Epidemic Spreading in Interdependent Networks

We introduce a dynamic process of epidemic spreading into interdependent networks because, in reality, interdependent networks are facing the threat of epidemic spreading, which still lacks research. With our model, we reveal that (i) interdependent networks are more fragile than an isolated single network when facing the threat of an epidemic; (ii) an epidemic is not massively blocked by cascading failure, even if cascading failure spreads much faster; (iii) interdependent networks are more fragile, with a larger value of average degree compared with epidemic spreading. Moreover, we propose an iterative method for estimating the fraction of removed or failed nodes in a system of interdependent networks using percolation theory. This research can improve the comprehension of interdependent networks from the point of view of epidemic spreading.


Introduction
With deeper research in network science, people have begun to realize that many interdependent relations exist among networks [1][2][3].One of the core features of interdependent networks is that the existence or functioning of nodes in one network highly interacts with and depends on the existence or functioning of nodes in another network and vice versa [4][5][6].For example, in a typical system of interdependent networks formed by a communication network and power grid network, communication nodes depend on power stations for electricity supply, and power stations depend on communication nodes for control.Motivated by the exploration for the robustness of interdependent networks, current research on interdependent networks mainly focuses on the robustness of a system under the conditions of removing or deleting a fraction of nodes, which then leads to a chain reaction of node failure [7][8][9][10][11].
In reality, interdependent networks, especially those systems of interdependent networks formed by communication networks and other infrastructure networks, face many other threats, such as failure of overload [12][13][14] and epidemic spreading [15].The interdependent relationship among networks is considered as a major issue in diffusion and spreading processes over a multilayer networks system [16,17].In recent years, some researchers have explored the mechanism of epidemic spreading in multilayer networks, especially in interdependent networks according to the features of virus spreading in real multilayer network system [18,19].Most of these researches focus on revealing the epidemic threshold in interconnected and interdependent networks [20,21].However, until now, there are still only very few studies on these threats of epidemic spreading in interdependent networks.The spreading of an epidemic is one of the threats in interdependent networks.Some reports have shown that systems of interdependent networks coupled with a communication network and power grid network becomes vulnerable to the threat of computer viruses, leaving the power grid networks vulnerable to fail [22,23].Interdependent networks can lead to cascading failure after only a few nodes in a communication network become infected and no longer work.
Therefore, it is necessary to explore the dynamic process of epidemic spreading in interdependent networks.Although interdependent networks are threatened by epidemics, to the best of our knowledge, this subject still has largely unexplored issues.
In this study, we introduce the dynamic process of epidemic spreading into interdependent networks and explore the complicated dynamical interplay between epidemic spreading and cascading failure in interdependent networks using percolation theory.The remainder of this article is organized as follows.In Section 2, we introduce the network model and epidemic spreading model.In Section 3.1, we qualitatively analyze the results of epidemic spreading in interdependent networks.In Section 3.2, we propose an iterative method for estimating the fraction of removed or failed nodes in a system of interdependent networks using percolation theory and validate it via simulations.Section 4 provides concluding remarks.

Models
2.1.Network Model.We first present an interdependent network model to describe the coupled patterns system structure.To simplify and clarify the results, we make reference to the model presented in [4] to construct interdependent networks.A system of interdependent networks Π consisting of two networks,  and , is considered.Without loss of generality, networks  and , with the same size  =   =   and the same average degree ⟨⟩ = ⟨  ⟩ = ⟨  ⟩, follow degree distributions of   () and   (), respectively.Note that we make an important assumption, i.e., that the time scale of the cascading failure is much smaller than that of epidemic propagation.This is because we consider the duration of cascading failures as much longer than that of epidemic spreading.For example, the cascading overload of the power grid network in the United States and Canada in 2003 only lasted for 8 minutes [24], while the spread of a computer virus can usually last for several months [25,26].Therefore, in our model, every single substep of epidemic spreading is executed until the chain reaction of node failure stops.

