ k −3 using the Barabási-Albert (BA) model [41].We set k ≈ 12 by setting the parameters m 0 and m of the BA model to 6 [41].Major statistics for the generated BA model are listed in Tab. 1.The relative size of the LCC is plotted against the node occupation probability p in Fig. 2(a).If S is very small for a large value of p, an immunization strategy is considered to be efficient.The Mod-C, Mod-B, and Mod-R strategies are as efficient as the D strategy.
These three strategies are superceded by the RD, Res, and RB strategies, as expected.
The inefficiency of the Mod strategies is presumably caused by the lack of the modular structure in the BA model.In general, a large Q-value indicates the presence of modular structure in a network [21,22,36] (but see [42]).The Q-values of this network determined by the three community detection algorithms are equal to 0.249 (Clauset), 0.258 (Blondel), and 0.184 (Rosvall) and are considered to be small.For a systematic comparison, we compare these Q-values with those of the networks generated by random rewiring of edges with the degree of each node preserved.The generated networks do not have particular structure except that the degrees are heterogeneous.The Q-values of the rewired networks are almost the same as those of the BA model (Tab.1), which indicates the absence of modular structure in the BA model.
Next, we apply the Mod strategy to ad hoc networks with modular structure.There are various algorithms that produce benchmark networks with modular structure [20,22].We generate two networks as follows.The following numerical results do not critically depend on the method of construction of the modular network.
Consider N M modules of the same size N/N M .In the first ad hoc network, a module is the Erdős-Rényi random graph with the connection probability p ℓ = k ℓ /(N/N M − 1), such that the mean degree within a module is equal to k ℓ = 8.Then, we generate the coarse-grained network among N M modules in the form of the random graph with a mean degree of 6.Any pair of node i in module M I and node j in another module M J (J = I) may be connected if M I and M J are connected in the coarse-grained network.When this is the case, we connect nodes i and j with probability k g /(6N/N M ).Then, for each node, the expected number of neighbors in different modules is equal to k g = 1.We set N = 5000 and N M = 25 and run the algorithm until we obtain a connected network.The mean degree of the generated network The results for different immunization strategies are compared in Fig. 2  modules; if we remove links in the decreasing order of the recalculated betweenness centrality of the links, the network is partitioned into modules efficiently [20].The drawback of the RB strategy with respect to the Mod strategy is the former's high computation time; we cannot apply the RB strategy to larger networks.We discuss this point in Sec. 4. We also carry out numerical simulations on a heterogeneous ad hoc modular network.We generate each module using the BA model with k ℓ ≈ 8 (i.e., m = m 0 = 4).The coarsegrained network among modules is assumed to be the BA model with a mean degree of 6 (i.e., m = m 0 = 3).Pairs of nodes in different modules are connected in the same way as in the previous network, such that k g = 1.We set N = 5000 and N M = 100.The mean degree of the generated network k = 8.59 ≈ k ℓ + k g .The generated network is a connected network.The immunization results for this modular scale-free network are shown in Fig. 2(c).
The results are qualitatively the same as those in Fig. 2(b).

Results for real-world networks
We investigate the application of the Mod strategy to four real-world networks.The statistics for each network including the number of modules and the Q-values are listed in Tab. 1.The first example is a high energy particle (HEP) citation network [43].We use this network as a representative of a relatively dense network.This network is used in a previous study of immunization [17].Because of its large mean degree, a relatively large fraction of nodes have to be removed to fragment this network.The Q-value for the partition using the three algorithms are large.They are also much larger than the Q-values for the networks generated by rewiring the edges without changing the degree of each node.Therefore, the HEP network has major modular structure.
Note that the rewiring sometimes makes the network disconnected.However, the Q-value does not differ much between connected and disconnected rewired networks.Therefore, we do not explore the effect of disconnectedness of the rewired networks.We do the same omission for the three other real-world networks examined later.
The immunization results for the HEP network are shown in Fig. 3(a).The results for the RB strategy are not shown because N is too large for us to employ the RB strategy.This limitation with regard to the RB strategy is also true for the three other networks.It can be observed from Fig. 3(a) that the Mod-R strategy outperforms all the other strategies including the Res strategy.The improvement obtained by employing the Mod-R strategy, which is quantified by the amount of shift of the percolation threshold is approximately as large as that obtained from the recently proposed strategy using graph partitioning [17].This strategy [17] divides the network into equal-sized groups; it is distinct from the Mod strategy.
The LCC for the Mod-C strategy is small when p is large.However, below p ≈ 0.76, S decreases slowly with a decrease in p.At p ≈ 0.76, all the modules are already separated.The LCC for the Mod-C strategy occupies a significant fraction of the original network at p ≈ 0.76 and is represented by the largest module in the network.Because we have not optimized the Mod strategy after all the modules are separated, the Mod-C strategy does not perform well below p ≈ 0.76.The Mod-B strategy yields a similar result; below p ≈ 0.69, the LCC is the largest module in the network.However, the LCC is smaller than that for the Mod-C strategy because the size of the largest module detected by Blondel's algorithm is smaller than that detected by Clauset's algorithm.The performance of the Mod strategies can be enhanced if we improve the immunization strategy after all the modules are separated.However, we do not explore this aspect in the present study.
The second example is a social network called the Pretty Good Privacy (PGP) network [44].
A link is formed when two persons share confidential information using the PGP encryption algorithm on the Internet.This network has a prominent community structure (see Tab. 1 for the Q-values).The immunization results are shown in Fig. 3(b).For this network, the Mod-C, Mod-B, and Mod-R strategies outperform the D, RD, and Res strategies.
The third example is the LCC of a dataset of the World Wide Web [45].We ignore the direction of the links.The numerical results for this LCC are shown in Fig. 3(c).The Mod-C, Mod-B, and Mod-R strategies perform better than the D, RD, and Res strategies at least in terms of the percolation threshold.
The fourth example is an email-based social network [46].The results shown in Fig. 3(d) indicate that, for this network, the Mod-C, Mod-B, and Mod-R strategies do not outperform the other strategies.The performance of the Mod-R strategy is superior to those of the other methods near the percolation threshold, but this superiority is only marginal.The performance of the Mod-C strategy is inferior to those of the other methods over the entire range of p.The Mod-B and Mod-R strategies are more inefficient than the D, RD, and Res strategies when p is large.
The three community detection algorithms result in large Q-values for the email social network.However, this network may not be as modular as indicated by the large Q-values for two reasons.First, the rewired networks also have relatively large Q-values, although they are significantly smaller than the Q-values for the original network (Tab.1).Second, generally speaking, networks with small mean degree tend to have large Q-values even if the modular structure is absent [42].The email social network may not have sufficient modular structure, which may have caused the inefficiency of the Mod strategy for this network.

