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Asymmetrically interacting spreading dynamics on complex layered networks

The spread of disease through a physical-contact network and the spread of information about the disease on a communication network are two intimately related dynamical processes. We investigate the asymmetrical interplay between the two types of spreading dynamics, each occurring on its own layer, by focusing on the two fundamental quantities underlying any spreading process: epidemic threshold and the final infection ratio. We find that an epidemic outbreak on the contact layer can induce an outbreak on the communication layer, and information spreading can effectively raise the epidemic threshold. When structural correlation exists between the two layers, the information threshold remains unchanged but the epidemic threshold can be enhanced, making the contact layer more resilient to epidemic outbreak. We develop a physical theory to understand the intricate interplay between the two types of spreading dynamics.


S1. Spreading dynamics on uncorrelated doubule-layer networks
We adopt the heterogeneous mean-field theory [1] to uncorrelated double-layer networks. Let P A (k A ) and P B (k B ) be the degree distributions of layers A and B, with mean degree ⟨k A ⟩ and ⟨k B ⟩, respectively. We assume that the subnetworks associated with both layers are random with no degree correlation. The time evolution of the epidemic spreading is described by the variables s A kA (t), ρ A kA (t), and r A kA (t), which are the densities of the susceptible, infected, and recovered nodes of degree k A in layer A at time t, respectively. Similarly, s B kB (t), ρ B kB (t), r B kB (t), and v B kB (t) respectively denote the susceptible, infected, recovered, and vaccinated densities of nodes of degree k B in layer B at time t.

A. Mean-field rate equations
The mean-field rate equations of the information spreading in layer A are then The mean-field rate equations of epidemic spreading in layer B are thus given by where Θ A (t) [Θ B (t)] is the probability that a neighboring node in layer A (layer B) is in the infected state. From Eqs. (S1)-(S7), the density associated with each distinct state in layer A or B is given by where h ∈ {A, B}, X ∈ {S, I, R, V }, and k h,max denotes the largest degree of layer h. The final densities of the whole system can be obtained by taking the limit t → ∞.

B. Linear analysis for the information threshold
On an uncorrelated layered network, at the outset of the spreading dynamics, the whole system can be regarded as consisting of two coupled SI-epidemic subsystems [2] with the time evolution described by Eqs. (S2) and (S5). For t → 0, we have s A kA (t) ≈ 1 and s B kB (t) ≈ 1, which reduce Eqs. (S2) and (S5) to For convenience, Eq. (S9) can be written concisely as where the vector of infected density is defined as and C is a block matrix in the following form: with matrix elements given by In general, information spreading on layer A can be facilitated by the outbreak of the epidemic on layer B, as an infected node in layer B instantaneously makes its counterpart node in layer A "infected" with the information about the disease. This coupling effect, in combination with the intrinsic spreading dynamics on layer A, leads to more informed nodes in the communication layer than infected nodes on layer B. If the maximum eigenvalue Λ C of matrix C is greater than 1, an outbreak of the information will occur in the system [3]. We then have where max{} denotes the greater of the two, and are the maximum eigenvalues of matrices C A and C B [4], respectively. The outbreak threshold of information spreading in layer A is given by denote the outbreak threshold of information spreading on layer A when it is isolated from layer B, and that of epidemic spreading on layer B when the coupling between the two layers is absent, respectively.

