Modularity affects the robustness of scale-free model and real-world social networks under betweenness and degree-based node attack
Quang Nguyen, Tuan Van Vu, Hanh Duyen Dinh, Davide Cassi, Francesco Scotognella, Roberto Alfieri, Michele Bellingeri
11 Modularity affects the robustness of scale-free model and real-world social networks under betweenness and degree-based node attack
Q. Nguyen a,b , T.V. Vu a , H.-D. Dinh c , D. Cassi d , F. Scotognella e , R. Alfieri d , M. Bellingeri d,e a Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam b Faculty of Finance and Banking, Ton Duc Thang University, Ho Chi Minh City, Vietnam c John Von Neumann Institute, Vietnam National University Ho chi minh City, Ho chi minh City, Vietnam d Dip. Scienze Matematiche, Fisiche e Informatiche, Università di Parma, Parco Area delle Scienze, 7/A, 43124 Parma e Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milano, Italy Corresponding author email address: [email protected]
Abstract
In this paper we investigate how the modularity of model and real-world social networks affect their robustness and the efficacy of node attack (removal) strategies based on node degree (ID) and node betweenness (IB). We build Barabasi-Albert model networks with different modularity by a new ad hoc algorithm that rewire links forming networks with community structure. We traced the network robustness using the largest connected component (
LCC ). We find that higher level of modularity decreases the model network robustness under both attack strategies, i.e. model network with higher community structure showed faster
LCC disruption when subjected to node removal. Very interesting, we find that when model networks showed non-modular structure or low modularity, the degree-based (ID) is more effective than the betweenness-based node attack strategy (IB). Conversely, in the case the model network present higher modularity, the IB strategies becomes clearly the most effective to fragment the
LCC . Last, we investigated how the modularity of the network structure evaluated by the modularity indicator ( Q ) affect the robustness and the efficacy of the attack strategies in 12 real-world social networks. We found that the modularity Q is negatively correlated with the robustness of the real-world social networks under IB node attack strategy ( p -value< 0.001). This result indicates how real-world networks with higher modularity (i.e. with higher community structure) may be more fragile to betwenness-based node attack. The results presented in this paper unveil the role of modularity and community structure for the robustness of networks and may be useful to select the best node attack strategies in network. Introduction
The study of real-world complex networks has attracted much attention in recent decades because a large number of complex systems in the real world can be considered as complex networks, such as social (Borgatti et al. 2009, Bellingeri et al. 2020), technological (Albert et al. 1999, Faloutsos et al. 1999), biological (Jeong et al. 2000, Barra et al. 2010), ecological complex systems (Bellingeri and Bodini 2013; Bellingeri and Vincenzi 2013). Many real-world networks show a scale-free structure, making them resilient to random node failure (Cohen et al. 2000) but can disintegrate quickly when a small proportion of important nodes are removed (Albert et al. 1999). The network’s robustness, which evaluates the capability of network to hold its functioning under such failures or attacks has drawn extensive attention in recent years (Albert and Barabási 2002; Cohen et al. 2000; Callaway et al. 2000; Iyer et al. 2013; Bellingeri et al. 2015; Bellingeri et al. 2014; Dall’Asta et al. 2006; Nguyen and Nguyen 2018; Wandelt et al. 2018; Bellingeri et al. 2019;2020). Usually, Monte-Carlo simulation is used to evaluate the network robustness: for random failure, nodes/edges are removed with the same probability (random removal), while for intentional attack, nodes/edges are removed according to different structural properties of the network and a robustness measure is then computed during the node/edge removal simulation (Albert et al. 2000; Cohen et al. 2000, 2001; Bellingeri et al. 2020; Lekha and Balakrishnan 2020). To identify the node/edge removal strategy that triggers the greatest amount of damage in the system is also highly important for revealing the links/nodes that act as key players in network functioning with many practical applications (Bellingeri