Discussion and Conclusions
We have proposed an efficient algorithm called the Mod strategy for immunizing networks with modular structure.This strategy combines a community detection algorithm and the identification of nodes with crucial intermodular links.We have validated the effectiveness of the Mod strategy with artificial and real networks using two community detection algorithms.
The Mod strategy is applicable to networks in which the size of modules is heterogeneous, as is the case in real modular networks [22].
The Mod strategy can be extended to the case of networks with more than two hierarchical levels, which are often found in real data [47,48].In such a network, we first remove the nodes responsible for the formation of the most global connection.If modules at the most global level have been fragmented, we apply the community detection algorithm to each module such that the nodes responsible for connecting different submodules in a module are preferentially removed.
For networks with bipartite modular structure [22], the Mod strategy is inefficient.This is because the Mod strategy is based on the conventional concept of modular structure, i.e., there are relatively more links within a module than across different modules.This property is not satisfied by networks with bipartite modular structure.Dealing with bipartite modular structure and also overlapping modular structure (see [22] for a review) is beyond the scope of the present paper.
The Mod strategy does not outperform the RB strategy.This is as expected because the RB strategy provides a useful algorithm for community detection [20].The heart of the algorithm lies in fragmenting a network into modules with a small number of links (not nodes) that are removed in the decreasing order of the betweenness centrality.However, carrying out community detection on the basis of the RB strategy [20] is computationally formidable; this strategy requires O(N 3 ) time for sparse networks.This fact has led to the development of faster algorithms for community detection that are independent of the recalculated betweenness centrality [21,22,[36][37][38].The RB strategy of immunization also requires O(N 3 ) time.
An immunization strategy developed by the adaptation of a faster community detection algorithm that sequentially removes links (so-called divisive algorithms) would outperform the Mod strategies examined in the present study (i.e., Mod-C, Mod-B, and Mod-R).However, such a community detection algorithm seems to be unknown [22].We state that the Mod strategy outperforms the RB strategy when the network is large.We have implicitly used fast community detection algorithms so that the Mod strategy performs faster than the RB strategy.For sparse networks, Rosvall's algorithm runs comfortably fast.Clauset's algorithm runs faster than Rosvall's algorithm on our data and requires only O(N log 2 N) time [37].Blondel's algorithm is even faster in general [38].
In the so-called out-of-the-neighborhood (OUT) immunization strategies [16], one picks a neighbor of node that has largest degrees out of the neighborhood of the original node.This is an efficient immunization strategy that uses only the local information about the network.
The ring vaccination [15] stands on a similar spirit.In contrast to these strategies, the Mod strategy has an important limitation that one needs global information about the connectivity among modules.Nevertheless, the Mod and OUT strategies are complementary with regard to the information needed for implementation.The Mod strategy requires coarse but global information about the network, plus the degree of each node.The OUT strategies require only the local information about the network, but with the information about the degree of the neighbors included.
Our results are consistent with the finding that nodes in a network can be classified according to their global and local roles [28][29][30].This is particularly true when the betweenness centrality is not predicted from the degree [28], which is typical for modular networks.The deviation of the global importance of a node from the local importance of the same node in modular networks is also reported for the PageRank and other similar centrality measures [49,50].
In this situation, the Mod strategy preferentially immunizes globally important nodes having important intermodular links rather than locally important ones such as local hubs.The general idea of targeting globally important nodes in modular networks has potential applications in other dynamical phenomena on networks, such as epidemic dynamics, synchronization, opinion formation, and traffic.
(b).The results labeled as Mod in Fig. 2(b) are based on the predefined modular structure with the number of modules N M = 25, because all the three algorithms for community detection identify the correct modular structure.Figure 2(b) indicates that the Mod strategy substantially outperforms the Res strategy.This is presumably because the Res strategy detects LCCs contained in a single module or a small number of modules as a signature of a global epidemic, as discussed in Sec.2.2, whereas the Mod strategy does not.

Figure 2 (
Figure 2(b) indicates that the RB strategy outperforms the Mod strategy.This is as expected because a link version of the RB strategy is used to partition the network efficiently into

1 ε 1 Figure 1 :Figure 2 :
Figure 1: Approximated modular network with N = 8 and N M = 2.The solid and dotted lines represent links with weights 1 and ǫ, respectively.

Table 1 :
Statistics of networks.The number of nodes and links are those of the LCC of the network.N M is the number of modules detected by each algorithm.For the rewired networks, the average and the standard deviation of the Q-values are shown for each community detection algorithm.To this end, we generate 100 rewired networks from each original network.± 0.001 ± 0.001 ± 0.000 ± 0.000