C. Competing percolation theory for epidemic threshold
To elucidate the interplay between epidemic and vaccination spreading, we must first determine which one is the faster "disease." At the early time of the epidemic outbreak on the isolated layer B, the average number of infected nodes grows exponentially as N e = n 0 R t e = n 0 e t ln Re , (S14) where R e = β B /β Bu is the basic reproductive number for the disease on the isolated layer B [5], and n 0 denotes the number of initially infected nodes. Similarly, for information spreading on the isolated layer A, the average number of informed nodes at the early time is where R i = β A /β Au is the reproductive number for information spreading on the isolated layer A. The resulting number of vaccinated nodes on layer B is Since both epidemic and vaccination spreading processes exhibit exponential growth, we can obtain the ratio of their growth rates as (S17) For θ > 1, i.e., β B β Au > β A β Bu , the epidemic disease spreads faster than the vaccination. In this case, the vaccination spread is insignificant and can be neglected.
To uncover the impact of information spreading on epidemic outbreak, we focus on the case of faster vaccination, i.e., θ < 1, in accordance with the fact that information always tends to spread much faster than epidemic in a modern society. Given that vaccination and epidemic can be treated successively and separately, the threshold of epidemic outbreak can be derived by a bond percolation analysis [6,7].
Firstly, when information spreading on layer A is over, the density of informed population is given by [5] x kA is the generating function for the degree distribution of layer A, and u is the probability that a node is not connected to the giant cluster via a particular one of its edges, which can be solved by Since p is the probability that an informed node in layer A makes its counterpart node in layer B vaccinated, the number of vaccinated or removed nodes in layer B is pS A . A necessary condition for the outbreak of epidemic is the existence of a giant residual cluster in layer B [8]. We have of layer B, and the prime denotes derivative. From Eq. (S20), we see that epidemic outbreak can occur only if pS . The degree distribution of the residual network of layer B is given by [9,10] where f = 1 − pS A is the probability that a node is in the residual network. The generating function for the degree distribution of the residual network is then [6] x kB is the generating function for the degree distribution of layer B. The generating function for its excess degree distribution is .
The basic reproductive number for a disease spreading over the residual network of layer B is then given by [5] The epidemic threshold corresponds to the point R i = 1, and thus we have where S A is the density of the informed population, which can be obtained by solving Eqs. (S18) and (S19).

S2. Spreading dynamics on correlated double-layer networks
We assume that layer A has the same degree distribution as layer B. After a certain fraction q of pairs of nodes, one from each layer, have been randomly rematched, the conditional probability P (k B |k A ) can be written as A. Mean-field rate equations Using Eqs. (S1)-(S3), we can write the mean-field rate equations for information spreading on layer A as Similarly, the mean-field rate equations for epidemic spreading on layer B are Substituting Eqs. (S28)-(S34) into Eq. (S8), we can get the density associated with each distinct state in layer A or B.

B. Linear analysis for the information threshold
At the outset of the spreading dynamics, the whole system can be regarded as two coupled SI-epidemic subsystems [2] with the time evolution described by Eqs. (S29) and (S32). In the limit t → 0, we have s A kA (t) ≈ 1 and s B kB (t) ≈ 1. Equations (S29) and (S32) can then be reduced to which can be written concisely as where the matrix C has the same form as in Eq. (S11) and The threshold of information outbreak is given by which is the same as Eq. (9) in the main text. As described in uncorrelated networks, there are two distinct mechanisms that can lead to the outbreak of information on layer A, and these hold for correlated layered-networks as well. For β B ≤ β Bu , only a small number of nodes in layer B are infected, so the impact of the disease on information-outbreak threshold on layer A is negligible. For β B > β Bu , epidemic spreading can result in the outbreak of information. In this case, the information-outbreak threshold is zero.

C. Competing percolation theory for epidemic threshold
For β A ≤ β Au , information itself cannot spread through the population. There is thus hardly any effect of the information layer on the epidemic spreading on layer B, and we have β Bc ≈ β Bu . But for β A > β Au , the effect of information spreading on the epidemic threshold cannot be ignored. To assess quantitatively the influence, we focus on the case of faster information spread, i.e., β A β Bu > β B β Au , rendering applicable a bond percolation analysis similar to uncorrelated networks. Specifically, after information spreads on layer A, the percentage of nodes that get the information is S A , and the density of recovered nodes of degree k A is r A kA = 1 − u kA , where u is the probability that a node is not connected to the giant cluster by a particular edge [Eq. (S19)]. Vaccinating a number of counterpart nodes results in the random removal of some edges which connect the vaccinated nodes with the remaining nodes [9,10]. The probability h of an edge linking to a vaccinated node is The new degree distribution of the residual network on layer B is thus given by The requirement that a giant residual cluster exists is where ⟨ k B ⟩ and ⟨ k 2 B ⟩ are the first and second moments of the degree distribution, respectively. Finally, we obtain the epidemic threshold as

S3. Simulation results
We first describe the simulation process of the two spreading dynamics on double-layer networks, and then demonstrate the validity of the theoretical analysis on uncorrelated networks with different network sizes and degree exponents, finally, we present results for SF-SF correlated networks.

