Abstract
Although an increasing amount of research is being done on the dynamical processes on interdependent spatial networks, knowledge of how interdependent spatial networks influence the dynamics of social contagion in them is sparse. Here we present a novel nonMarkovian social contagion model on interdependent spatial networks composed of two identical twodimensional lattices. We compare the dynamics of social contagion on networks with different fractions of dependency links and find that the density of final recovered nodes increases as the number of dependency links is increased. We use a finitesize analysis method to identify the type of phase transition in the giant connected components (GCC) of the final adopted nodes and find that as we increase the fraction of dependency links, the phase transition switches from secondorder to firstorder. In strong interdependent spatial networks with abundant dependency links, increasing the fraction of initial adopted nodes can induce the switch from a firstorder to secondorder phase transition associated with social contagion dynamics. In networks with a small number of dependency links, the phase transition remains secondorder. In addition, both the secondorder and firstorder phase transition points can be decreased by increasing the fraction of dependency links or the number of initiallyadopted nodes.
Introduction
Realworld networks are often interdependent and embedded in physical space^{1,2,3,4}. For example, the worldwide seaport network is strongly coupled to the worldwide airport network, and both are spatially embedded^{5}. The nodes in a communications network are strongly coupled to the nodes in the power grid network and both are spatially embedded^{2}. The Internet is a network of routers connected by wires in which the routers are grouped as autonomous systems (AS), and at this level the Internet itself can be seen as a set of interconnected AS embedded in physical space^{1}.
We know that these interdependent spatial networks can significantly influence the dynamical processes in them^{3,4,6,7,8,9,10}. The percolation transition can change from discontinuous to continuous when the distance in space between the interdependent nodes is reduced^{11}, and the system can collapse in an abrupt transition when the fraction of dependency links increases to a certain value^{12}. The universal propagation features of cascading overloads, which are characterized by a finite linear propagation velocity, exist on spatially embedded networks^{13}. In particular, a localized attack can cause substantially more damage to spatially embedded systems with dependencies than an equivalent random attack^{14}. Spatial networks are typically described as lattices^{15,16}. Studies of the dynamics in interdependent lattices have found that asymmetric coupling between interdependent lattices greatly promotes collective cooperation^{17}, and the transmission of disease in interconnected lattices differs as infection rates differ^{18}. Recent works demonstrated a change in the type of phase transition on related social dynamics in interdependent multilayer networks^{19,20,21,22}. Systematic computations revealed that in networks with interdependent links so that the failure of one node causes the immediate failures of all nodes connected to it by such links, both first and secondorder phase transitions and the crossover between the two can arise when the coupling strength is changed^{23}. The results of ref. 24 demonstrated that these phenomena can occur in the more general setting where no interdependent links are present.
Social contagions^{25,26,27,28,29,30}, which include the adoption of social innovations^{31,32,33}, healthy behaviors^{34}, and the diffusion of microfinance^{35}, are another typical dynamical process. Research results show that multiple confirmations of the credibility and legitimacy of a piece of news or a new trend are ubiquitous in social contagions, and the probability that an individual will adopt a new social behavior depends upon previous contacts, i.e., the social reinforcement effect^{34,36,37,38,39}. A classical model for describing the reinforcement effect in social contagions is the threshold model^{40} in which an individual adopts the social behavior only if the number or fraction of its neighbors who have already adopted the behavior exceeds an adoption threshold. Using this threshold model, network characteristics affecting social contagion such as the clustering coefficient^{41}, community structure^{42,43}, and multiplexity^{44,45,46} have been explored, but the existing studies paid little attention to the dynamics of social contagion on interdependent spatial networks.
