# Epidemic Spreading in Interdependent Networks

Authors: Lurong Jiang; Qiaoyu Xu; Bo Ouyang; Yicong Lang; Yanyun Dai; Jijun Tong
Journal: Mathematical Problems in Engineering
 (2018)

---

## Abstract

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.


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## Body

## 1. Introduction

With deeper research in network science, people have begun to realize that many interdependent relations exist among networks [1–3]. 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 [4–6]. 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 [7–11].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 [12–14] 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 Section2, 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.


## 2. 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, A and B, is considered. Without loss of generality, networks A and B, with the same size N
=
N
A
=
N
B and the same average degree 〈
k
〉
=
〈
k
A
〉
=
〈
k
B
〉, follow degree distributions of p
A
(
k
) and p
B
(
k
), respectively. These two networks are coupled with random bidirectional interdependent links: (i) the functioning of an arbitrary node a
i in network A depends and only depends on a randomly picked node b
i in network B and vice versa; (ii) if a
i stops functioning, b
i also stops functioning and vice versa. Here, we say that node a
i depends on node b
i and vice versa.


### 2.2. Dynamical Behavior Model

For example, in a system of interdependent networksΠ, network A is a communication network through which the epidemic spreads, while network B is a power grid network. Combining the SIR epidemic spreading model and interdependent networks model, nodes statuses transition relations are revealed, as in Figure 1. In network A, an arbitrary node is only in one of the four statuses: susceptible, infected, removed, and failed. Susceptible nodes are those healthy and functioning nodes that can be infected. Infected nodes have the ability to infect their susceptible neighbors. Removed nodes are those nonfunctioning nodes from infected status, which cannot be infected again. A susceptible node becomes infected with a probability β for each of its links, leading to an infected node, while an infected node becomes removed with probability γ for every time step. The epidemic effective spreading rate is λ
=
β
/
γ. Without loss of generality, we let γ
=
1. A node in network A fails due to three possible reasons: (i) an infected node loses its dependent node in network B; (ii) a susceptible node loses its dependent node in network B; (iii) a susceptible node is not in the mutually connected clusters (MCC). Both removed and failed nodes in network A are nonfunctioning and cannot provide support for nodes in network B. These failed nodes also no longer interact with other nodes in the consequent process either.Figure 1
Node status transition in interdependent networks. The statuses S, I, R, and F are abbreviations for susceptible, infected, removed, and failed, respectively.
(a)
Network A
(b)
Network BIn networkB, each node is in only one of two statuses: susceptible and failed. A susceptible node in network 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 interdependent networks is detailed in the following steps:(1)
First, all nodes in the system of interdependent networksΠ are susceptible. Randomly select a few nodes in network A as the initial infected nodes.
(2)
Substep 2.1. In network A, for every link connecting an infected node to a susceptible node, the susceptible node becomes infected with probability λ, and, for every infected node, it becomes removed with probability 1.
Substep 2.2. The removal of nodes in network A may cause chain reaction of nodes failure in both networks A and B. Repeat these cascades until all remaining nodes in network A are in the MCC and supported by nodes in network B and vice versa.
(3)
If there still exist infected nodes in networkA, go back to (
2
) to start the next stage of epidemic spreading; otherwise, the process halts.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.


### 2.3. An Example

To better understand the model, we give a simple example here, as shown in Figure2. Networks A and B are coupled with each other to form a system of interdependent networks Π. Node a
i in network A depends on node b
i in network B and vice versa. First, all nodes in networks A and B are susceptible and functioning. Then, node a
1 in network A becomes infected as shown in Figure 2(a) and the system Π falls into an unstable dynamic process. In stage 1, node a
3 is infected by node a
1 with probably λ, and then a
1 is removed due to infection. Node a
2 fails because it does not connect to the MCC. In network B, nodes b
1 and b
2 fail because they lose their dependent nodes a
1 and a
2 in network A. Then, all survival nodes (including node a
3) in network A connect to the MCC and are supported by nodes in network B and vice versa. Therefore, stage 1 finishes as illustrated in Figure 2(b). However, there still exists an infected node in network A; thus, the dynamic process enters into stage 2. Node a
5 becomes infected with probably λ as it connects to node a
3; then, node a
3 is removed. Node b
3 fails because it loses its support node a
3. Then, all survival nodes in network A connect to MCC and are supported by nodes in network B and vice versa. Therefore, stage 2 finishes, as shown in Figure 2(c). Because there still exists an infected node a
5 in network A, the dynamic process enters the next stage. In stage 3, none of the neighbors of a
5 in network A become infected, and node a
5 is removed after a time step. Node a
4 fails because it does not connect to the MCC in network A. Nodes b
4 and b
5 fail because a
4 and a
5 either fail or are removed in this stage. Now, there is no infected node in network A, and the dynamic process halts; thus stage 3 is the final status after epidemic spreading in interdependent networks.Figure 2
An example of epidemic spreading in a system of interdependent networksΠ. Links inside networks A and B are shown as arcs, while interdependent links between networks A and B are shown as horizontal straight lines. Susceptible nodes in networks A and B are marked as ◯ and ⬜, respectively. Infected nodes in network A are marked as ●. Removed and failure nodes are no longer functioning; thus, these nodes and the links to them are not presented in this figure.
(a)
Start
(b)
Stage 1
(c)
Stage 2
(d)
Stage 3