A. Simulation process
To initiate an epidemic spreading process, a node in layer B is randomly infected and its counterpart node in layer A is thus in the informed state, too. The updating process is performed with parallel dynamics, which is widely used in statistical physics [11]. At each time step, we first calculate the informed (infected) probability that each susceptible node in layer A (B) may be informed (infected) by its informed (infected) neighbors, where n A I (n B I ) is the number of its informed (infected) neighboring nodes. According to the dynamic mechanism, once node A i is in the susceptible state, its counterpart node B i will be also in the susceptible state. Considering the asymmetric coupling between the two layers in this case, both the information-transmission and disease-transmission events can hardly occur at the same time. Thus, with probability π A /(π A + π B ), node A i have a probability π A to get the information from its informed neighbors in layer A. If node A i is informed, its counterpart node B i will turn into the vaccination state with probability p. With probability π B /(π A + π B ), node B i have a probability π B to get the infection from its infected neighbors in layer B, and then node A i also get the information about the disease.
In the other case that node B i and its corresponding node A i are in the susceptible state and the informed (or refractory) state respectively, only the disease-transmission event can occur at the time step. Thus, node B i will be infected with probability π B .
After renewing the states of susceptible nodes, each informed (infected) node can enter the recovering phase with probability µ A = 1.0 (µ B = 1.0). The spreading dynamics terminates when all informed (or infected) nodes in both layers are recovered, and the final densities R A , R B , and V B are then recorded. The simulations are implemented using 30 different two-layer network realizations. The network size of N A = N B = 2 × 10 4 and average degrees ⟨k A ⟩ = ⟨k B ⟩ = 8 are used for all subsequent numerical results, unless otherwise specified.

B. Uncorrelated double-layer networks
The effect of network size N on the information and epidemic outbreak thresholds is first studied. According to Eq. (S13), the behavior of the information threshold can be classified into two classes. For β B ≤ β Bu , the disease transmission on layer B has little impact on the information threshold, as we have β Ac ≈ β Au = ⟨k A ⟩/(⟨k 2 A ⟩ − ⟨k A ⟩); while β Ac = 0.0 for β B > β Bu . We here focus on the information threshold for β B ≤ β Bu . From Figs. S1(a) and (c), we see that the theoretical predictions are basically accordant with the simulated thresholds for different network sizes. With the growth of network size, the information threshold decreases as ⟨k 2 ⟩ of layer A increases [12]. According to Eq. (S25), the theoretical epidemic threshold can be predicted. For SF-ER double-layer networks, Figs. S1(b) and (d) shows that the simulated epidemic thresholds deviate slightly from the theoretical predictions. However, the larger deviations occur for the larger values of the vaccination rate p, e.g., p = 0.9 in Fig. S2, because the basic assumption of competing percolation theory is not strictly correct for the finite-size networks. As pointed out by Karrer and Newman [7], in the limit of large network size N , the vaccination and epidemic processes can be treated successively and separately. On the double-layer networks with finite network size, the effect of information spreading is somewhat over-emphasized. From Figs. S1 and S2, we also see that the discrepancy between the simulated and theoretical thresholds decreases with network size N .
We then investigate how the degree heterogeneity of layer A influences the information and epidemic outbreak thresholds by adjusting the exponent γ A . The information thresholds for the different exponents of layer A are compared in Fig. S3(a), and the stronger heterogeneity of layer A (i.e., smaller γ A ) can more easily make the information outbreak. Fig. S3(b) shows that increasing the heterogeneity of layer A can slightly raise the epidemic  threshold β Bc at a small information-transmission rate β A , while making for the epidemic outbreak at a large β A . This phenomenon results from the different effects of the heterogeneity on the information spreading under different transmission rates. The more homogeneous degree distribution does not always hinder the diffusion of information, especially at a large transmission rate [10,13].
To further demonstrate the validity of the theoretical analysis, we consider the case of SF-SF double-layer networks. Similar to the case of SF-ER networks, the gap between the theoretical and simulated thresholds is narrowing with the increase of network size [see Figs. S1(d) and S2], which implies the reasonability of the assumption in the thermodynamic limit. The final dynamical state of the SF-SF spreading system is also shown in Fig. S4, and it displays a similar phenomenon to the case of SF-ER networks. We also see that the theoretical predictions from mean-field rate equations are in good agreement with the simulation results.

C. Correlated double-layer networks
On SF-SF correlated networks, we investigate the effect of positive inter-layer correlation on the two types of spreading dynamics. As shown in Figs. S5, S6 and S7, with the increase of the correlation m s (by reducing the rematching probability q), the information threshold remains unchanged but the epidemic threshold can be enhanced, making the contact layer more robust to epidemic outbreak, which is consistent with the results for ER-ER correlated networks.