Disease Localization in Multilayer Networks

We present a continuous formulation of epidemic spreading on multilayer networks using a tensorial representation, extending the models of monoplex networks to this context. We derive analytical expressions for the epidemic threshold of the susceptible-infected-susceptible (SIS) and susceptible-infected-recovered dynamics, as well as upper and lower bounds for the disease prevalence in the steady state for the SIS scenario. Using the quasistationary state method, we numerically show the existence of disease localization and the emergence of two or more susceptibility peaks, which are characterized analytically and numerically through the inverse participation ratio. At variance with what is observed in single-layer networks, we show that disease localization takes place on the layers and not on the nodes of a given layer. Furthermore, when mapping the critical dynamics to an eigenvalue problem, we observe a characteristic transition in the eigenvalue spectra of the supra-contact tensor as a function of the ratio of two spreading rates: If the rate at which the disease spreads within a layer is comparable to the spreading rate across layers, the individual spectra of each layer merge with the coupling between layers. Finally, we report on an interesting phenomenon, the barrier effect; i.e., for a three-layer configuration, when the layer with the lowest eigenvalue is located at the center of the line, it can effectively act as a barrier to the disease. The formalism introduced here provides a unifying mathematical approach to disease contagion in multiplex systems, opening new possibilities for the study of spreading processes.

Popular Summary

Networks are all around. They describe the flow of information, the movement of people and goods via multiple modes of transportation, and the spread of disease across interconnected populations. Traditionally, networks have been studied as if they were a single layer, which flattens out hierarchies such as social circles. Multilayer networks, which consider each of those circles as a layer, are more accurate descriptions of real-world networks and their use can have deep implications for understanding the dynamics of the system. Using the spread of disease as a model, we have developed a mathematical framework that accounts for the multilayer structure, and we have identified several behaviors that emerge from this analysis.

The framework relies on tensors, mathematical objects that allow us to represent multidimensional data in a compact way. Through mathematical analysis and numerical simulations, we find a number of interesting features such as the existence of multiple epidemic thresholds and transmission rates beyond which the number of individuals that catch a disease is non-negligible. We also show the existence of disease localization, a scenario in which the disease cannot escape a layer and jump to another.

Our work provides a unifying mathematical approach to studying disease transmission among multilayered populations. There are still many aspects to investigate such as how to use these results to help contain an epidemic as well as how the picture changes in more complex scenarios. Disease-like models can also be used to explore other networks such as the propagation of information.

Figure 1

Schematic illustration of the three multilayer network cases considered as examples. The left panels represent the original networks, which give rise to three distinct configurations for the networks of layers (right panels). See the text for more details.

Figure 2

Phase diagrams over a two-layer multiplex system, where each layer is a scale-free network with $n=10^4$ nodes, for a fixed value of $\mu =1$. (a) Density of spreaders as a function of the parameters $\eta$ and $\lambda$. (b) Density of recovered individuals as a function of the parameters $\eta$ and $\lambda$. Colors represent the fraction of spreaders, and the white line is the threshold calculated using Eq. (6).

Figure 3

Individual layer behavior over a two-layer multiplex system. Each layer has $n=10^4$ for a fixed value of $\mu =1$. The results considering both layers are shown in (a), while the dynamics in the individual layers are shown in (b) [$P(k)\sim k^{-2.5}$] and (c) [$P(k)\sim k^{-4.5}$]. The arrows indicate the leading eigenvalues of the layers.

Figure 4

Spectral properties of the tensor $\mathcal{R}(\lambda ,\eta )$ as a function of the ratio $\eta /\lambda$ for a multiplex with two layers, the first with $\gamma \approx 2.2$ and the second with $\gamma \approx 2.8$. Both have $⟨k⟩\approx 8$. In the top panel, we present the inverse participation ratio of the two larger eigenvalues and the individual-layer contributions, while in the bottom panel, we show the leading eigenvalues. Every curve is composed of $10^3$ log spaced points, in order to have enough resolution.