## 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, A and B, is considered. Without loss of generality, networks A and B, with the same size N
=
N
A
=
N
B and the same average degree 〈
k
〉
=
〈
k
A
〉
=
〈
k
B
〉, follow degree distributions of p
A
(
k
) and p
B
(
k
), respectively. These two networks are coupled with random bidirectional interdependent links: (i) the functioning of an arbitrary node a
i in network A depends and only depends on a randomly picked node b
i in network B and vice versa; (ii) if a
i stops functioning, b
i also stops functioning and vice versa. Here, we say that node a
i depends on node b
i and vice versa.


## 2.2. Dynamical Behavior Model

For example, in a system of interdependent networksΠ, network A is a communication network through which the epidemic spreads, while network B is a power grid network. Combining the SIR epidemic spreading model and interdependent networks model, nodes statuses transition relations are revealed, as in Figure 1. In network A, an arbitrary node is only in one of the four statuses: susceptible, infected, removed, and failed. Susceptible nodes are those healthy and functioning nodes that can be infected. Infected nodes have the ability to infect their susceptible neighbors. Removed nodes are those nonfunctioning nodes from infected status, which cannot be infected again. A susceptible node becomes infected with a probability β for each of its links, leading to an infected node, while an infected node becomes removed with probability γ for every time step. The epidemic effective spreading rate is λ
=
β
/
γ. Without loss of generality, we let γ
=
1. A node in network A fails due to three possible reasons: (i) an infected node loses its dependent node in network B; (ii) a susceptible node loses its dependent node in network B; (iii) a susceptible node is not in the mutually connected clusters (MCC). Both removed and failed nodes in network A are nonfunctioning and cannot provide support for nodes in network B. These failed nodes also no longer interact with other nodes in the consequent process either.Figure 1
Node status transition in interdependent networks. The statuses S, I, R, and F are abbreviations for susceptible, infected, removed, and failed, respectively.
(a)
Network A
(b)
Network BIn networkB, each node is in only one of two statuses: susceptible and failed. A susceptible node in network 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 interdependent networks is detailed in the following steps:(1)
First, all nodes in the system of interdependent networksΠ are susceptible. Randomly select a few nodes in network A as the initial infected nodes.
(2)
Substep 2.1. In network A, for every link connecting an infected node to a susceptible node, the susceptible node becomes infected with probability λ, and, for every infected node, it becomes removed with probability 1.
Substep 2.2. The removal of nodes in network A may cause chain reaction of nodes failure in both networks A and B. Repeat these cascades until all remaining nodes in network A are in the MCC and supported by nodes in network B and vice versa.
(3)
If there still exist infected nodes in networkA, go back to (
2
) to start the next stage of epidemic spreading; otherwise, the process halts.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.


## 2.3. An Example

To better understand the model, we give a simple example here, as shown in Figure2. Networks A and B are coupled with each other to form a system of interdependent networks Π. Node a
i in network A depends on node b
i in network B and vice versa. First, all nodes in networks A and B are susceptible and functioning. Then, node a
1 in network A becomes infected as shown in Figure 2(a) and the system Π falls into an unstable dynamic process. In stage 1, node a
3 is infected by node a
1 with probably λ, and then a
1 is removed due to infection. Node a
2 fails because it does not connect to the MCC. In network B, nodes b
1 and b
2 fail because they lose their dependent nodes a
1 and a
2 in network A. Then, all survival nodes (including node a
3) in network A connect to the MCC and are supported by nodes in network B and vice versa. Therefore, stage 1 finishes as illustrated in Figure 2(b). However, there still exists an infected node in network A; thus, the dynamic process enters into stage 2. Node a
5 becomes infected with probably λ as it connects to node a
3; then, node a
3 is removed. Node b
3 fails because it loses its support node a
3. Then, all survival nodes in network A connect to MCC and are supported by nodes in network B and vice versa. Therefore, stage 2 finishes, as shown in Figure 2(c). Because there still exists an infected node a
5 in network A, the dynamic process enters the next stage. In stage 3, none of the neighbors of a
5 in network A become infected, and node a
5 is removed after a time step. Node a
4 fails because it does not connect to the MCC in network A. Nodes b
4 and b
5 fail because a
4 and a
5 either fail or are removed in this stage. Now, there is no infected node in network A, and the dynamic process halts; thus stage 3 is the final status after epidemic spreading in interdependent networks.Figure 2
An example of epidemic spreading in a system of interdependent networksΠ. Links inside networks A and B are shown as arcs, while interdependent links between networks A and B are shown as horizontal straight lines. Susceptible nodes in networks A and B are marked as ◯ and ⬜, respectively. Infected nodes in network A are marked as ●. Removed and failure nodes are no longer functioning; thus, these nodes and the links to them are not presented in this figure.
(a)
Start
(b)
Stage 1
(c)
Stage 2
(d)
Stage 3