Here we numerically study social contagion on interdependent spatial networks using a novel nonMarkovian social contagion model. A node adopts a new behavior if the cumulative pieces of information received from adopted neighbors in the same lattice exceeds an adoption threshold, or if its dependency node becomes adopted. We compare the dynamics of social contagion in networks when we vary the fraction of dependency links and find that the density of final recovered nodes increases greatly in networks when the number of dependency links is high. We also find that the fraction of dependency links can change the type of the phase transition. We use a finitesize analysis method^{47} to identify the type of phase transition and find that the phase transition is secondorder when the fraction of dependency links is small and firstorder when the fraction is large. In interdependent spatial networks the fraction of initiallyadopted nodes ρ_{0} may also affect the phase transition. Concretely, when we increase ρ_{0} the type of phase transition does not change in networks with a small fraction of dependency links, but changes from firstorder to secondorder in networks with a large fraction of dependency links. The phase transition points decrease when the fraction of dependency links or initiallyadopted nodes increases.
Results
NonMarkovian social contagion model on interdependent spatial networks
Our spatial network model consists of two identical twodimensional lattices A and B of linear size L and N = L × L nodes with periodic boundaries, as shown in Fig. 1(a). In each lattice, p fraction of nodes are randomly chosen as dependency nodes with two types of link, connectivity links (i.e., links between two nodes in the same lattice) and dependency links (i.e., links between nodes in lattice A and nodes in lattice B). The remaining 1 − p fraction of nodes only have connectivity links. More details of the interdependent spatial networks can be found in the Method section.
We divide the interdependent network population into three compartments, susceptible (S), adopted (A), and recovered (R) nodes. We generalize the cascading threshold model^{40} to the interdependent spatial network, describe the dynamics of social contagion using the susceptibleadoptedrecovered (SAR) model, and add social reinforcement through considering individual memory. Within the same lattice, nodes can retain their memory of previous information received from neighbors and adopt the new behavior if the cumulative pieces of information received from their neighbors exceeds an adoption threshold T [see Fig. 1(b)]. We designate this type of behavior adoption connected infection. A node can also adopt the new behavior when its corresponding dependency node becomes adopted. We designate this type of behavior adoption dependency infection [see Fig. 1(c)].
The simulations of the social contagion dynamics are implemented by using synchronous updating methods^{48}. Initially, ρ_{0} fraction of nodes are randomly selected to be adopted (i.e., to serve as seeds) in lattice A, and we leave all other nodes in the susceptible state. Each node has a record m_{i} of the pieces of received information from its neighbors. Initially, m_{i} = 0 for every node. At each time step, each adopted node transmits the behavior information to its susceptible neighbors in the same lattice with probability λ through the connectivity links. Once a susceptible node i is exposed to the information from an adopted neighbor, its m_{i} increases by one. If m_{i} is greater than or equal to the adoption threshold T, the susceptible node i will become an adopted node (Here connected infection happens). Once node i becomes an adopted one, its susceptible dependency nodes also become adopted at the same time (Here dependency infection happens). Infected nodes may also lose interest in the social behavior and become recovered with a probability u. When an adopted node becomes a recovered node it no longer takes part in the propagation of the social behavior. The time step is discrete and increases by Δt = 1. The dynamics of social contagion evolve until there are no more adopted nodes in the interdependent spatial network. In this paper, T is set to 3, unless otherwise specified. Note that our model is similar to the susceptibleinfectedrecovered (SIR) epidemic model^{49,50} but differs in that we add the memory of received information^{34,35,36,47,51,52}. Our proposed model of social contagion may describe the adoption of realworld social behavior. For example, a couple can discuss household products they use with their circle of friends. A wife or husband may adopt a new product if many of their friends have adopted it, or if either wife or husband adopts it then the other immediately adopts it as well.
Effects of the fraction of dependency links
Figure 2 shows a plot of the spatiotemporal pattern of the dynamical process at different stages. At t = 0 each node is either susceptible or adopted. After several steps (e.g., t = 8) susceptible, adopted, and recovered nodes can coexist. As t increases (e.g., t = 15 and t = 30) recovered nodes gradually dominate. Figure 2 also shows the time evolution of the population densities in which the density of susceptible (recovered) nodes decreases (increases) with time and ultimately reaches some value. The density of the adopted individuals decreases initially due to the fact that the number of individuals who newly adopt the behavior is less than that of individuals who become recovered. Then it is advanced with the growth of newly adopted individual and reaches the maximum value at t ≈ 12.
