Effect of the interconnected network structure on the epidemic threshold

Most real-world networks are not isolated. In order to function fully, they are interconnected with other networks, and this interconnection influences their dynamic processes. For example, when the spread of a disease involves two species, the dynamics of the spread within each species (the contact network) differs from that of the spread between the two species (the interconnected network). We model two generic interconnected networks using two adjacency matrices, A and B, in which A is a $2N\times 2N$ matrix that depicts the connectivity within each of two networks of size $N$, and B a $2N\times 2N$ matrix that depicts the interconnections between the two. Using an $N$-intertwined mean-field approximation, we determine that a critical susceptible-infected-susceptible (SIS) epidemic threshold in two interconnected networks is $1/{\lambda }_1(A+\alpha B)$, where the infection rate is $\beta$ within each of the two individual networks and $\alpha \beta$ in the interconnected links between the two networks and ${\lambda }_1(A+\alpha B)$ is the largest eigenvalue of the matrix $A+\alpha B$. In order to determine how the epidemic threshold is dependent upon the structure of interconnected networks, we analytically derive ${\lambda }_1(A+\alpha B)$ using a perturbation approximation for small and large $\alpha$, the lower and upper bound for any $\alpha$ as a function of the adjacency matrix of the two individual networks, and the interconnections between the two and their largest eigenvalues and eigenvectors. We verify these approximation and boundary values for ${\lambda }_1(A+\alpha B)$ using numerical simulations, and determine how component network features affect ${\lambda }_1(A+\alpha B)$. We note that, given two isolated networks $G_1$ and $G_2$ with principal eigenvectors $x$ and $y$, respectively, ${\lambda }_1(A+\alpha B)$ tends to be higher when nodes $i$ and $j$ with a higher eigenvector component product $x_i{y}_j$ are interconnected. This finding suggests essential insights into ways of designing interconnected networks to be robust against epidemics.

Figure 1

A plot of ${\lambda }_1\left(W\right)$ as a function of $\alpha$ for both simulation results (symbol) and its (a) perturbation approximation (7) for small $\alpha$ (dashed line) and (b) perturbation approximation (9) for large $\alpha$ (dashed line). The interconnected network is composed of an ER random network and a BA scale-free network both with $N=1000$ and link density $p=0.006$, randomly interconnected with density $p_I$. All the results are averages of 100 realizations.

Figure 2

Plot ${\lambda }_1\left(W\right)$ as a function of $\alpha$ for both simulation results (symbol) and its (a) lower bound (3) (dashed line) and (b) upper bound (4) (dashed line). The interconnected network is composed of an ER random network and a BA scale-free network both with $N=1000$ and link density $p=0.006$, randomly interconnected with density $p_I$. All the results are averages of 100 realizations.

Figure 3

Plot ${\lambda }_1\left(W\right)$ as a function of $\alpha$ for both simulation results (symbol) and its lower bound $\alpha {\lambda }_1\left(B\right)$ (dashed line). The interconnected network is composed of an ER random network and a BA scale-free network both with $N=1000$ and link density $p=0.006$, randomly interconnected with density $p_I$. All the results are averages of 100 realizations.

Figure 4

A plot of ${\lambda }_1\left(W\right)$ as a function of $\alpha$ for both simulation results (symbol) and its (a) perturbation approximation (8) for small $\alpha$ (dashed line) and (b) perturbation approximation (9) for large $\alpha$ (dashed line). The interconnected network is composed of two identical BA scale-free networks with $N=1000$ and link density $p=0.006$, randomly interconnected with density $p_I$. All the results are averages of 100 realizations.

Figure 5

Plot ${\lambda }_1\left(W\right)$ as a function of $\alpha$ for both simulation results (symbol) and (a) its lower bound (3) (dashed line) and (b) upper bound (4) (dashed line). The interconnected network is composed of two identical BA scale-free networks $N=1000$ and link density $p=0.006$, randomly interconnected with density $p_I$. All the results are averages of 100 realizations.