## 3. Analysis

### 3.1. Qualitative Analysis

In this section, we qualitatively analyze the epidemic spreading in interdependent networks using the simulation results. Figures3 and 4 exhibit the fractions of removed nodes σ
r
A and failed nodes σ
f
A in network A and the fraction of failed nodes σ
f
B in network B with varied epidemic effective spreading rates λ, in coupled ER (Erdös Rényi) and BA (Barabási Albert) networks. Notice that the fraction of failed nodes in network B equals the sum of the fractions of removed and failed nodes in network A, which indicates that σ
f
B
=
σ
r
A
+
σ
f
A. All the results shown are averaged over 50 realizations. The performances of results are enumerated qualitatively as follows.Figure 3
Numerical simulations of coupled ER networks with〈
k
〉
=
6,8
,
10,12 and N
=
5000.
(a)
σ
r
A
(b)
σ
f
A
(c)
σ
f
B or σ
f
A
+
r
AFigure 4
Numerical simulations of coupled BA networks with〈
k
〉
=
6,8
,
10,12 and N
=
5000.
(a)
σ
r
A
(b)
σ
f
A
(c)
σ
f
B or σ
f
A
+
r
A(1)
Interdependent networks are more fragile than an isolated single network with the same value of average degreek when compared with epidemic spreading. Both σ
r
A
+
σ
f
A and σ
f
B exist as two critical points, which vary by λ. First, there is a small critical value of λ, at which σ
r
A
+
σ
f
A and σ
f
B increase from zero to nonzero. This indicates that the epidemic starts to spread out in network A and begins to lead to a chain reaction of cascading failure in both networks A and B. We believe this critical value of λ is the same as the critical value of the epidemic effective spreading rate in an isolated single network with the same structure as network A. This is because the cascading failure in interdependent networks is caused by the spreading of the epidemic. When λ 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 λ, σ
r
A
+
σ
f
A and σ
f
B 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. Whenλ is sufficiently large, an epidemic can still infect most nodes in network A, 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 A are removed from infected status and cause another small fraction of nodes to fail in both networks A and B. 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〈
k
〉 compared with epidemic spreading. In contrast, interdependent networks with the same network structure as our model are more fragile, with a smaller value of 〈
k
〉 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
-
p
) fraction of nodes in network A at first. With the increase of 〈
k
〉, 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 〈
k
〉, it is easier for the epidemic in network A to spread out, leading to a more intense cascading failure.Figure 5
In each time step of epidemic spreading, the fraction of newly removed and failed nodes in networkA (Δ
ϕ
r
A and Δ
ϕ
f
A, respectively) and newly failed nodes in network B (Δ
ϕ
f
B) in a coupled ER network with 〈
k
〉
=
10 and λ
=
0.18 is shown. All the results shown are averaged over 50 realizations.


### 3.2. Quantitative Analysis

In this section, we quantitatively analyze the proportion of removed nodesσ
r
A and failed nodes σ
f
A in network A and the proportion of failed nodes σ
f
B in network B.


#### 3.2.1. Two Types of Percolation Processes

The spreading of an epidemic is abond percolation process [27, 28]. The process of epidemic spreading is equivalent to selecting a link from a network with probability λ, and the outbreak of an epidemic indicates that the set of chosen links forms a connectivity cluster. Cascading failure is equivalent to asite percolation [29, 30]. In our model, when an epidemic spreads out, there exist several nodes in network A participating in both bond percolation and site percolation. We assume that these nodes in network A 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 theQualitative Analysis.We denotes
k
A and b
k
A as the probability that a node of degree k in network A participates in site percolation and bond percolation, respectively. We denote f
k
A and r
k
A as the probability that a node of degree k in network A fails or is removed, respectively, after the dynamic process terminates in steady state. We denote s
k
B and f
k
B as the probability that a node of degree k in network B 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(1)
f
k
A
=
1
-
b
k
A
s
k
A
,
(2)
r
k
A
=
b
k
A
,
(3)
f
k
B
=
s
k
B
.In a coupled degree-degree uncorrelated interdependent system of networks, the neighbor of an arbitrary node in networkA becomes a failed node or removed node with probability ϕ
A or ρ
A, respectively, and the neighbor of an arbitrary node in network B becomes a failed node with probability ϕ
B, which are calculated as follows:(4)
ϕ
A
=
∑
j
j
p
j
f
j
A
k
A
=
∑
j
j
p
j
1
-
b
j
A
s
j
A
k
A
,
(5)
ρ
A
=
∑
j
j
p
j
r
j
A
k
A
=
∑
j
j
p
j
b
j
A
k
A
,
(6)
ϕ
B
=
∑
j
j
p
j
f
j
B
k
B
=
∑
j
j
p
j
s
j
B
k
B
.