An Example.
To better understand the model, we give a simple example here, as shown in Figure 2. Networks  and  are coupled with each other to form a system of interdependent networks Π. Node   in network A depends on node   in network  and vice versa.First, all nodes in networks  and  are susceptible and functioning.Then, node  1 in network  becomes infected as shown in Figure 2(a) and the system Π falls into an unstable dynamic process.In stage 1, node  3 is infected by node  1 with probably , and then  1 is removed due to infection.Node  2 fails because it does not connect to the MCC.In network , nodes  1 and  2 fail because they lose their dependent nodes  1 and  2 in network .Then, all survival nodes (including node  3 ) in network  connect to the MCC and are supported by nodes in network  and vice versa.Therefore, stage 1 finishes as illustrated in Figure 2(b).However, there still exists an infected node in network ; thus, the dynamic process enters into stage 2. Node  5 becomes infected with probably  as it connects to node  3 ; then, node  3 is removed.Node  3 fails because it loses its support node  3 .Then, all survival nodes in network  connect to MCC and are supported by nodes in network  and vice versa.Therefore, stage 2 finishes, as shown in Figure 2(c).Because there still exists an infected node  5 in network , the dynamic process enters the next stage.In stage 3, none of the neighbors of  5 in network  become infected, and node  5 is removed after a time step.Node  4 fails because it does not connect to the MCC in network .Nodes  4 and  5 fail because  4 and  5 either fail or are removed in this stage.Now, there is no infected node in network , and the dynamic process halts; thus stage 3 is the final status after epidemic spreading in interdependent networks.

Analysis
3.1.Qualitative Analysis.In this section, we qualitatively analyze the epidemic spreading in interdependent networks using the simulation results.Figures 3 and 4    which    +    and    increase from zero to nonzero.This indicates that the epidemic starts to spread out in network  and begins to lead to a chain reaction of cascading failure in both networks  and .We believe this critical value of  is the same as the critical value of the epidemic effective spreading rate in an isolated single network with the same structure as network .This is because the cascading failure in interdependent networks is caused by the spreading of the epidemic.When  is extremely small, only very few nodes become infected, and it is too far from the threshold to lead to a chain reaction of cascading failure.Second, by further increasing ,    +    and    increase to 1.This critical moment implies that epidemic and cascading failure kill all nodes in system Π.The appearance of these two critical points indicates the fragility of interdependent networks compared with epidemic spreading.
(2) An epidemic is not blocked massively by a cascading failure; even the speed of a cascading failure is much faster than that of epidemic spreading.When  is sufficiently large, an epidemic can still infect most nodes in network , although the strength of the cascading failure is much more intense than that of epidemic spreading.Figure 5 exhibits the dynamic process of epidemic spreading and cascading failure in the time domain.In each time step of epidemic spreading, only a small fraction of nodes in network  are removed from infected status and cause another small fraction of nodes to fail in both networks  and .Therefore, the intensity of cascading failure is not strong enough to block an epidemic, although cascading failure spreads much faster than an epidemic at each time step.
(3) Interdependent networks are more fragile, with a larger value of ⟨⟩ compared with epidemic spreading.In contrast, interdependent networks with the same network structure as our model are more fragile, with a smaller value of ⟨⟩ compared with cascading failure [4].This contradictory performance is caused by the different mechanisms of these two dynamic processes.In [4], cascading failure is first caused by the removal of (1 − ) fraction of nodes in network  at first.With the increase of ⟨⟩, the probability that a functioning node is not in the MCC decreases, and the robustness of interdependent networks against cascading failure is improved.In our model, with a higher value of ⟨⟩, it is easier for the epidemic in network  to spread out, leading to a more intense cascading failure.

Quantitative Analysis.
In this section, we quantitatively analyze the proportion of removed nodes    and failed nodes    in network  and the proportion of failed nodes    in network .

Two Types of Percolation
Processes.The spreading of an epidemic is a bond percolation process [27,28].The process of epidemic spreading is equivalent to selecting a link from a network with probability , and the outbreak of an epidemic indicates that the set of chosen links forms a connectivity cluster.Cascading failure is equivalent to a site percolation [29,30].In our model, when an epidemic spreads out, there exist several nodes in network  participating in both bond percolation and site percolation.We assume that these nodes in network  belong to the bond percolation process rather than to the site percolation process, because, in each time step, the spreading of the epidemic causes a further dynamic process of cascading failure and cascading failure is not strong enough to block an epidemic as mentioned in the Qualitative Analysis.
We denote    and    as the probability that a node of degree  in network  participates in site percolation and bond percolation, respectively.We denote    and    as the probability that a node of degree  in network  fails or is removed, respectively, after the dynamic process terminates in steady state.We denote    and    as the probability that a node of degree  in network  participates in site percolation or fails in steady state, respectively.According to our assumption regarding those nodes for both participating bond percolation and site percolation, we have In a coupled degree-degree uncorrelated interdependent system of networks, the neighbor of an arbitrary node in network  becomes a failed node or removed node with probability   or   , respectively, and the neighbor of an arbitrary node in network  becomes a failed node with probability   , which are calculated as follows:

The Proportion of Removed Nodes in Network 𝐴.
In network , a susceptible node will be infected with probability  for each link connecting to an infected node.Therefore, considering the locally tree-like approximation in percolation, the probability that nodes of degree  in network  participate in bond percolation is Substituting ( 7) into (2), we get the probability that nodes of degree  in network  will become removed nodes is

The Proportion of Failed
Nodes in Network .From a microscopic view, the failure of a coupled pair of nodes   and   starts from one side to the other side.Thus, we define two parameters describing the direction of reaction between this coupled pair of nodes: (i) : the probability that the failure of node   in network  is caused by losing the dependent node   in network  directly.(ii) : the probability that the failure of node   in network  is caused by losing the dependent node   in network  directly.
Then, an arbitrary node   of degree  in network  who participates in site percolation can be divided into two situations: (i) The probability that the failure of   is caused by losing the dependent node   directly is defined as (ii) The probability that node   fails and the failure is NOT caused by losing the dependent node   directly is defined as Substituting ( 8), (9), and ( 10) into (1), we can get the probability that nodes with degree  in network  fail 3.2.4.The Proportion of Failed Nodes in Network .Similarly, an arbitrary node   of degree  in network  that participates in site percolation can be divided into two situations: (i) The probability that the failure of   is caused by losing the dependent node   directly is defined as (ii) The probability that node   fails and the failure is NOT caused by losing the dependent node   directly is defined as Substituting ( 12) and ( 13) into (3), we can get the probability that nodes with degree  in network  fail: 3.2.5.The Relationship between Parameters  and .According to the definition of , it is equivalent to the probability that a node of network  fails first and then leads to the failure of a random node in network  as shown in the following equation: is equivalent to the probability that a node of network  is removed or fails first and then leads to the failure of a random node in network  as follows: 3.2.6.The Proportion of All Nodes.Now we achieve the derivation by using (4), ( 5), ( 6), ( 8), (11), and (14).For clarity, we relist them as follows: Obviously, these six equations depend on each other.Thus we can get the fraction of removed nodes   and the fraction of failed nodes   of network  and the fraction of failed nodes   of network  in a steady state from these six equations via the following iteration process: (1) Assign a very small initial value to    min , where  min represents the minimum degree of nodes in network .Let    = 0( ≤  min ),    = 0, and    = 0. Substitute    ,    , and    into ( 17), (18), and ( 19) and get the initial values of   ,   ,   , , and .

Simulations and Discussion
We test the validity of the analytical predictions/results in coupled ER random networks and coupled BA scale-free networks with varied average degree ⟨⟩ = 6, 8, 10 The reason for these deviations is that nodes that participate both in the bond percolation (epidemic spreading) and in site percolation (cascading failure) processes are supposed to belong to the former one in the theoretical prediction.Therefore, in network , the fraction of removed nodes is     In order to understand the dynamic behaviors closer to the real network topology structure, we further constructed an scene that a wireless network coupled with a scalefree network.Assume that epidemic spreads in the wireless network, which causes the entire coupled networks into cascading failure.We select DNG (Delaunay triangulation graph), KNN (k-nearest neighbor), and LAEE (local-area and energy-efficient) topologies for wireless networks [31].In the experiment, the number of nodes in each network is 4000, and the transmission range is 100 meters.Each value is the average of 50 dependent realizations.As Figure 12(    shows, with the epidemic effective spreading rate  increases, only some nodes in the wireless network eventually become removed.This shows that the spread of epidemic has been significantly hampered in these coupled structure.The reason for this phenomenon is that when  is very large, a large number of nodes in the network are removed and failed, and then some of the epidemic transmission paths are cut off, as shown in Figure 12(b).Compared with separate wireless networks, coupled networks of wireless networks and other networks are more vulnerable to epidemic spreading.This is because there are both epidemic transmission and cascade failure in these coupled networks, as shown in Figure 12(c).