Figure 5

Diagram of the contribution of each layer to the $\mathrm{IPR}(\mathrm{\Lambda })$ for different values of the spreading ratio $\eta /\lambda$. The dashed line represents the case where both layers have the same contribution, i.e., a line with slope one. In the inset, we show the angle $\theta$ between the vector composed of the contributions of each layer to the $\mathrm{IPR}(\mathrm{\Lambda })$, $v={[\mathrm{IPR}({\mathrm{\Lambda }}_1^1),\mathrm{IPR}({\mathrm{\Lambda }}_1^2)]}^T$, and the $x$ axis. The multiplex network used here is composed of two Erdös-Rényi networks, both with $n=5\times 10^4$—the first layer is $⟨k⟩=16$ [$({\mathrm{\Lambda }}_1^1{)}^{-1}\approx 0.0625$], while the second is $⟨k⟩=12$ [$({\mathrm{\Lambda }}_1^2{)}^{-1}\approx 0.0833$].

Figure 6

Susceptibility $\chi$ as a function of the spreading rate $\lambda$ for different ratios of interlayer and intralayer spreading ratings, $\eta /\lambda$, for a fixed value of $\mu =1$ over a two-layer multiplex system, where each layer has $n=10^3$—the first with $\gamma \approx 2.2$ and the second with $\gamma \approx 2.8$. Both have $⟨k⟩\approx 8$. The simulated values are $(\eta /\lambda )=0.1$, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30.

Figure 7

Time evolution of the fraction of infected nodes in the second layer for $\mu =1$, different values of $\eta$ (${10}^{-4}$, ${10}^{-3}$, ${10}^{-2}$, and ${10}^{-1}$), and different values of the spreading rate: (a) $\lambda =0.078$, (b) $\lambda =0.083$, (c) $\lambda =0.085$, and (d) $\lambda =0.088$. The multiplex network used is composed of two Erdös-Rényi networks, both with $n=5\times 10^4$—the first layer with $⟨k⟩=16$ [$({\mathrm{\Lambda }}_1^1{)}^{-1}\approx 0.0625$] and the second with $⟨k⟩=12$ [$({\mathrm{\Lambda }}_1^2{)}^{-1}\approx 0.0833$].

Figure 8

Spectral properties of the tensor $\mathcal{R}(\lambda ,\eta )$ as a function of the ratio $\eta /\lambda$ for a multiplex with two layers with the same degree distribution (different random realizations of the configuration model) and connected to its counterpart on the other layer. In the top panel, we present the $\mathrm{IPR}(\mathrm{\Lambda })$ of the two larger eigenvalues and the individual-layer contributions, while in the bottom panel, we show the leading eigenvalues. Every curve is composed by $10^3$ log spaced points, in order to have enough resolution. In panel (a), we show the line $(2.3+2.9+2.6)$, while panel (b) is the multiplex case.

Figure 9

Susceptibility $\chi$ as a function of $\lambda$ considering all three-layer configurations and many different ratios $\eta /\lambda$, which is represented by the color of the lines. The recovering rate is $\mu =1$. The simulated values are $(\eta /\lambda )=0.05$, 0.06, 0.07, 0.08, 0.09, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20. In panel (a), we show the line $(2.3+2.9+2.6)$, while panel (b) is the multiplex case.

Figure 10

Spectral properties of the tensor $\mathcal{R}(\lambda ,\eta )$ as a function of the ratio $\eta /\lambda$ for a multiplex with two layers with the exact same degree distribution and connected to its counterpart on the other layer. In the top panel, we present the inverse participation ratio of the three larger eigenvalues, while in the bottom panel, we show the leading eigenvalues. Every curve is composed of $10^3$ log spaced points, in order to have enough resolution.

Figure 11

Spectral properties of the tensor $\mathcal{R}(\lambda ,\eta )$ as a function of the ratio $\eta /\lambda$ for a multiplex with two layers with the same degree distribution (different random realizations of the configuration model) and connected to its counterpart on the other layer. In the top panel, we present the inverse participation ratio of the two larger eigenvalues and the individual-layer contributions, while in the bottom panel, we show the leading eigenvalues. Every curve is composed of $10^3$ log spaced points, in order to have enough resolution.