#### 3.2.2. The Proportion of Removed Nodes in NetworkA

In networkA, a susceptible node will be infected with probability λ for each link connecting to an infected node. Therefore, considering the locally tree-like approximation in percolation, the probability that nodes of degree k in network A participate in bond percolation is(7)
b
k
A
=
1
-
1
-
λ
∑
j
j
p
j
b
j
A
k
A
k
-
1
=
1
-
1
-
λ
ρ
A
k
-
1
.Substituting (7) into (2), we get the probability that nodes of degree k in network A will become removed nodes is(8)
r
k
A
=
b
k
A
=
1
-
1
-
λ
ρ
A
k
-
1
.


#### 3.2.3. The Proportion of Failed Nodes in NetworkA

From a microscopic view, the failure of a coupled pair of nodesa
i and b
i 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)
α: the probability that the failure of node a
i in network A is caused by losing the dependent node b
i in network B directly.
(ii)
β: the probability that the failure of node b
i in network B is caused by losing the dependent node a
i in network A directly.Then, an arbitrary nodea
i of degree k in network A who participates in site percolation can be divided into two situations:(i)
The probability that the failure ofa
i is caused by losing the dependent node b
i directly is defined as(9)
s
k
A
1
=
α
.
(ii)
The probability that nodea
i fails and the failure is NOT caused by losing the dependent node b
i directly is defined as(10)
s
k
A
2
=
1
-
α
ρ
A
+
ϕ
A
k
.Substituting (8), (9), and (10) into (1), we can get the probability that nodes with degree k in network A fail(11)
f
k
A
=
1
-
r
k
A
s
k
A
1
+
s
k
A
2
=
1
-
r
k
A
α
+
1
-
α
ρ
A
+
ϕ
A
k
.


#### 3.2.4. The Proportion of Failed Nodes in NetworkB

Similarly, an arbitrary nodeb
j of degree k in network B that participates in site percolation can be divided into two situations:(i)
The probability that the failure ofb
j is caused by losing the dependent node a
j directly is defined as(12)
s
k
B
1
=
β
.
(ii)
The probability that nodeb
j fails and the failure is NOT caused by losing the dependent node a
j directly is defined as(13)
s
k
B
2
=
1
-
β
ϕ
B
k
.Substituting (12) and (13) into (3), we can get the probability that nodes with degree k in network B fail:(14)
f
k
B
=
s
k
B
=
s
k
B
1
+
s
k
B
2
=
β
+
1
-
β
ϕ
B
k
.


#### 3.2.5. The Relationship between Parametersα and β

According to the definition ofα, it is equivalent to the probability that a node of network B fails first and then leads to the failure of a random node in network A as shown in the following equation:(15)
α
=
∑
j
p
j
s
j
B
2
=
∑
j
p
j
1
-
β
ϕ
B
j
.β is equivalent to the probability that a node of network A is removed or fails first and then leads to the failure of a random node in network B as follows:(16)
β
=
∑
j
p
j
b
j
A
+
1
-
b
j
A
s
j
A
2
=
∑
j
p
j
r
j
A
+
1
-
r
j
A
1
-
α
ρ
A
+
ϕ
A
j
.


#### 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:(17)
ϕ
A
=
∑
j
j
p
j
f
j
A
k
A
,
(18)
ρ
A
=
∑
j
j
p
j
r
j
A
k
A
,
(19)
ϕ
B
=
∑
j
j
p
j
f
j
B
k
B
,
(20)
r
k
A
=
1
-
1
-
λ
ρ
A
k
-
1
,
(21)
f
k
A
=
1
-
r
k
A
α
+
1
-
α
ρ
A
+
ϕ
A
k
,
(22)
f
k
B
=
β
+
1
-
β
ϕ
B
k
.Obviously, these six equations depend on each other. Thus we can get the fraction of removed nodesρ
A and the fraction of failed nodes ϕ
A of network A and the fraction of failed nodes ρ
B of network B in a steady state from these six equations via the following iteration process:(1)
Assign a very small initial value tor
k
m
i
n
A, where k
m
i
n represents the minimum degree of nodes in network A. Let r
k
A
=
0
(
k
≤
k
m
i
n
), f
k
A
=
0, and f
k
B
=
0. Substitute r
k
A, f
k
A, and f
k
B into (17), (18), and (19) and get the initial values of ϕ
A, ρ
A, ϕ
B, α, and β.
(2)
Substituteϕ
A, ρ
A, ϕ
B, α, and β into (20), (21), and (22) to get the values of r
k
A, f
k
A, and f
k
B. Next, substitute the new values of r
k
A, f
k
A, and f
k
B into (17), (18), and (19) and then update ϕ
A, ρ
A, ϕ
B, α, and β. Repeat this step until convergence.
(3)
Get the fraction of removed and failed nodes in networkA by using σ
r
A
=
∑
k
p
k
A
r
k
A and σ
f
A
=
∑
k
p
k
A
f
k
A and the fraction of failed nodes in network B by using s
i
g
m
a
f
B
=
∑
k
p
k
B
f
k
B.