Conclusion
Researching interdependent networks has been one of the hotspots of complex network science in recent years.Current studies on interdependent networks mainly focus on the  robustness of a system with the condition of deleting a fraction of nodes.In reality, interdependent networks face other threats, such as the spread of epidemics, which still lacks research.Therefore, we propose a model that introduces the dynamic process of epidemic spreading into interdependent networks.With our model, we reveal that (i) interdependent networks are more fragile than an isolated single network when facing the threat of an epidemic, (ii) an epidemic is not massively blocked by cascading failure, even if cascading failure spreads much faster, and (iii) interdependent networks are more fragile, with a larger value of ⟨⟩ compared with epidemic spreading.Moreover, we propose an iterative method for estimating the fraction of removed or failed nodes in a system of interdependent networks using percolation theory and validate it via simulations.We believe this research can improve the comprehension of interdependent networks from the point of view of epidemic spreading.

Figure 1 :
Figure 1: Node status transition in interdependent networks.The statuses S, I, R, and F are abbreviations for susceptible, infected, removed, and failed, respectively.

Figure 5 :
Figure 5: In each time step of epidemic spreading, the fraction of newly removed and failed nodes in network  (Δ   and Δ   , respectively) and newly failed nodes in network  (Δ   ) in a coupled ER network with ⟨⟩ = 10 and  = 0.18 is shown.All the results shown are averaged over 50 realizations.
(a),(b).In, the fraction of nonfunctioning nodes in networks  and  is well estimated.

Figure 12 :
Figure 12: Numerical simulations of coupled wireless network and BA network with ⟨⟩ = 6.

each node is in only one of two statuses: susceptible and failed. A susceptible node in net- work 𝐵 fails if (i) it loses its dependent node in network 𝐴 or (ii) it is not in the giant component formed by susceptible nodes. The dynamic process of epidemic spreading in inter- dependent networks is detailed in the following steps:
These two networks are coupled with random bidirectional interdependent links: (i) the functioning of an arbitrary node   in network  depends and only depends on a randomly picked node   in network  and vice versa; (ii) if   stops functioning,   also stops functioning and vice versa.Here, we say that node   depends on node   and vice versa.An example of epidemic spreading in a system of interdependent networks Π. Links inside networks  and  are shown as arcs, while interdependent links between networks  and  are shown as horizontal straight lines.Susceptible nodes in networks  and  are 2.2.Dynamical Behavior Model.For example, in a system of interdependent networks Π, network  is a communication network through which the epidemic spreads, while network  is a power grid network.Combining the SIR epidemic spreading model and interdependent networks model, nodes statuses transition relations are revealed, as in Figure1.In network , an arbitrary node is only in one of the four statuses: susceptible, infected, removed, and failed.marked as ⃝ and r, respectively.Infected nodes in network  are marked as e.Removed and failure nodes are no longer functioning; thus, these nodes and the links to them are not presented in this figure.
exhibit the fractions of removed nodes    and failed nodes    in network  and the fraction of failed nodes    in network  with varied epidemic effective spreading rates , in coupled ER (Erdös Rényi) and BA (Barabási Albert) networks.Notice that the fraction of failed nodes in network  equals the sum of the fractions of removed and failed nodes in network , (1) Interdependent networks are more fragile than an isolated single network with the same value of average degree  when compared with epidemic spreading.Both    +   and    exist as two critical points, which vary by .First, there is a small critical value of , at ,    , and    .Next, substitute the new values of    ,    , and    into (17), (18), and (19) and then update   ,   ,   , , and .Repeat this step until convergence.
. The simulation results are illustrated in.Each data point is the average of 50 dependent realizations.It can be seen from Figures 6-11 that, on the whole, our theoretical values are in good, though not perfect, agreement with the simulation results.In Figures6-11 (a), the epidemic breaks out if  is larger than a critical value.When  is large enough for the epidemic to break out, the theoretical value of    is a little smaller than the simulation result of    .As ⟨⟩ increases, the deviation of    between the theoretical value and simulation result becomes smaller.In contrast, the theoretical value of    becomes larger than the simulation result of    , which is larger than the simulation results of    when the epidemic breaks out, and the deviation of decreases as ⟨⟩ increases, as shown in.