Figure 3 compares the dynamics of social contagion on interdependent spatial networks when p = 0.1 and p = 0.9. Figure 3(a) shows that when p = 0.9 the average density of final recovered nodes R_{A} in lattice A grows more rapidly than when p = 0.1. When p = 0.9 the behavior information from lattice A can easily propagates to lattice B because the abundant dependency links allow nodes in lattice A to adopt behavior through both connected infections from neighbors in the same lattice and dependency infections from the many dependent nodes in lattice B. The asymmetry of results in lattice A and B is due to the asymmetry of the initial condition. When p = 0.9 the propagation in lattice B is approximately the same as that in lattice A. When p = 0.1 the prevalence in lattice B is much lower than in lattice A because there are relatively few dependency links, the propagation from lattice A to lattice B is difficult, and the small number of seeds disallow outbreaks of behavior information in lattice B. Figure 3(b) shows the normalized sizes of the giant connected component (GCC) of final recovered nodes and on lattices A and B, respectively. Note that the trends of the giant connected components versus the transmission probability λ are similar to those of the density of final recovered nodes. Unlike when p = 0.1, both and increase abruptly at some λ when p = 0.9. These results indicate that the behaviors of and versus λ may be a secondorder phase transition when p = 0.1 and a firstorder phase transition when p = 0.9.
Figure 4 shows a finitesize analysis^{47} of lattice A of the type of phase transition described above. The average density of recovered nodes R_{A} are nearly the same for different linear size L values, especially when the interdependent network is weak [see Fig. 4(a,c)]. When p = 0.1, the normalized size giant connected component for different L values begin to converge after λ ≈ 0.915 [see Fig. 4(b)], which indicates that the behavior of GCC versus λ is a secondorder phase transition^{23,24}. When p = 0.9, all the curves intersect at one point [see Fig. 4(d)], and thus the type of phase transition will become firstorder as N → ∞^{23,24}. Here the abundant dependency links enable the dependent node B_{i} of an adopted node A_{i} to immediately adopt the new behavior. Node B_{i} transmits the information to one of its susceptible neighbors B_{u}, which becomes adopted when the cumulative pieces of received information exceed the adoption threshold and causes the behavior to be adopted by its dependency node A_{u}. This phenomenon induces cascading effects in adopting behavior, causes a large number of nodes to become adopted simultaneously, and contributes to the appearance of a firstorder phase transition. These results indicate that the parameter p is a key factor in social contagion on interdependent spatial networks. We also perform a finitesize analysis of lattice B and find a similar phenomenon (see the Supplemental Material for details).
Variability methods^{53,54} can numerically determine the epidemic threshold^{55,56} in SIR epidemiological models. To determine the firstorder phase transition point in a complex social contagion process, we calculate the number of iterations (NOI) required for the dynamical process to reach a steady state^{16,24,57} and count only the iterations during which at least one new node becomes adopted. For a secondorder phase transition, we calculate the normalized size of the second giant connected component (SGCC) of the final recovered nodes after the dynamical process is complete^{16,24,58}. In the thermodynamic limit, we obtain the secondorder transition point for p = 0.1 and the firstorder transition point for p = 0.9 (see the Methods for details). We also present some critical phenomena in the Method section.
Figure 5 shows the dependency of and on different p and λ values. Both and increase with p because many dependency links enhance the ability of the nodes to access the behavior information. Using the behavior of GCC versus λ, we divide the λ − p plane into different regions. Figure 5(a) shows that in lattice A there is a critical fraction p^{s} of dependency links that divides the phase diagram into a secondorder phase transition region (region II) and a firstorder phase transition region (region I). In region II most of the behavior information in lattice A propagates through contacts between neighbors. The dependency infection from lattice B is small because there are few dependency links and there is no abrupt increase of with λ. In region I the large number of dependency links cause cascading effects in adopting behavior, cause a large number of nodes to simultaneously become adopted nodes, and cause a firstorder phase transition. In lattice B, the λ − p plane is divided into three different regions in which regions I and II indicate that the behaviors of GCC versus λ are firstorder and secondorder phase transitions, respectively [see Fig. 5(b)]. In contrast to lattice A, when p < p^{*} there is an additional region III within which the social behavior cannot widely propagate no matter how large the λ value. This is because here the few dependency links produce only a few initiallyadopted nodes in lattice B, and they can not provide sufficient contacts with adopted neighbors for susceptible nodes to adopt the behavior. Note that both and decrease as p increases, which indicates that the strong interdependent spatial networks are promoting the social contagion.