Figure 12

The final number of infected nodes in the second layer (with lowest individual eigenvalue) as a function of the size of the layers on the main panels. In the insets, we present the fraction of infected nodes (on the left) and the standard deviation in the steady state (on the right). The parameters used in the simulations are shown in the tile of each panel. They are a combination of the parameters $\lambda =0.078$, 0.083, 0.085, 0.088 and $\eta ={10}^{-4}$, ${10}^{-3}$, ${10}^{-2}$, ${10}^{-1}$. Furthermore, the layer sizes are $n=2\times 10^3$, $3\times 10^3$, $4\times 10^3$, $5\times 10^3$, $6\times 10^3$, $7\times 10^3$, $8\times 10^3$, $9\times 10^3$, $10^4$, $2\times 10^4$, $3\times 10^4$, $4\times 10^4$, and $5\times 10^4$, and $m=2$ in all cases.

Figure 13

The final fraction of infected nodes in the layer with the lowest individual eigenvalue as a function of the size of the layers. The colors represent different values of $\eta$. We show $\lambda =0.078$ in panel (a), $\lambda =0.083$ in panel (b), $\lambda =0.085$ in panel (c), and $\lambda =0.088$ in panel (d). Furthermore, the layer sizes are $n=2\times 10^3$, $3\times 10^3$, $4\times 10^3$, $5\times 10^3$, $6\times 10^3$, $7\times 10^3$, $8\times 10^3$, $9\times 10^3$, $10^4$, $2\times 10^4$, $3\times 10^4$, $4\times 10^4$, and $5\times 10^4$, and $m=2$ in all cases. Each curve is the result of a parameter $\eta$; from bottom to top, $\eta ={10}^{-4},{10}^{-3},{10}^{-2},{10}^{-1}$.

Figure 14

Finite-size analysis of the susceptibility. In the main panel, we have the susceptibility as a function of $\lambda$ for different sizes of two-layer multiplex networks, where the first layer has $⟨k⟩=16$ and the second $⟨k⟩=12$. In this experiment, we fixed the ratio $(\eta /\lambda )=0.01$. In the inset, we show the susceptibility of the two peaks as a function of the layer size, where the blue symbols refer to the first peaks while the green symbols refer to the second peak. In addition, the red lines are a linear fitting of those points. The layer sizes evaluated are $n=3\times 10^3$, $4\times 10^3$, $5\times 10^3$, $6\times 10^3$, $7\times 10^3$, $8\times 10^3$, $9\times 10^3$, $10^4$, $2\times 10^4$, $3\times 10^4$, $4\times 10^4$, $5\times 10^4$, $10^5$.

Figure 15

Distribution of the eigenvalues. In the rows, from top to bottom, we show the interconnected networks of lines $2.3+2.6+2.9$, $2.3+2.9+2.6$, $2.6+2.3+2.9$ and the multiplex. In the columns, from left to right, we varied the ratios $(\eta /\lambda )=1$, 10, 100, and 1000, respectively. All histograms were built with 100 bins.

Figure 16

Evaluation of the eight first eigenvalues of $\mathcal{R}(\lambda ,\eta )$ for the multiplex configuration as a function of the ratio $\eta /\lambda$. It is noteworthy that such a plot is visually equivalent for all the layer topologies composed of three layers. The dashed lines represent the individual-layer leading eigenvalues.

Figure 17

Spectral properties of the tensor $\mathcal{R}(\lambda ,\eta )$ as a function of the ratio $\eta /\lambda$ for a multiplex with two layers with the same degree distribution (different random realizations of the configuration model) and connected to its counterpart on the other layer. In the top panel, we present the inverse participation ratio of the two larger eigenvalues and the individual-layer contributions, while in the bottom panel, we show the leading eigenvalues. Every curve is composed of $10^3$ log spaced points, in order to have enough resolution.

Figure 18

Susceptibility $\chi$ as a function of $\lambda$ considering all three-layer configurations and many different ratios $\eta /\lambda$, which is represented by the color of the lines. The recovering rate is $\mu =1$. The simulated values are $(\eta /\lambda )=0.05$, 0.06, 0.07, 0.08, 0.09, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20.