Introduction

Real-world networks are often interdependent and embedded in physical space1,2,3,4. For example, the world-wide seaport network is strongly coupled to the world-wide airport network, and both are spatially embedded5. The nodes in a communications network are strongly coupled to the nodes in the power grid network and both are spatially embedded2. 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 space1.

We know that these interdependent spatial networks can significantly influence the dynamical processes in them3,4,6,7,8,9,10. The percolation transition can change from discontinuous to continuous when the distance in space between the interdependent nodes is reduced11, and the system can collapse in an abrupt transition when the fraction of dependency links increases to a certain value12. The universal propagation features of cascading overloads, which are characterized by a finite linear propagation velocity, exist on spatially embedded networks13. In particular, a localized attack can cause substantially more damage to spatially embedded systems with dependencies than an equivalent random attack14. Spatial networks are typically described as lattices15,16. Studies of the dynamics in interdependent lattices have found that asymmetric coupling between interdependent lattices greatly promotes collective cooperation17, and the transmission of disease in interconnected lattices differs as infection rates differ18. Recent works demonstrated a change in the type of phase transition on related social dynamics in interdependent multilayer networks19,20,21,22. 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 second-order phase transitions and the crossover between the two can arise when the coupling strength is changed23. The results of ref. 24 demonstrated that these phenomena can occur in the more general setting where no interdependent links are present.

Social contagions25,26,27,28,29,30, which include the adoption of social innovations31,32,33, healthy behaviors34, and the diffusion of microfinance35, 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 effect34,36,37,38,39. A classical model for describing the reinforcement effect in social contagions is the threshold model40 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 coefficient41, community structure42,43, and multiplexity44,45,46 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 non-Markovian 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 finite-size analysis method47 to identify the type of phase transition and find that the phase transition is second-order when the fraction of dependency links is small and first-order when the fraction is large. In interdependent spatial networks the fraction of initially-adopted nodes ρ0 may also affect the phase transition. Concretely, when we increase ρ0 the type of phase transition does not change in networks with a small fraction of dependency links, but changes from first-order to second-order in networks with a large fraction of dependency links. The phase transition points decrease when the fraction of dependency links or initially-adopted nodes increases.

Results

Non-Markovian social contagion model on interdependent spatial networks

Our spatial network model consists of two identical two-dimensional 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.

Figure 1: Illustration of the social contagion on the interdependent spatial network.
figure 1

(a) Interdependent spatial network composed of two 2-dimensional periodic square lattices A and B, where a node Ai in lattice A is randomly interconnected with a node Bj in lattice B. (b) Connected propagation with T = 3: In lattice A, the node Ai becomes adopted after exposing three times to the social behavior from its adopted neighbors. Here ti, tj and tk are any three different time steps of the dynamics confined with ti < tj < tk. (c) Dependency propagation: At some step the node Bj becomes adopted, and then the corresponding dependency node Ai adopts the social behavior.

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We divide the interdependent network population into three compartments, susceptible (S), adopted (A), and recovered (R) nodes. We generalize the cascading threshold model40 to the interdependent spatial network, describe the dynamics of social contagion using the susceptible-adopted-recovered (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 methods48. Initially, ρ0 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 mi of the pieces of received information from its neighbors. Initially, mi = 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 mi increases by one. If mi 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 susceptible-infected-recovered (SIR) epidemic model49,50 but differs in that we add the memory of received information34,35,36,47,51,52. Our proposed model of social contagion may describe the adoption of real-world 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 spatio-temporal 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 co-exist. 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 2: Spatio-temporal pattern of the dynamical process and time evolution of the population densities on interdependent spatial networks.
figure 2

The paraments are chosen as N = 104, p = 0.9, ρ0 = 0.1, λ = 0.8, μ = 0.5, and T = 3. The colors green, red and blue represent susceptible, adopted and recovered states, respectively.

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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 RA 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 second-order phase transition when p = 0.1 and a first-order phase transition when p = 0.9.