Effects of the fraction of initial seeds
All of the above results depend on the initial condition in which there are ρ_{0} = 0.1 fraction of adopted nodes. Here we further explore the effects of the initial adopted fraction on social contagion on interdependent spatial networks.
Figure 6 shows the propagation when there are ρ_{0} = 0.5 fraction of initiallyadopted nodes. Figure 6(a,c) show that R_{A} are approximately the same for different L values, especially when p = 0.1. Figure 6(b) shows that for different L values begin to converge after λ ≈ 0.334. Here the large ρ_{0} value provides many opportunities for susceptible nodes to receive the information. After receiving sufficient information they become adopted, and this eventually induces a secondorder phase transition. Figure 6(b) shows that the analogy between ρ_{0} = 0.5 and ρ_{0} = 0.1 indicates that the type of phase transition does not change with ρ_{0} when p = 0.1. Note that all curves of also begin to converge after λ ≈ 0.25 when p = 0.9, as shown in Fig. 6(d). This is because there are sufficient initial seeds to raise the probability of susceptible nodes becoming adopted through connected infection. The cascading effects from dependency links are somewhat weakened, and this leads to a secondorder phase transition. The differences between the behaviors of versus λ for ρ_{0} = 0.5 and ρ_{0} = 0.1 indicate that the phase transition is no longer firstorder as ρ_{0} is increased when p = 0.9. The similar phenomena are also found in lattice B (see the Supplemental Material for details). According to the method of determining the secondorder phase transition point, we obtain for p = 0.1 and for p = 0.9 in the thermodynamic limit (see the Methods for details). Some critical phenomena are presented in the Method section.
Figure 7 shows the dependency of and on different ρ_{0} and λ values when p = 0.9. Note that both and increase with ρ_{0} because there are many initiallyadopted nodes to promote the propagation of behavior information among neighbors. Figure 7(a) uses the behavior of GCC versus λ to show that the phase diagram is divided into two different regions. When , the cascading effect caused by abundant dependency links strongly promotes information propagation and leads to the firstorder phase transition region (region I). When , the secondorder phase transition region (region II) appears, since the susceptible nodes adopt the behavior mainly through connected infection within the same lattice and the cascading effects are weakened. These phenomena indicate that on strongly interdependent spatial networks the phase transition changes from firstorder to secondorder as ρ_{0} is increased. In addition, both the secondorder and firstorder phase transition points decrease with ρ_{0}. This supports the findings shown in Figs 4(d) and 6(d) and indicates the important role of the initiallyadopted fraction. Figure 7(b) shows that as in lattice A the λ − ρ_{0} plane in lattice B is divided into two regions in which region I corresponds to the firstorder phase transition and region II corresponds to the secondorder phase transition. The phase transition points also decrease as ρ_{0} increases.
Discussion
We have studied in detail the social contagion on interdependent spatial networks consisting of two finite lattices that have dependency links. We first propose a nonMarkovian social contagion model in which a node adopts a new behavior when the cumulative pieces of information received from adopted neighbors in the same lattice exceed an adoption threshold, or if its dependency node becomes adopted. The effects of dependency links on this social contagion process are studied. Unlike networks with a small fraction p of dependency links, networks with abundant dependency links greatly facilitate the propagation of social behavior. We investigate the normalized sizes of GCC of final recovered nodes on networks of different linear sizes L and find that the phase transition changes from secondorder to firstorder as p increases. The firstorder and secondorder phase transitions points are determined by calculating the number of iterations and the normalized size of the second giant connected component, respectively. Using interdependent spatial networks, we further investigate how the fraction of initiallyadopted nodes influences the social contagion process. We find that increasing the fraction of initiallyadopted nodes ρ_{0} causes the behavior of GCC versus λ to change from a firstorder phase transition to a secondorder phase transition on networks with a large p value. If the p value of the network is small the phase transition remains secondorder even when there are abundant initial seeds. In addition, both the firstorder and secondorder phase transition points decrease as p or ρ_{0} increases.