et al. 2020). For example, the understanding of how the node/edge removal affects real social systems may predict how the abandoning of individuals affects the information spread in the social network, thus individuating the “influential spreaders” in the network, such as most important scholars or influencers (Ahajjam and Badir 2018; Bellingeri et al. 2020). On the other hand, in social contact network on which a disease can spread, it is critical to understand how node removal through vaccination affects the spread of the disease to efficiently prevent an epidemic (Holme 2004; Wang et al. 2015; Bellingeri et al. 2020). One of the most important measure of network robustness is the size of the largest connected component (
LCC ), i.e. the
LCC is the highest number of connected nodes in the network (Albert et al. 2000). The
LCC gives us a simple interpretation of the system robustness when subjected to node/edge removal accounting the largest functioning part of the network. For example, if the Internet is attacked, all nodes (servers) within the
LCC can still transfer information mutually and indicating the largest networked structure still active. Another example, in a social contact network, the
LCC represents the highest number of individuals that can be affected by a disease spreading (Bellingeri et al. 2019). For this reason, the most efficient node attack strategy is the one that is able to induce the fastest
LCC decrease (Figure 1), and numerical simulations have shown that attack strategies based on network’s nodes centrality measures can effectively individuating the most important nodes to reduce the size of the
LCC (Albert et al. 2000, Cohen et al. 2000, 2001, Callaway et al. 2000; Iyer et al. 2013; Bellingeri et al 2018; Bellingeri et al. 2014; Nguyen and Nguyen 2018; Wandelt et al. 2018). In specific, overall findings showed that nodes attack strategies based on betweenness centrality are highly efficient to dismantle the
LCC (Iyer et al. 2013, Bellingeri et al. 2014; Sun et al. 2017, Nguyen and Nguyen 2018; Wandelt et al. 2018), especially for real-world networks. However, the difference in the effectiveness varied considerably among different real-world networks (Iyer et al. 2013; Bellingeri et al. 2014; Wandelt et al. 2020).
Figure 1 : Schematic behavior of the size of the
LCC as a function of the proportion of remaining nodes p during a node removal process: A) LCC shows a continuous 2 nd order decreases without abrupt decrease and B) LCC is subjected to first-order percolation phase transition (Achilioptas et al. 2009) showing an abrupt decrease in correspondence of p = p c . The node attack strategy in B is able to dismantle B) p L CC p c A) p L CC p c the network with a small proportion of removed nodes, i.e. is the most effective to decrease the LCC , thus individuating the most important nodes in network.
The mechanism that gives rise to such an abrupt decrease is studied using percolation theory and is assigned to the first-order percolation phase transition (Achilioptas et al. 2009, Riordan et al. 2011, Cho et al. 2013). However, the question whether such an abrupt decrease occurs for a certain real network under attack remain unclear. This question is of great importance from two aspects: on one hand, if we want to break a network using node removal, we would find strategies that remove nodes that can cause such abrupt and fast decrease in
LCC ’s size. On the other hand, if we want to protect a network, we must design it in a way that such abrupt decrease should not happen. Since the network robustness must depend on its topology, several studies have investigated the relationship between topological metrics and the robustness of a network. Iyer et al. (2013) studied robustness of model networks with power-law and exponential degree distribution, with various node clustering coefficient (or node transitivity) level. They found that increasing the clustering coefficient of the network nodes results in decreasing robustness to node attack with the most dramatic effect being displayed for node attack based on their degree and betweenness. The authors also suggested for increasing the robustness, it is necessary to design topological structures with low clustering coefficient as is consistent with the functional requirements of the network. Their simulation on real-world networks also show that the difference in the effectiveness between strategies varied across networks. Nguyen and Trang (2019) studied the