Figure 3: Comparison of the average outbreak size R and the giant connected components of recovered nodes G1 among different interdependent spatial networks.
figure 3

(a) RA and RB vs. λ for p = 0.1 (solid and empty circles) and p = 0.9 (solid and empty squares). (b) and vs. λ for p = 0.1 (solid and empty circles) and p = 0.9 (solid and empty squares). The parameters are chosen as L = 100, ρ0 = 0.1 and μ = 0.5. The results are averaged over 102 × 104 independent realizations in 102 different configurations of dependency links.

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Figure 4 shows a finite-size analysis47 of lattice A of the type of phase transition described above. The average density of recovered nodes RA 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 second-order phase transition23,24. When p = 0.9, all the curves intersect at one point [see Fig. 4(d)], and thus the type of phase transition will become first-order as N → ∞23,24. Here the abundant dependency links enable the dependent node Bi of an adopted node Ai to immediately adopt the new behavior. Node Bi transmits the information to one of its susceptible neighbors Bu, 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 Au. 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 first-order phase transition. These results indicate that the parameter p is a key factor in social contagion on interdependent spatial networks. We also perform a finite-size analysis of lattice B and find a similar phenomenon (see the Supplemental Material for details).

Figure 4
figure 4

For ρ0 = 0.1, the finite-size effects on interdependent spatial networks with p = 0.1 (a,b) and p = 0.9 (c,d). (a) RA vs. λ for p = 0.1. (b) vs. λ for p = 0.1. (c) RA vs. λ for p = 0.9. (d) vs. λ for p = 0.9. The solid lines, dash lines, dot lines, dash dot lines and dash dot dot lines respectively represent L = 50, 100, 200, 400 and 600. We perform 102 × 104 independent realizations on 102 different networks.

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Variability methods53,54 can numerically determine the epidemic threshold55,56 in SIR epidemiological models. To determine the first-order 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 state16,24,57 and count only the iterations during which at least one new node becomes adopted. For a second-order phase transition, we calculate the normalized size of the second giant connected component (SGCC) of the final recovered nodes after the dynamical process is complete16,24,58. In the thermodynamic limit, we obtain the second-order transition point for p = 0.1 and the first-order 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 ps of dependency links that divides the phase diagram into a second-order phase transition region (region II) and a first-order 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 first-order 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 first-order and second-order 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 initially-adopted 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.

Figure 5: Dependency of the normalized size of giant connected components on p and λ for ρ0 = 0.1.
figure 5

The colors represents the normalized size of GCC. (a) vs. p and λ. (b) vs. p and λ. ps indicates the critical fraction of dependency links that separates the second-order phase transition from first-order phase transition. p* indicates the critical fraction of dependency links below which the behavior information could not propagate. We perform 102 × 104 independent realizations on 102 different networks.

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Effects of the fraction of initial seeds

All of the above results depend on the initial condition in which there are ρ0 = 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 ρ0 = 0.5 fraction of initially-adopted nodes. Figure 6(a,c) show that RA 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 ρ0 value provides many opportunities for susceptible nodes to receive the information. After receiving sufficient information they become adopted, and this eventually induces a second-order phase transition. Figure 6(b) shows that the analogy between ρ0 = 0.5 and ρ0 = 0.1 indicates that the type of phase transition does not change with ρ0 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 second-order phase transition. The differences between the behaviors of versus λ for ρ0 = 0.5 and ρ0 = 0.1 indicate that the phase transition is no longer first-order as ρ0 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 second-order 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 6
figure 6

For ρ0 = 0.5, the finite-size effects on interdependent spatial networks with p = 0.1 (a,b) and p = 0.9 (c,d). (a) RA vs. λ for p = 0.1. (b) vs. λ for p = 0.1. (c) RA vs. λ for p = 0.9. (d) vs. λ for p = 0.9. The solid lines, dash lines, dot lines, dash-dot lines and dash-dot-dot lines respectively represent L = 50, 100, 200, 400 and 600. The results are averaged over 102 × 104 independent realizations.