We have numerically studied the dynamics of social contagion on interdependent spatial networks. The results show that both the fractions of dependency links and initiallyadopted node can influence the type of phase transition. Our results extend existing studies of interdependent spatial networks and help us understand phase transitions in the social contagion process. The social contagion models including other individual behavior mechanisms, e.g., limited contact ability^{27} or heterogenous adopted threshold^{28}, should be further explored. Further theoretical studies of our model are very important and full of challenges since the nonMarkovian character of our model and nonlocaltree like structure of the lattice make it extremely difficult to describe the strong dynamical correlations among the states of neighbors.
Methods
Generation of the interdependent spatial networks
To establish an interdependent spatial network, we first generate two identical lattices A and B with the same linear size L. In each lattice all nodes are arranged in a matrix of L × L, and each node is connected to its four neighbors in the same lattice via connectivity links. We then randomly choose p fraction of nodes in lattice A to be dependency nodes. Once a node A_{i} in lattice A is chosen as a dependency node, it will be connected to one and only one node B_{j} randomly selected in lattice B via a dependency link [see Fig. 1(a)]. Thus, a dependency link connects two random nodes respectively located in lattice A and B with probability p. Each dependency node has only one dependency link. The number of dependency links in the interdependent spatial network is determined by the parameter p. For simplicity, the interdependent networks with a large p value are defined as the strong interdependent networks, and those with a small p value are defined as the weak ones.
Determination of phase transition points
To locate the transition points and as a function of the network size N = L × L, we study the location of the peak of SGCC and NOI, respectively. On a network with finite size N, NOI reaches its peak at the firstorder phase transition point and SGCC reaches its peak at the secondorder phase transition point^{24}. In the thermodynamic limit (i.e., N → ∞), the critical point and should fulfill with α > 0 and with β > 0, respectively^{59}. Then, from the finitesize scaling theory one should obtain the scaling G^{1} ~ N^{−δ} (with δ > 0) only at the secondorder phase transition point , and a power law relation NOI ~ N^{γ} (with γ > 0) only at the firstorder phase transition point .
Figure 8(a) shows that when p = 0.1, the peak of the normalized size of the second giant connected component in lattice A (i.e., ) versus λ gradually shifts to the right as L is increased. In Fig. 8(b) we plot versus N = L × L for fixed λ. We obtain a power law relation at . Then we fit versus 1/L by using the leastsquaresfit method in Fig. 8(c). We find that . Figure 8(d) shows that when p = 0.9, the peak of NOI in lattice A (i.e., NOI_{A}) versus λ gradually shifts to the left as L is increased. In Fig. 8(e) we plot NOI_{A} versus N for fixed λ, and obtain a power law relation NOI_{A} ~ N^{0.2026} at . We further fit versus 1/L by using the leastsquaresfit method in Fig. 8(f), and find that .
We perform the similar analyses for ρ_{0} = 0.5, as shown in Fig. 9. Figure 9(a) shows that when p = 0.1, the peak of versus λ gradually shifts to the right as L is increased. In Fig. 9(b) we plot versus N = L × L for fixed λ. We obtain a power law relation at . Then we fit versus 1/L in Fig. 9(c). We find that . Fig. 9(d) shows that when p = 0.9, the trend of versus λ as L is increased is similar to that when p = 0.1. In Fig. 9(e) we plot versus N = L × L for fixed λ, and obtain a power law relation at . We further fit versus 1/L in Fig. 9(f), and find that .
Additional Information
How to cite this article: Shu, P. et al. Social contagions on interdependent lattice networks. Sci. Rep. 7, 44669; doi: 10.1038/srep44669 (2017).