Facebook social networks and found those networks with higher modularity Q have lower robustness to node removal. The modularity indicator Q introduced by Newman and Girvan (2004) measures how well a network breaks into communities, (i.e. a community or module in a network is a well-connected group of nodes which have sparser connections with the nodes outside the group). Networks with high modularity Q have dense connections (more links) among the nodes within modules but sparse connections (few links) among nodes from different modules. Therefore, the modularity Q is higher in networks with marked community structure, which are called modular networks (Girvan and Newman 2002). Using percolation theory, Dong et al. (2018) pointed out that in a modular network, a small fraction of nodes that connect nodes of different modules, called ‘interconnected nodes’, is critical to the robustness of the network. By analyzing the LCC size during node removal process by varying the fraction of interconnected nodes ( r ) in the network, they found that LCC scale with r by a power-law with universal criticality. This result suggests that modular network with higher fraction of interconnected nodes (therefore low modularity Q because the fraction of links between nodes in the same modules is lower) will result in a lower LCC decrease during node removal and consequently higher network robustness. Shai et al. (2015) developed both analytical and simulation analyses for evaluating the robustness of random and scale-free model networks with modular structure (Shai et al. 2015). They simulate the attack of interconnected nodes, i.e. nodes that connect to neighbors that are in other modules, and analyze the critical node occupation probability p c , i.e. the fraction of remaining p when a large decrease in LCC occurs, as a function of the number of modules m and the ratio between probabilities for an intra- and inter-module link α . They found that percolation phase transition falls into two regimes depending on the number of modules m for a fixed α: - For m < m ∗ the network presents very high modularity and collapses abruptly under node removal (i.e. 1 st order phase transition) as a result of the modules becoming disconnected from each another, while their internal structure is almost unaffected. - In contrast, for m > m ∗ , the network presents low modularity and the interconnected nodes play an important role to maintain the whole network connected when nodes are removed. Therefore, the node attack causes lower damage breaking continuously the entire system without sharp LCC decrease (i.e. 2 nd order phase transition). Put another way, m ∗ represents the threshold above which the network modular structure vanishes and the network returns to behaving as a non-modular network. In this work, we analyze how the modularity of scale-free model and real-world social networks affects their robustness and the efficacy of the node attack strategies. Using model network, we vary the level of modularity by changing the ratio of intra-modules links over inter-modules links ( κ ). We found that the attack strategy based on node betweenness, which was found to be the most effective strategy to break the LCC of real-world networks (Wandelt et al. 2018; Nguyen et al. 2019), is the best strategy to disrupt the
LCC only when κ is higher than a given value κ c , i.e. when the network has high modularity. Below, when network has low modularity Q , or even no modular structure, the attack strategy based on node degree is more effective. In addition, the type of the network percolation phase transition when nodes are removed change from a continuous 2 nd order (in which LCC has no abrupt decrease) to an abrupt 1 st order transition (with abrupt LCC decrease) when κ increases. We also examine the effect of network’s density (i.e. the average number of links per node) and the number of modules on network robustness and found that those parameters affect the network robustness, but not the type of the network percolation phase transition (1 st or 2 nd order) which only depends on κ . Finally, we study those effects for a variety of real social networks and we found that real social networks with higher modularity Q are less robust when subjected to the attack strategy based on nodes betweenness. In other words, the efficacy of the attack strategy based on nodes betweenness is higher for real social networks showing higher modularity Q . Methods