## 3.1. Qualitative Analysis

In this section, we qualitatively analyze the epidemic spreading in interdependent networks using the simulation results. Figures3 and 4 exhibit the fractions of removed nodes σ
r
A and failed nodes σ
f
A in network A and the fraction of failed nodes σ
f
B in network B with varied epidemic effective spreading rates λ, in coupled ER (Erdös Rényi) and BA (Barabási Albert) networks. Notice that the fraction of failed nodes in network B equals the sum of the fractions of removed and failed nodes in network A, which indicates that σ
f
B
=
σ
r
A
+
σ
f
A. All the results shown are averaged over 50 realizations. The performances of results are enumerated qualitatively as follows.Figure 3
Numerical simulations of coupled ER networks with〈
k
〉
=
6,8
,
10,12 and N
=
5000.
(a)
σ
r
A
(b)
σ
f
A
(c)
σ
f
B or σ
f
A
+
r
AFigure 4
Numerical simulations of coupled BA networks with〈
k
〉
=
6,8
,
10,12 and N
=
5000.
(a)
σ
r
A
(b)
σ
f
A
(c)
σ
f
B or σ
f
A
+
r
A(1)
Interdependent networks are more fragile than an isolated single network with the same value of average degreek when compared with epidemic spreading. Both σ
r
A
+
σ
f
A and σ
f
B exist as two critical points, which vary by λ. First, there is a small critical value of λ, at which σ
r
A
+
σ
f
A and σ
f
B increase from zero to nonzero. This indicates that the epidemic starts to spread out in network A and begins to lead to a chain reaction of cascading failure in both networks A and B. We believe this critical value of λ is the same as the critical value of the epidemic effective spreading rate in an isolated single network with the same structure as network A. This is because the cascading failure in interdependent networks is caused by the spreading of the epidemic. When λ 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 λ, σ
r
A
+
σ
f
A and σ
f
B 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. Whenλ is sufficiently large, an epidemic can still infect most nodes in network A, 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 A are removed from infected status and cause another small fraction of nodes to fail in both networks A and B. 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〈
k
〉 compared with epidemic spreading. In contrast, interdependent networks with the same network structure as our model are more fragile, with a smaller value of 〈
k
〉 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
-
p
) fraction of nodes in network A at first. With the increase of 〈
k
〉, 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 〈
k
〉, it is easier for the epidemic in network A to spread out, leading to a more intense cascading failure.Figure 5
In each time step of epidemic spreading, the fraction of newly removed and failed nodes in networkA (Δ
ϕ
r
A and Δ
ϕ
f
A, respectively) and newly failed nodes in network B (Δ
ϕ
f
B) in a coupled ER network with 〈
k
〉
=
10 and λ
=
0.18 is shown. All the results shown are averaged over 50 realizations.


## 3.2. Quantitative Analysis

In this section, we quantitatively analyze the proportion of removed nodesσ
r
A and failed nodes σ
f
A in network A and the proportion of failed nodes σ
f
B in network B.


### 3.2.1. Two Types of Percolation Processes

The spreading of an epidemic is abond percolation process [27, 28]. The process of epidemic spreading is equivalent to selecting a link from a network with probability λ, and the outbreak of an epidemic indicates that the set of chosen links forms a connectivity cluster. Cascading failure is equivalent to asite percolation [29, 30]. In our model, when an epidemic spreads out, there exist several nodes in network A participating in both bond percolation and site percolation. We assume that these nodes in network A 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 theQualitative Analysis.We denotes
k
A and b
k
A as the probability that a node of degree k in network A participates in site percolation and bond percolation, respectively. We denote f
k
A and r
k
A as the probability that a node of degree k in network A fails or is removed, respectively, after the dynamic process terminates in steady state. We denote s
k
B and f
k
B as the probability that a node of degree k in network B 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(1)
f
k
A
=
1
-
b
k
A
s
k
A
,
(2)
r
k
A
=
b
k
A
,
(3)
f
k
B
=
s
k
B
.In a coupled degree-degree uncorrelated interdependent system of networks, the neighbor of an arbitrary node in networkA becomes a failed node or removed node with probability ϕ
A or ρ
A, respectively, and the neighbor of an arbitrary node in network B becomes a failed node with probability ϕ
B, which are calculated as follows:(4)
ϕ
A
=
∑
j
j
p
j
f
j
A
k
A
=
∑
j
j
p
j
1
-
b
j
A
s
j
A
k
A
,
(5)
ρ
A
=
∑
j
j
p
j
r
j
A
k
A
=
∑
j
j
p
j
b
j
A
k
A
,
(6)
ϕ
B
=
∑
j
j
p
j
f
j
B
k
B
=
∑
j
j
p
j
s
j
B
k
B
.