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Figure 7 shows the dependency of and on different ρ0 and λ values when p = 0.9. Note that both and increase with ρ0 because there are many initially-adopted 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 first-order phase transition region (region I). When , the second-order 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 first-order to second-order as ρ0 is increased. In addition, both the second-order and first-order phase transition points decrease with ρ0. This supports the findings shown in Figs 4(d) and 6(d) and indicates the important role of the initially-adopted fraction. Figure 7(b) shows that as in lattice A the λ − ρ0 plane in lattice B is divided into two regions in which region I corresponds to the first-order phase transition and region II corresponds to the second-order phase transition. The phase transition points also decrease as ρ0 increases.

Figure 7: Dependency of the normalized size of giant connected component on parameters ρ0 and λ for p = 0.9.
figure 7

The colors represents the normalized size of GCC. (a) vs. ρ0 and λ. (b) vs. ρ0 and λ. indicates the critical fraction of initial adopted nodes that separates the second-order phase transition from first-order phase transition. The results are averaged over 102 × 104 independent realizations.

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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 non-Markovian 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 second-order to first-order as p increases. The first-order and second-order 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 initially-adopted nodes influences the social contagion process. We find that increasing the fraction of initially-adopted nodes ρ0 causes the behavior of GCC versus λ to change from a first-order phase transition to a second-order phase transition on networks with a large p value. If the p value of the network is small the phase transition remains second-order even when there are abundant initial seeds. In addition, both the first-order and second-order phase transition points decrease as p or ρ0 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 initially-adopted 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 ability27 or heterogenous adopted threshold28, should be further explored. Further theoretical studies of our model are very important and full of challenges since the non-Markovian character of our model and non-local-tree 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 Ai in lattice A is chosen as a dependency node, it will be connected to one and only one node Bj 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 first-order phase transition point and SGCC reaches its peak at the second-order phase transition point24. In the thermodynamic limit (i.e., N → ∞), the critical point and should fulfill with α > 0 and with β > 0, respectively59. Then, from the finite-size scaling theory one should obtain the scaling G1 ~ N−δ (with δ > 0) only at the second-order phase transition point , and a power law relation NOI ~ Nγ (with γ > 0) only at the first-order 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 least-squares-fit 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., NOIA) versus λ gradually shifts to the left as L is increased. In Fig. 8(e) we plot NOIA versus N for fixed λ, and obtain a power law relation NOIA ~ N0.2026 at . We further fit versus 1/L by using the least-squares-fit method in Fig. 8(f), and find that .

Figure 8
figure 8

For ρ0 = 0.1, the determination of phase transition point on interdependent spatial networks with p = 0.1 (a–c) and p = 0.9 (d–f). (a) vs. λ for p = 0.1. (b) vs. N = L × L for p = 0.1. (c) vs. 1/L for p = 0.1. (d) NOIA vs. λ for p = 0.1. (e) vs N for p = 0.9. (f) vs. 1/L for p = 0.9. In figures (a,d), the solid lines, dash lines, dot lines, dash dot lines and dash dot dot lines respectively represent L = 50, 100, 200, 400 and 600. We perform 102 × 104 independent realizations on 102 different networks.

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We perform the similar analyses for ρ0 = 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 .

Figure 9
figure 9

For ρ0 = 0.5, the determination of phase-transition point on interdependent spatial networks with p = 0.1 (a–c) and p = 0.9 (d–f). (a) vs. λ for p = 0.1. (b) vs. N = L × L for p = 0.1. (c) vs. 1/L for p = 0.1. (d) vs. λ for p = 0.9. (e) vs. N = L × L for p = 0.9. (f) vs. 1/L for p = 0.9. In figures (a,d), the solid lines, dash lines, dot lines, dash-dot lines and dash-dot-dot lines respectively represent L = 50, 100, 200, 400 and 600. The results are averaged over 102 × 104 independent realizations.

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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).

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