A network can be represented as a graph G = ( V , E ), where V = {1,2,..., N } is the set of N nodes (vertices), and E = { e ij | i , j ∈ V , i ≠ j } is the set of E links (edges). Networks can be undirected when the links have no specified direction, or directed, in the case links present directionality. Network are unweighted when only the presence-absence of the links is considered, or weighted, in the case some interaction value is associated to the link, i.e. the link weight. Undirected and unweighted networks can be abstracted by an adjacency an NxN matrix A where element a i,j =1 when there is a link between node i and j and a i,j =0 otherwise. In this paper, only undirected and unweighted networks are considered. Generation of model scale-free network Model scale-free networks with size of N = 10000 nodes are generated using the well-known Barabási- Albert (BA) model (Barabasi and Albert 1999). The BA model starts from a small clique (a completely connected graph) of N nodes. At each successive time step, a new node is added and connected to M different existing nodes ( M < N ) with the probability of connect an existing node is proportional to its degree (i.e the number of links to the node). The network then has a power-law degree distribution P(k)=k -γ with degree exponent γ = 3 (Barabasi and Albert 1999). We chose the average node degree
𝑁〈𝑘〉 and 𝑁〈𝑘〉 , respectively. - After the rewiring process, the number of inter-modules links become (1 −𝑤) (𝑚−1)𝑚
𝑁〈𝑘〉 and the number of intra-modules links become ( + 𝑤 (𝑚−1)𝑚 )𝑁〈𝑘〉 - The ratio between the number of intra-module links ( L intra ) and inter-modules links ( L inter ) become 𝜅 = 𝐿 𝑖𝑛𝑡𝑟𝑎 𝐿 𝑖𝑛𝑡𝑒𝑟 = 1 + 𝑤(𝑚 − 1)(1 − 𝑤)(𝑚 − 1) = 𝑚(1 − 𝑤)(𝑚 − 1) − 1 which is a monotone function of w when m > 1. - We derive α , the ratio between the probability for a given link to be intra-link ( p intra ) over that for a given link to be inter-link ( p inter ) as in (Shai et al 2015) by: 𝛼 = 𝑝 𝑖𝑛𝑡𝑟𝑎 𝑝 𝑖𝑛𝑡𝑒𝑟 ~ (𝑚 − 1) 𝐿 𝑖𝑛𝑡𝑟𝑎 𝐿 𝑖𝑛𝑡𝑒𝑟 = 𝑚(1 − 𝑤) − (𝑚 − 1) which is also a monotone function of w when m > 1. The monotone change of κ and α as function of w was confirmed with simulation results which are shown in Appendix B. Thus increasing w , we increase the modularity of the network, i.e. increasing w we marked the network community structure. Figure 2 presents example of modular network with m = 5 and different value w , created from the initial network with N = 10000 and the average degree of
1 ( ) ( , )2 2 i jij i ji j k kQ a c cL L where L is the number of links, a ij is the element of the A adjacency matrix in row i and column j , k i is the degree of i , k j is the degree of j , c i is the module (or community) of i , c j that of j , the sum goes over all i and j pairs of nodes, and δ (x, y) is 1 if x = y and 0 otherwise (Clauset et al. 2004). - LCC : the largest connected component (also called ‘giant cluster’) represents the maximum number of connected nodes in the network (Boccaletti et al. 2006; Bellingeri et al. 2020). Considering all the network clusters, i.e. the sub-networks of connected nodes, the LCC can be defined: max ( ) j j LCC S
1 where S j is the size (number of nodes) of the j -th cluster. - Diameter: the diameter of the network ( D ) is the longest shortest path length of all pairs of nodes in the network, also called the longest geodesic (Newman 2013) - Transitivity: the transitivity ( C ) is based on triplets of nodes. A triplet is three nodes that are connected by either two (open triplet) or three (closed triplet) undirected edges. The transitivity is the number of closed triplets (or 3 x triangles) over the total number of triplets (both open and closed). In formula: closedtotal C where λ closed is the number of closed triples and λ total is the number of all possible triples in the network. Transitivity represents the overall probability for the network to have adjacent nodes interconnected, thus making more tightly connected modules (Newman et al. 2002) - Link Density: the link density (Density) is number of links divided by the total number of possible links (Boccaletti et al. 2006).
Networks
N L
LCC LCC(%) < k > D C Density Q TV Shows
Politician
Government
Public Figures
Athletes
Company
New sites
Artist
SP500_1
315 8,706 315 100% 27.6 6.0 0.511 0.08802 0.253
SP500_2
371 10,636 369 99% 28.7 6.0 0.718 0.07748 0.373
NetScience
Email_EU Table 1 : Structural statistics of the real-world social networks: nodes ( N ), links ( L ), size of the LCC , size of the