### 3.2.2. The Proportion of Removed Nodes in NetworkA

In networkA, a susceptible node will be infected with probability λ for each link connecting to an infected node. Therefore, considering the locally tree-like approximation in percolation, the probability that nodes of degree k in network A participate in bond percolation is(7)
b
k
A
=
1
-
1
-
λ
∑
j
j
p
j
b
j
A
k
A
k
-
1
=
1
-
1
-
λ
ρ
A
k
-
1
.Substituting (7) into (2), we get the probability that nodes of degree k in network A will become removed nodes is(8)
r
k
A
=
b
k
A
=
1
-
1
-
λ
ρ
A
k
-
1
.


### 3.2.3. The Proportion of Failed Nodes in NetworkA

From a microscopic view, the failure of a coupled pair of nodesa
i and b
i 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)
α: the probability that the failure of node a
i in network A is caused by losing the dependent node b
i in network B directly.
(ii)
β: the probability that the failure of node b
i in network B is caused by losing the dependent node a
i in network A directly.Then, an arbitrary nodea
i of degree k in network A who participates in site percolation can be divided into two situations:(i)
The probability that the failure ofa
i is caused by losing the dependent node b
i directly is defined as(9)
s
k
A
1
=
α
.
(ii)
The probability that nodea
i fails and the failure is NOT caused by losing the dependent node b
i directly is defined as(10)
s
k
A
2
=
1
-
α
ρ
A
+
ϕ
A
k
.Substituting (8), (9), and (10) into (1), we can get the probability that nodes with degree k in network A fail(11)
f
k
A
=
1
-
r
k
A
s
k
A
1
+
s
k
A
2
=
1
-
r
k
A
α
+
1
-
α
ρ
A
+
ϕ
A
k
.


### 3.2.4. The Proportion of Failed Nodes in NetworkB

Similarly, an arbitrary nodeb
j of degree k in network B that participates in site percolation can be divided into two situations:(i)
The probability that the failure ofb
j is caused by losing the dependent node a
j directly is defined as(12)
s
k
B
1
=
β
.
(ii)
The probability that nodeb
j fails and the failure is NOT caused by losing the dependent node a
j directly is defined as(13)
s
k
B
2
=
1
-
β
ϕ
B
k
.Substituting (12) and (13) into (3), we can get the probability that nodes with degree k in network B fail:(14)
f
k
B
=
s
k
B
=
s
k
B
1
+
s
k
B
2
=
β
+
1
-
β
ϕ
B
k
.


### 3.2.5. The Relationship between Parametersα and β

According to the definition ofα, it is equivalent to the probability that a node of network B fails first and then leads to the failure of a random node in network A as shown in the following equation:(15)
α
=
∑
j
p
j
s
j
B
2
=
∑
j
p
j
1
-
β
ϕ
B
j
.β is equivalent to the probability that a node of network A is removed or fails first and then leads to the failure of a random node in network B as follows:(16)
β
=
∑
j
p
j
b
j
A
+
1
-
b
j
A
s
j
A
2
=
∑
j
p
j
r
j
A
+
1
-
r
j
A
1
-
α
ρ
A
+
ϕ
A
j
.


### 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:(17)
ϕ
A
=
∑
j
j
p
j
f
j
A
k
A
,
(18)
ρ
A
=
∑
j
j
p
j
r
j
A
k
A
,
(19)
ϕ
B
=
∑
j
j
p
j
f
j
B
k
B
,
(20)
r
k
A
=
1
-
1
-
λ
ρ
A
k
-
1
,
(21)
f
k
A
=
1
-
r
k
A
α
+
1
-
α
ρ
A
+
ϕ
A
k
,
(22)
f
k
B
=
β
+
1
-
β
ϕ
B
k
.Obviously, these six equations depend on each other. Thus we can get the fraction of removed nodesρ
A and the fraction of failed nodes ϕ
A of network A and the fraction of failed nodes ρ
B of network B in a steady state from these six equations via the following iteration process:(1)
Assign a very small initial value tor
k
m
i
n
A, where k
m
i
n represents the minimum degree of nodes in network A. Let r
k
A
=
0
(
k
≤
k
m
i
n
), f
k
A
=
0, and f
k
B
=
0. Substitute r
k
A, f
k
A, and f
k
B into (17), (18), and (19) and get the initial values of ϕ
A, ρ
A, ϕ
B, α, and β.
(2)
Substituteϕ
A, ρ
A, ϕ
B, α, and β into (20), (21), and (22) to get the values of r
k
A, f
k
A, and f
k
B. Next, substitute the new values of r
k
A, f
k
A, and f
k
B into (17), (18), and (19) and then update ϕ
A, ρ
A, ϕ
B, α, and β. Repeat this step until convergence.
(3)
Get the fraction of removed and failed nodes in networkA by using σ
r
A
=
∑
k
p
k
A
r
k
A and σ
f
A
=
∑
k
p
k
A
f
k
A and the fraction of failed nodes in network B by using s
i
g
m
a
f
B
=
∑
k
p
k
B
f
k
B.


