Competition of simple and complex adoption on interdependent networks

We consider the competition of two mechanisms for adoption processes: a so-called complex threshold dynamics and a simple susceptible-infected-susceptible (SIS) model. Separately, these mechanisms lead, respectively, to first-order and continuous transitions between nonadoption and adoption phases. We consider two interconnected layers. While all nodes on the first layer follow the complex adoption process, all nodes on the second layer follow the simple adoption process. Coupling between the two adoption processes occurs as a result of the inclusion of some additional interconnections between layers. We find that the transition points and also the nature of the transitions are modified in the coupled dynamics. In the complex adoption layer, the critical threshold required for extension of adoption increases with interlayer connectivity whereas in the case of an isolated single network it would decrease with average connectivity. In addition, the transition can become continuous depending on the detailed interlayer and intralayer connectivities. In the SIS layer, any interlayer connectivity leads to the extension of the adopter phase. Besides, a new transition appears as a sudden drop of the fraction of adopters in the SIS layer. The main numerical findings are described by a mean-field type analytical approach appropriately developed for the threshold-SIS coupled system.

Figure 1

Schematic representation of the model's dynamics in two layers forming an interdependent network in which a node can interact with neighbors from both layers. In the upper layer (Thr) nodes change their states through a complex adoption process following a threshold model rules. In the lower layer (SIS) nodes evolve by a simple adoption process following the SIS type of dynamics.

Figure 2

Phase diagram in the independent layer case $m=0$. Left-side panel shows the case of the threshold layer for $k_1=10, N_1=1000$. Here the dashed area and the line $x_1=1$ is the set of stable fixed points. The right panel displays the steady state fraction of adopters for the $SIS$ model $k_2=10, N_2=1000$. Dashed line presents results for usual model of infection, solid line presents our model of simple adoption.

Figure 3

Phase diagram in the coupled layer cases $k_1=12, k_2=3$ (left) and $k_1=3, k_2=12$ (right). In both cases it is $m=10, N_1=N_2=1000$.

Figure 4

Fractions of adopters in threshold layer $〈s_1〉$ (left column), and in SIS layer $〈s_2〉$ (right column) as a function of $\theta$ and $\lambda$ for $m=10$ and $k_1=3$ and $k_2=12$ (top), $k_1=k_2=6$ (middle), $k_1=12$ and $k_2=3$ (bottom). Results come after computer simulations performed for a network of $N_1=N_2=1000$ nodes, where both layers are ER random networks, and are averaged over 500 realizations of networks and 500 realizations of the dynamics for each network configuration. The continuous lines displayed in the SIS column come from the full stability analysis of the mean-field dynamical equations; the horizontal line is $\lambda =1/k_2$. The arrows indicate transitions between the regions that will be discussed in other figures.

Figure 5

Fractions of adopters in threshold layer (left column) and in SIS layer (right column) as a function of $\theta$ ($\lambda$) when $\lambda$ ($\theta$) is fixed and for $m=10$. Top panel shows results when $\lambda =0.05$, middle panel when $\lambda =0.5$, and bottom panel when $\theta =0.25$. Red circles correspond to the case $k_1=k_2=6$, green triangles to $k_1=12>k_2=3$, and blue squares to $k_1=3<k_2=12$. Symbols present the results from computer simulations and solid lines with corresponding colors stand for analytical solutions.

Figure 6

Critical value of the threshold ${\theta }_c$ as a function of the interconnectivity $m$ for $\lambda =0.5$. Same symbols and line meanings than in figure 5.

Figure 7

Jump of the fraction of adopters in threshold layer $\mathrm{\Delta }\langle s_1\rangle$ at the transition point ${\theta }_c$ as a function of $m$ and for $\lambda =0.5$. Values of the ${\theta }_c\left(m\right)$ are shown in Fig. 6. Same symbols and line meanings as in Fig. 5.

Figure 8

Fraction of adopters right above ${\lambda }_c^1=0$ as a function of $m$. Different colors represent different values of $k_1$ and $k_2$. Red circles correspond to the case $k_1=k_2$; green triangles, $k_1>k_2$; and blue squares, $k_1<k_2$. Symbols present the results from computer simulations for a network of $N_1+N_2=2000$ nodes, where both layers are ER random networks. Results are averaged over 500 realizations of networks and 500 realizations of dynamics for each network configuration. We took $\theta =\lambda$ in computer simulations. Solid lines with corresponding colors stand for analytical solutions.

Figure 9

Maxima of the fraction of adopters in threshold layer ${\langle s_1\rangle }_{max}$ as a function of $\theta$ for fixed values of $\lambda$, different panels present results for different intralayer connectivities, i.e., (a) $k_1=k_2=6$, (b) $k_1=12$, and $k_2=3$, (c) $k_1=3$ and $k_2=12$ and as a function of $\lambda$ for fixed $\theta$, (d). In all cases the number of interlinks is set to $m=10$.

Figure 10

Maxima of the fraction of adopters in threshold layer ${\langle s_1\rangle }_{max}$ as a function of $\theta$ when $k_1=6, m=10$, and $\lambda =0.5$ for different values of $k_2$.