LCC as % with respect the total number of network nodes, average node degree < k >, diameter, transitivity, the edge density, modularity Q . Results
Robustness of non-modular scale-free network We simulate scale-free network of size N = 10000 nodes and average degree
LCC shrinks to quasi zero) at a critical occupation probability p c (0.62 and 0.56 for ID and IB, respectively), as seen in Figure 3A. At this transition, we also found that the 2 nd LCC has its maximum value as shown in Figure 3B. Such phase transitions are called ‘continuous phase transitions’ or ‘second-order phase transitions’ and denote robust network (see Mnyukh 2013). Interestingly, while overall findings showed that nodes attack strategies based on betweenness centrality are highly efficient for most real-world networks (Bellingeri et al. 2014; Iyer et al, 2013; Nguyen and Nguyen 2018; Wandelt et al. 2018), our results shown different conclusion. For scale-free network without modular structures (lower value of parameter w ), betweenness-based strategy IB does not perform better than the degree-based strategy ID. For scale-free network with significant modular structures (higher value of parameter w ), betweenness-based strategy IB clearly performs better than the degree-based strategy ID. It is therefore arguable that the presence of modular structure in networks is an important factor enhancing the efficacy of betweenneess-based attack strategy for breaking the 1 st LCC , as shown in the next sub-section. 3
Figure 3 : Simulation result for the non-modular network with N = 10000 nodes and average degree < k > = 4: A) Size of the first largest connected component (1 st LCC ) and B) the second largest connected component (2 nd LCC ) as a function of the occupation probability p . Robustness of modular scale-free network
We first present the robustness of the network of different modularity by varying the re-wiring ratio w , then we discuss the robustness of the network with different node average degree
In this work we study the robustness of model scale-free networks and real-world social networks with different modularity. The model networks are generated from BA model with a novel method for tuning their modular structure. Using Monte-Carlo simulation we simulate two node attack strategies, IB and ID based on node’s betweenness and degree, respectively. With both attack strategies, we found two types of percolation phase transitions take place. The 1 st order abrupt phase transition happens when the model network has high modularity, representing by κ > κ c with κ c ~ 23.8 for both IB and ID attack strategies. Also at and above this critical point, the model network is more fragile under betweenness-based strategy attack: R IB < R ID , as found in many real-world complex networks. When κ < κ c or when the model network has no modular structure, the network experiences a continuous 2 nd order phase transition under both nodes attack strategies. Interestingly, under this regime, the network is more robust against the betweenness-based attack strategy IB than the degree-based attack strategy ID, contrary to most of results on real-world networks. In addition, our work showed that the ratio κ is the main factor for the type of phase transition: small κ corresponds to 2 nd order continuous phase transition while high κ corresponds to 1 st order abrupt phase transition. Further, we investigate how the modularity affect the robustness of the system against node removal in 12 real-world social networks and find a similar R IB decrease with modularity Q ( p -value< 0.001) that we observe in model networks varying the modularity. This result indicate how network with higher modularity (i.e. with higher community structure) may be more fragile to betwenness based node attack. At the same, this result show how the betwenness based node attack (IB) is highly effective when attacking network with marked community structure (higher modularity Q ). Differently, in the case the network shows very low modularity (or no modularity), the degree-based node attack ID may perform better than IB. 1 This result helps to understand the role of modularity and community structure for the robustness of networks, to select most effective node removal in networks, and may shed light to the design of robust networks. References