## 3.2.1. Two Types of Percolation Processes

The spreading of an epidemic is abond percolation process [27, 28]. The process of epidemic spreading is equivalent to selecting a link from a network with probability λ, and the outbreak of an epidemic indicates that the set of chosen links forms a connectivity cluster. Cascading failure is equivalent to asite percolation [29, 30]. In our model, when an epidemic spreads out, there exist several nodes in network A participating in both bond percolation and site percolation. We assume that these nodes in network A 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 theQualitative Analysis.We denotes
k
A and b
k
A as the probability that a node of degree k in network A participates in site percolation and bond percolation, respectively. We denote f
k
A and r
k
A as the probability that a node of degree k in network A fails or is removed, respectively, after the dynamic process terminates in steady state. We denote s
k
B and f
k
B as the probability that a node of degree k in network B 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(1)
f
k
A
=
1
-
b
k
A
s
k
A
,
(2)
r
k
A
=
b
k
A
,
(3)
f
k
B
=
s
k
B
.In a coupled degree-degree uncorrelated interdependent system of networks, the neighbor of an arbitrary node in networkA becomes a failed node or removed node with probability ϕ
A or ρ
A, respectively, and the neighbor of an arbitrary node in network B becomes a failed node with probability ϕ
B, which are calculated as follows:(4)
ϕ
A
=
∑
j
j
p
j
f
j
A
k
A
=
∑
j
j
p
j
1
-
b
j
A
s
j
A
k
A
,
(5)
ρ
A
=
∑
j
j
p
j
r
j
A
k
A
=
∑
j
j
p
j
b
j
A
k
A
,
(6)
ϕ
B
=
∑
j
j
p
j
f
j
B
k
B
=
∑
j
j
p
j
s
j
B
k
B
.


## 3.2.2. The Proportion of Removed Nodes in NetworkA

In networkA, a susceptible node will be infected with probability λ for each link connecting to an infected node. Therefore, considering the locally tree-like approximation in percolation, the probability that nodes of degree k in network A participate in bond percolation is(7)
b
k
A
=
1
-
1
-
λ
∑
j
j
p
j
b
j
A
k
A
k
-
1
=
1
-
1
-
λ
ρ
A
k
-
1
.Substituting (7) into (2), we get the probability that nodes of degree k in network A will become removed nodes is(8)
r
k
A
=
b
k
A
=
1
-
1
-
λ
ρ
A
k
-
1
.


## 3.2.3. The Proportion of Failed Nodes in NetworkA

From a microscopic view, the failure of a coupled pair of nodesa
i and b
i 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)
α: the probability that the failure of node a
i in network A is caused by losing the dependent node b
i in network B directly.
(ii)
β: the probability that the failure of node b
i in network B is caused by losing the dependent node a
i in network A directly.Then, an arbitrary nodea
i of degree k in network A who participates in site percolation can be divided into two situations:(i)
The probability that the failure ofa
i is caused by losing the dependent node b
i directly is defined as(9)
s
k
A
1
=
α
.
(ii)
The probability that nodea
i fails and the failure is NOT caused by losing the dependent node b
i directly is defined as(10)
s
k
A
2
=
1
-
α
ρ
A
+
ϕ
A
k
.Substituting (8), (9), and (10) into (1), we can get the probability that nodes with degree k in network A fail(11)
f
k
A
=
1
-
r
k
A
s
k
A
1
+
s
k
A
2
=
1
-
r
k
A
α
+
1
-
α
ρ
A
+
ϕ
A
k
.


## 3.2.4. The Proportion of Failed Nodes in NetworkB

Similarly, an arbitrary nodeb
j of degree k in network B that participates in site percolation can be divided into two situations:(i)
The probability that the failure ofb
j is caused by losing the dependent node a
j directly is defined as(12)
s
k
B
1
=
β
.
(ii)
The probability that nodeb
j fails and the failure is NOT caused by losing the dependent node a
j directly is defined as(13)
s
k
B
2
=
1
-
β
ϕ
B
k
.Substituting (12) and (13) into (3), we can get the probability that nodes with degree k in network B fail:(14)
f
k
B
=
s
k
B
=
s
k
B
1
+
s
k
B
2
=
β
+
1
-
β
ϕ
B
k
.


## 3.2.5. The Relationship between Parametersα and β

According to the definition ofα, it is equivalent to the probability that a node of network B fails first and then leads to the failure of a random node in network A as shown in the following equation:(15)
α
=
∑
j
p
j
s
j
B
2
=
∑
j
p
j
1
-
β
ϕ
B
j
.β is equivalent to the probability that a node of network A is removed or fails first and then leads to the failure of a random node in network B as follows:(16)
β
=
∑
j
p
j
b
j
A
+
1
-
b
j
A
s
j
A
2
=
∑
j
p
j
r
j
A
+
1
-
r
j
A
1
-
α
ρ
A
+
ϕ
A
j
.