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
References
 1.
Barthélemy, M. Spatial networks. Phys. Rep. 499, 1–101 (2011).
 2.
Li, D., Kosmidis, K., Bunde, A. & Havlin, S. Dimension of spatially embedded networks. Nat. Phys. 7, 481–484 (2011).
 3.
Boccaletti, S., Bianconi, G., Criado, R., del Genio, C. I., GómezGardeñes, J., Romance, M., SendiñaNadal, I., Wang, Z. & Zanin, M. The structure and dynamics of multilayer networks. Phys. Rep. 544, 1–122 (2014).
 4.
Balcana, D., Colizza, V., GonÇalves, B., Hu, H., Ramasco, J. J. & Vespignani, A. Multiscale mobility networks and the spatial spreading of infectious diseases. Proc. Natl. Acad. Sci. 106, 21484–21489 (2009).
 5.
Parshani, R., Rozenblat, C., Ietri, D., Ducruet, C. & Havlin, S. Intersimilarity between coupled networks. Europhys. Lett. 92, 68002 (2011).
 6.
Son, S.W., Grassberger, P. & Paczuski, M. Percolation transitions are not always sharpened by making networks interdependent. Phys. Rev. Lett. 107, 195702 (2011).
 7.
Jiang, L.L. & Perc, M. Spreading of cooperative behaviour across interdependent groups. Sci. Rep. 3, 02483 (2013).
 8.
Shekhtman, L. M., Berezin, Y., Danziger, M. M. & Havlin, S. Robustness of a network formed of spatially embedded networks. Phys. Rev. E 90, 012809 (2014).
 9.
Wang, B., Tanaka, G., Suzuki, H. & Aihara, K. Epidemic spread on interconnected metapopulation networks. Phys. Rev. E 90, 032806 (2014).
 10.
Morris, R. G. & Barthelemy, M. Transport on Coupled Spatial Networks. Phys. Rev. Lett. 109, 128703 (2012).
 11.
Li, W., Bashan, A., Buldyrev, S. V., Stanley, H. E. & Havlin, S. Cascading Failures in Interdependent Lattice Networks: The Critical Role of the Length of Dependency Links. Phys. Rev. Lett. 108, 228702 (2012).
 12.
Bashan, A., Berezin, Y., Buldyrev, S. V. & Havlin, S. The extreme vulnerability of interdependent spatially embedded networks. Nat. Phys. 9, 667–672 (2013).
 13.
Zhao, J., Li, D., Sanhedrai, H., Cohen, R. & Havlin, S. Spatiotemporal propagation of cascading overload failures in spatially embedded networks. Nat. Commun. 7, 10094 (2015).
 14.
Berezin, Y., Bashan, A., Danziger, M. M., Li, D. & Havlin, S. Localized attacks on spatially embedded networks with dependencies. Sci. Rep. 5, 08934 (2015).
 15.
Kleinberg, J. M. Navigation in a small world. Nature 406, 845–845 (2000).
 16.
Gao, J., Zhou, T. & Hu, Y. Bootstrap percolation on spatial networks. Sci. Rep. 5, 14662 (2015).
 17.
Xia, C.Y., Meng, X.K. & Wang, Z. Heterogeneous Coupling between Interdependent Lattices Promotes the Cooperation in the Prisoner’s Dilemma Game. PLoS ONE 10, e0129542 (2015).
 18.
Li, D., Qin, P., Wang, H., Liu, C. & Jiang, Y. Epidemics on interconnected lattices. Europhys. Lett. 105, 68004 (2014).
 19.
Czaplicka A. & Toral R. & San Miguel M. Competition of simple and complex adoption on interdependent networks. Phys. Rev. E 94, 062301 (2016).
 20.
Rojas F. V. & Vazquez F. Interacting opinion and disease dynamics in multiplex networks: discontinuous phase transition and nonmonotonic consensus times. arXiv:1612.01003 (2016).
 21.
Zhao K. & Bianconi G. Percolation on interdependent networks with a fraction of antagonistic interactions. J. Stat. Phys. 152, 1069–1083 (2013).