Achlioptas D, D’souza RM, Spencer J (2009) Explosive percolation in random networks. Science 323(5920):1453–1455. Ahajjam S, Badir H. Identification of influential spreaders in complex networks using HybridRank algorithm. Sci Rep. (2018) 8:1–10. doi: 10.1038/s41598-018-30310-2 R. Albert, H. Jeong, A.-L. Barabasi, Nature 401 (1999) 130. R. Albert, H. Jeong, A.-L. Barabasi, Error and attack tolerance of complex networks, Nature 406 (2000) 378. Braunstein, A., Dall’Asta, L., Semerjian, G. & Zdeborová, L. Network dismantling. Proc. Natl. Acad. Sci. 201605083 (2016). A.-L. Barabási, R. Albert, Emergence of scaling in random networks, Science 286 (1999) 509–512. A. Barra, E. Agliari, Stochastic dynamics for idiotypic immune networks, Physica A 389 (2010) 5903–5911. M Bellingeri, S Vincenzi (2013), Robustness of empirical food webs with varying consumer's sensitivities to loss of resources, Journal of theoretical biology 333, 18-26 M Bellingeri, A Bodini (2013), Threshold extinction in food webs, Theoretical ecology 6 (2), 143-152 Bellingeri, M., Cassi, D. & Vincenzi, S. Efficiency of attack strategies on complex model and real-world networks. Phys. A Stat. Mech. its Appl. 414, 174–180 (2014). Bellingeri, M., Agliari, E. & Cassi, D. Optimization strategies with resource scarcity: from immunization of networks to the traveling salesman problem. Mod. Phys. Lett. B (2015). 2 Bellingeri M, Bevacqua D, Scotognella F, Alfieri R, Nguyen Q, Montepietra D and Cassi D (2020) Link and Node Removal in Real Social Networks: A Review. Front. Phys. 8:228. doi: 10.3389/fphy.2020.00228 M Bellingeri, D Bevacqua, F Scotognella, R Alfieri, D Cassi (2020), A comparative analysis of link removal strategies in real complex weighted networks, Scientific reports 10 (1), 1-15 Bellingeri, M., D. Bevacqua, F. Scotognella, and D. Cassi. (2019a). The heterogeneity in link weights may decrease the robustness of real-world complex weighted networks; Scientific Reportsvolume 9, Article number: 10692 (2019a) M. Bellingeri, D. Cassi, Robustness of weighted networks, Physica A 489 (2018) 47–55. S.P. Borgatti, et al., Network analysis in the social sciences., Science 323 (2009) 892. S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, D. Hwang, Complex networks, structure and dynamics, Phys. Rep. 424 (2006) 175–308. Boccaletti S, et al. (2014) The structure and dynamics of multilayer networks. Physics Reports 544(1):1–122. Brian Karrer and M. E. J. Newman, Stochastic blockmodels and community structure in networks. PHYSICAL REVIEW E 83, 016107 (2011) Buldyrev SV, Parshani R, Paul G, Stanley HE, Havlin S (2010) Catastrophic cascade of failures in interdependent networks. Nature 464(7291):1025–1028. D.S. Callaway, M.E.J. Newmann, S.H. Strogatz, D.J. Watts, Phys. Rev. Lett. 85 (2000) 5468. Cho YS, Hwang S, Herrmann HJ, Kahng B (2013) Avoiding a spanning cluster in percolation models. Science 339(6124):1185–1187. R. Cohen, K. Erez, D. ben Avraham, S. Havlin, Resilience of the internet to random breakdowns, Phys. Rev. Lett. 85 (2000) 4626. R. Cohen, K. Erez, D. ben Avraham, S. Havlin, Breakdown of the internet under intentional attack, Phys. Rev. Lett. 86 (2001) 3682. 3 Dong, Gaogao; Fan, Jingfang; Shekhtman, Louis M.; Shai, Saray; Du, Ruijin; Tian, Lixin; Chen, Xiaosong; Stanley, H. Eugene; Havlin, Shlomo. Resilience of networks with community structure behaves as if under an external field. PNAS 115, 27, 6911-6915 (2018) M. Faloutsos, P. Faloutsos, C. Faloutsos, Comput. Commun. Rev. 29 (1999) 251. Gao J, Buldyrev SV, Stanley HE, Havlin S (2012) Networks formed from interdependent networks.Nature Physics 8(1):40–48. Gao J, Liu X, Li D, Havlin S (2015) Recent progress on the resilience of complex networks. Energies 8(10):12187–12210. M. Girvan and M. E. J. Newman, “Community structure in social and biological networks,” Proceedings of the National Academy of Sciences of the United States of America, vol. 99, no. 12, pp. 7821–7826, 2002. J.L. Hao Yin, A.R. Benson, D.F. Gleich, Local higher-order graph clustering, in: In Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, ACM, 2017. Holme P. Efficient local strategies for vaccination and network attack. Europhys Lett. (2004) 68:908–14. doi: 10.1209/epl/i2004-10286-2 Iyer S, Killingback T, Sundaram B, Wang Z (2013) Attack Robustness and Centrality of Complex Networks. PLoS ONE 8(4): e59613. doi:10.1371/ journal.pone.0059613 H. Jeong, B. Tombor, R. Albert, Z.N. Oltvai, A.-L. Barabasi, Nature 407 (2000) 651. Kivel¨a M, et al. (2014) Multilayer networks. Journal of complex networks 2(3):203–271. DS Lekha, K Balakrishnan (2020), Central attacks in complex networks: A revisit with new fallback strategy, Physica A: Statistical Mechanics and its Applications J.K.J. Leskovec, C. Faloutsos, Graph evolution: Densification and shrinking diameters, in: ACM Transactions on Knowledge Discovery from Data (ACM TKDD), ACM, 2007, p. 1(1). 