## 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:(17)
ϕ
A
=
∑
j
j
p
j
f
j
A
k
A
,
(18)
ρ
A
=
∑
j
j
p
j
r
j
A
k
A
,
(19)
ϕ
B
=
∑
j
j
p
j
f
j
B
k
B
,
(20)
r
k
A
=
1
-
1
-
λ
ρ
A
k
-
1
,
(21)
f
k
A
=
1
-
r
k
A
α
+
1
-
α
ρ
A
+
ϕ
A
k
,
(22)
f
k
B
=
β
+
1
-
β
ϕ
B
k
.Obviously, these six equations depend on each other. Thus we can get the fraction of removed nodesρ
A and the fraction of failed nodes ϕ
A of network A and the fraction of failed nodes ρ
B of network B in a steady state from these six equations via the following iteration process:(1)
Assign a very small initial value tor
k
m
i
n
A, where k
m
i
n represents the minimum degree of nodes in network A. Let r
k
A
=
0
(
k
≤
k
m
i
n
), f
k
A
=
0, and f
k
B
=
0. Substitute r
k
A, f
k
A, and f
k
B into (17), (18), and (19) and get the initial values of ϕ
A, ρ
A, ϕ
B, α, and β.
(2)
Substituteϕ
A, ρ
A, ϕ
B, α, and β into (20), (21), and (22) to get the values of r
k
A, f
k
A, and f
k
B. Next, substitute the new values of r
k
A, f
k
A, and f
k
B into (17), (18), and (19) and then update ϕ
A, ρ
A, ϕ
B, α, and β. Repeat this step until convergence.
(3)
Get the fraction of removed and failed nodes in networkA by using σ
r
A
=
∑
k
p
k
A
r
k
A and σ
f
A
=
∑
k
p
k
A
f
k
A and the fraction of failed nodes in network B by using s
i
g
m
a
f
B
=
∑
k
p
k
B
f
k
B.


## 4. 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〈
k
〉
=
6,8
,
10. The simulation results are illustrated in Figures 6–11. Each data point is the average of 50 dependent realizations.Figure 6
Coupled ER networks with〈
k
〉
=
6.
(a)
σ
r
A
(b)
σ
f
A
(c)
σ
f
B or σ
f
A
+
r
AFigure 7
Coupled ER networks with〈
k
〉
=
8.
(a)
σ
r
A
(b)
σ
f
A
(c)
σ
f
B or σ
f
A
+
r
AFigure 8
Coupled ER networks with〈
k
〉
=
10.
(a)
σ
r
A
(b)
σ
f
A
(c)
σ
f
B or σ
f
A
+
r
AFigure 9
Coupled BA networks with〈
k
〉
=
6.
(a)
σ
r
A
(b)
σ
f
A
(c)
σ
f
B or σ
f
A
+
r
AFigure 10
Coupled BA networks with〈
k
〉
=
8.
(a)
σ
r
A
(b)
σ
f
A
(c)
σ
f
B or σ
f
A
+
r
AFigure 11
Coupled BA networks with〈
k
〉
=
10.
(a)
σ
r
A
(b)
σ
f
A
(c)
σ
f
B or σ
f
A
+
r
AIt can be seen from Figures6–11 that, on the whole, our theoretical values are in good, though not perfect, agreement with the simulation results. In Figures 6–11  (
a
), the epidemic breaks out if λ is larger than a critical value. When λ is large enough for the epidemic to break out, the theoretical value of σ
r
A is a little smaller than the simulation result of σ
r
A. As 〈
k
〉 increases, the deviation of σ
r
A between the theoretical value and simulation result becomes smaller. In contrast, the theoretical value of σ
f
A becomes larger than the simulation result of σ
f
A, which is larger than the simulation results of σ
f
A when the epidemic breaks out, and the deviation of σ
f
A decreases as 〈
k
〉 increases, as shown in Figures 6–11  (
b
). 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 A, the fraction of removed nodes is overvalued, while the fraction of failed nodes is undervalued. However, the theoretical values of σ
r
A and σ
f
A can describe the trend of simulation values of σ
r
A and σ
f
A with increasing λ, as shown in Figures 6–11
(
a
),(b). In Figures 6–11  (
c
), the fraction of nonfunctioning nodes in networks A and B is well estimated.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 scale-free 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(a) shows, with the epidemic effective spreading rate λ 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 λ 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).Figure 12
Numerical simulations of coupled wireless network and BA network with〈
k
〉
=
6.
(a)
σ
r
A
(b)
σ
f
A
(c)
σ
f
B or σ
f
A
+
r
A


## 5. 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〈
k
〉 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.



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*Source: 9374039-2018-07-05.xml*