 22.
Radicchi F. & Arenas A. Abrupt transition in the structural formation of interconnected networks. Nat. Phys. 9, 717–720 (2013).
 23.
Buldyrev, S. V., Parshani, R., Paul, G., Stanley, H. E. & Havlin, S. Catastrophic cascade of failures in interdependent networks. Nature 464, 1025–1028 (2010).
 24.
Liu, R. R., Wang, W. X., Lai, Y.C. & Wang, B. H. Cascading dynamics on random networks: Crossover in phase transition. Phys. Rev. E 85, 026110 (2012).
 25.
Bond, R. M., Fariss, C. J., Jones, J. J., Kramer, A. D. I., Marlow, C., Settle, J. E. & Fowler, J. H. A 61millionperson experiment in social influence and political mobilization. Nature 489, 295–298 (2012).
 26.
Wang, W., Tang, M., Zhang, H.F. & Lai, Y.C. Dynamics of social contagions with memory of nonredundant information. Phys. Rev. E 92, 012820 (2015).
 27.
Wang, W., Shu, P., Zhu, Y.X., Tang, M. & Zhang, Y.C. Dynamics of social contagions with limited contact capacity. Chaos 25, 103102 (2015).
 28.
Wang, W., Tang, M., Shu, P. & Wang, Z. Dynamics of social contagions with heterogeneous adoption thresholds: crossover phenomena in phase transition. New J. Phys. 18, 013029 (2016).
 29.
Ruan, Z., Iñiguez, G., Karsai, M. & Kertész, J. Kinetics of Social Contagion. Phys. Rev. Lett. 115, 218702 (2015).
 30.
Cozzo, E., Baños, R. A., Meloni, S. & Moreno Y. Contactbased social contagion in multiplex networks. Phys. Rev. E 88, 050801(R) (2013).
 31.
Hu, Y., Havlin, S. & Makse, H. A. Conditions for Viral Influence Spreading through Multiplex Correlated Social Networks. Phys. Rev. X 4, 021031 (2014).
 32.
Gallos, L. K., Rybski, D., Liljeros, F., Havlin, S. & Makse, H. A. How People Interact in Evolving Online Affiliation Networks. Phys. Rev. X 2, 031014 (2012).
 33.
Young, H. P. The dynamics of social innovation. Proc. Natl. Acad. Sci. USA 108, 21285–21291 (2011).
 34.
Centola, D. An Experimental Study of Homophily in the Adoption of Health Behavior. Science 334, 1269–1272 (2011).
 35.
Banerjee, A., Chandrasekhar, A. G., Duflo, E. & Jackson, M. O. The Diffusion of Microfinance. Science 341, 1236498 (2013).
 36.
Dodds, P. S. & Watts, D. J. Universal Behavior in a Generalized Model of Contagion. Phys. Rev. Lett. 92, 218701 (2004).
 37.
Dodds, P. S. & Watts, D. J. A generalized model of social and biological contagion. J. Theor. Biol. 232, 587–604 (2005).
 38.
Weiss, C. H., PoncelaCasasnovas, J., Glaser, J. I., Pah, A. R., Persell, S. D., Baker, D. W., Wunderink, R. G. & Amaral, L. A. N. Adoption of a HighImpact Innovation in a Homogeneous Population. Phys. Rev. X 4, 041008 (2014).
 39.
Centola, D. & Macy, M. Complex Contagions and the Weakness of Long Ties. Am. J. Sociol. 113, 702–734 (2007).
 40.
Watts, D. J. A simple model of global cascades on random networks. Proc. Natl. Acad. Sci. USA 99, 5766–5771 (2002).
 41.
Whitney, D. E. Dynamic theory of cascades on finite clustered random networks with a threshold rule. Phys. Rev. E 82, 066110 (2010).
 42.
Gleeson, J. P. Cascades on correlated and modular random networks. Phys. Rev. E 77, 046117 (2008).
 43.