4 Newman, M. E., Watts, D. J., & Strogatz, S. H. (2002). Random graph models of social networks. Proceedings of the national academy of sciences, 99 (suppl 1), 2566-2572. M. Newman, Networks: an introduction, Oxford university press, 2013. M.E.J. Newman, The structure and function of complex networks, SIAM Rev. 45 (2003) 167–256. M.E.J.Newman and M.Girvan (2004) Finding and evaluating community structure in networks. Physical Review E, 69:026113, 2004. Nguyen, Q., Pham, H. D., Cassi, D. & Bellingeri, M. Conditional attack strategy for real-world complex networks. Phys. A Stat. Mech. its Appl. 530, 121561 (2019). Q Nguyen, T. Trang Le (2019b) Structure and Robustness of Facebook's pages networks. Proceeding of the 2019 The 10th Conference on Network Modeling and Analysis (Marami 2019), Dijon, France Reis SD, et al. (2014) Avoiding catastrophic failure in correlated networks of networks. Nature Physics 10(10):762–767. Radicchi F, Arenas A (2013) Abrupt transition in the structural formation of interconnected networks. Nature Physics 9(11):717–720. Riordan O, Warnke L (2011) Explosive percolation is continuous. Science 333(6040):322–324. Shai, S. et al. Critical tipping point distinguishing two types of transitions in modular network structures. Phys. Rev. E 92, 062805 (2015). Shekhtman LM, Danziger MM, Havlin S (2016) Recent advances on failure and recovery in networks of networks. Chaos, Solitons & Fractals 90:28–36. S Wandelt, X Shi, X Sun, M Zanin (2020), Community Detection Boosts Network Dismantling on Real-World Networks, IEEE Access 8, 111954-111965 Wang Z, Zhao DW, Wang L, Sun GQ, Jin Z. Immunity of multiplex networks via acquaintance vaccination. EPL. (2015) 112:48002. doi: 10.1209/0295-5075/112/48002 5 Xiaoqian Sun, Volker Gollnick, Sebastian Wandelt, Robustness analysis metrics for worldwide airport network: A comprehensive . Volume 30, Issue 2, April 2017, Pages 500-512. Chinese Journal of Aeronautics. Yuan X, Hu Y, Stanley HE, Havlin S (2017) Eradicating catastrophic collapse in interdependent networks via reinforced nodes. Proceedings of the National Academy of Sciences p. 201621369. Yuri Mnyukh, Second-Order Phase Transitions, L. Landau and His Successors, American Journal of Condensed Matter Physics, Vol. 3 No. 2, 2013, pp. 25-30. doi: 10.5923/j.ajcmp.20130302.02.
Acknowledgement
This work is supported by the Vietnam’s Ministry of Science and Technology (MOST) under the Vietnam-Italy scientific and technological cooperation program for the period of 2021-2023. Many thanks to Dr. Vu-Lan Nguyen for useful comments on the paper.
APPENDIX
A. Prove that the rewired network statistically preserves the original node degree distribution
Given a node with degree k , the proportion of inter-modules links and intra-module links of this node before the rewiring process are approximated by (𝑚−1)𝑚 𝑘 and 𝑘 , respectively, where m is the number of modules. A proportion w of its inter-modules links will be rewired, thus the expected number of links that this node loses is: 𝑤 (𝑚 − 1)𝑚 𝑘 Similarly, this node can also be selected when links from nodes of the same modules are rewired. We compute the expected number of rewired links that this node can acquire as following: 6 - The total of rewired links in the network is 𝑤 (𝑚−1)𝑚 𝑁 < 𝑘 > - The total of rewired links that will be connected to nodes within the module of the node is: 𝑤 (𝑚−1)𝑚 𝑁 < 𝑘 >/𝑚 - The probability that the node is selected is proportioned to the ratio of its degree to the total degree of all nodes in the module (according to our method) and is: 𝑘/(𝑁 <𝑘 >/𝑚) - The expected number of rewired links that this node can be selected is therefore equal to: 𝑤 (𝑚−1)𝑚 𝑁 < 𝑘 >/𝑚 × 𝑘/(𝑁 < 𝑘 >/𝑚) = 𝑤 (𝑚−1)𝑚 𝑘 which is exactly equal to the expected number of links that this node loses. In consequence, the expected number of links of each node after rewiring process is equal to their initial degree, and the network’s degree distribution remain unchanged. B. Graph of 𝜅 and 𝛼 as function of rewiring probability w and number of modules m Comparison of analytical and simulation results for modular scale-free network for A) as a function of w and m to show and B) as a function of w and m . Both measures show the goodness of mathematical derivation in the Method section.