Nematzadeh, A., Ferrara, E., Flammini, A. & Ahn, Y.Y. Optimal Network Modularity for Information Diffusion. Phys. Rev. Lett. 113, 088701 (2014).
 44.
Lee, K.M., Brummitt, C. D. & Goh, K.I. Threshold cascades with response heterogeneity in multiplex networks. Phys. Rev. E 90, 062816 (2014).
 45.
Brummitt, C. D., Lee, K.M. & Goh, K.I. Multiplexityfacilitated cascades in networks. Phys. Rev. E 85, 045102(R) (2012).
 46.
Yağan, O. & Gligor, V. Analysis of complex contagions in random multiplex networks. Phys. Rev. E 86, 036103 (2012).
 47.
Marro, J. & Dickman, R. Nonequilibrium Phase Transitions in Lattice Models (Cambridge University Press, Cambridge, 1999).
 48.
Schönfisch B. & de Roos A. Synchronous and asynchronous updating in cellular automata. Bio. Syst. 51, 123–143 (1999).
 49.
Anderson, R. M. & May, R. M. Infectious Diseases of Humans: Dynamics and Control (Oxford University Press, Oxford, 1992).
 50.
Moreno, Y., PastorSatorras, R. & Vespignani, A. Epidemic outbreaks in complex heterogeneous networks. Eur. Phys. J. B 26, 521–529 (2002).
 51.
Chung, K., Baek, Y., Kim, D., Ha, M. & Jeong, H. Generalized epidemic process on modular networks. Phys. Rev. E 89, 052811 (2014).
 52.
Aral, S. & Walker, D. Identifying Influential and Susceptible Members of Social Networks. Science 337, 337–341 (2012).
 53.
Shu, P., Wang, W., Tang, M. & Do, Y. Numerical identification of epidemic thresholds for susceptibleinfectedrecovered model on finitesize networks. Chaos 25, 063104 (2015).
 54.
Shu, P., Wang, W., Tang, M., Zhao, P. & Zhang, Y.C. Recovery rate affects the effective epidemic threshold with synchronous updating. Chaos 26, 063108 (2016).
 55.
Bogũná, M., PastorSatorras, R. & Vespignani, A. Absence of Epidemic Threshold in ScaleFree Networks with Degree Correlations. Phys. Rev. Lett. 90, 028701 (2003).
 56.
Bogũná, M., Castellano, C. & PastorSatorras, R. Nature of the epidemic threshold for the susceptibleinfectedsusceptible dynamics in networks. Phys. Rev. Lett. 111, 068701 (2013).
 57.
Parshani, R., Buldyrev, S. V. & Havlin, S. Critical effect of dependency groups on the function of networks. Proc. Natl. Acad. Sci. USA 108, 1007–1010 (2011).
 58.
Li, D., Li, G., Kosmidis, K., Stanley, H. E., Bunde, A. & Havlin, S. Percolation of spatially constraint networks. Europhys. Lett. 93, 68004 (2011).
 59.
Radicchi, F. & Castellano, C. Breaking of the sitebond percolation universality in networks. Nat. Commun. 6, 10196 (2015).
Acknowledgements
This work was partially supported by National Natural Science Foundation of China (Grant Nos 61501358, 61673085), and the Fundamental Research Funds for the Central Universities.
Author information
Affiliations
School of Sciences, Xi’an University of Technology, Xi’an, 710054, China
 Panpan Shu
Web Sciences Center, University of Electronic Science and Technology of China, Chengdu, 610054, China
 Lei Gao
 & Wei Wang
School of Physics and Optoelectronic Engineering, Xidian University, Xi’an, 710071, China
 Pengcheng Zhao
Big data research center, University of Electronic Science and Technology of China, Chengdu 610054, China
 Wei Wang
Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts, 02215, USA
 Wei Wang
 & H. Eugene Stanley
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Contributions
P.S. and W.W. devised the research project. P.S., L.G., P.Z. and W.W. performed numerical simulations. P.S., L.G., P.Z., W.W. and H.E.S. analyzed the results. P.S., L.G., P.Z., W.W. and H.E.S. wrote the paper.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to Wei Wang.
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