Effect of resource constraints on intersimilar coupled networks

Most real-world networks do not live in isolation but are often coupled together within a larger system. Recent studies have shown that intersimilarity between coupled networks increases the connectivity of the overall system. However, unlike connected nodes in a single network, coupled nodes often share resources, like time, energy, and memory, which can impede flow processes through contention when intersimilarly coupled. We study a model of a constrained susceptible-infected-recovered (SIR) process on a system consisting of two random networks sharing the same set of nodes, where nodes are limited to interact with (and therefore infect) a maximum number of neighbors at each epidemic time step. We obtain that, in agreement with previous studies, when no limit exists (regular SIR model), positively correlated (intersimilar) coupling results in a lower epidemic threshold than negatively correlated (interdissimilar) coupling. However, in the case of the constrained SIR model, the obtained epidemic threshold is lower with negatively correlated coupling. The latter finding differentiates our work from previous studies and provides another step towards revealing the qualitative differences between single and coupled networks.

Figure 1

Fraction of infected nodes at the end of the epidemic outbreaks $R_{\infty }$ as a function of the infection rate $\lambda$ for constrained and nonconstrained epidemic spreading, running in a positively correlated [(black) squares], negatively correlated [(red) circles], and uncorrelated [(green) triangles] network consisting of two ER or BA networks of size $N={10}^4$ and mean degree 6. Symbols and error bars (if larger than the symbol size) correspond to the mean and standard deviation computed over 1000 simulation runs. (a) Nonconstrained epidemic in coupled ER networks. (b) Nonconstrained epidemic in coupled BA networks. (c) Constrained epidemic with interaction limit $m=12$ in coupled ER networks. (d) Constrained epidemic with interaction limit $m=12$ in coupled BA networks.

Figure 2

Degree distributions of the original and the interaction graphs, obtained from a positively correlated [(black) squares], a negatively correlated [(red) circles], and an uncorrelated [(green) triangles] network consisting of two ER or BA networks of size $N={10}^4$ and mean degree 6. Results obtained by averaging all the graphs. (a) Original graphs of coupled ER networks. (b) Original graphs of coupled BA networks. (c) Interaction graphs with interaction limit $m=12$ obtained from coupled ER networks. (d) Interaction graphs with interaction limit $m=12$ obtained from coupled BA networks.

Figure 3

Nearest-neighbors average connectivity (NNAC) as a function of node degree for a positively correlated [(black) squares], a negatively correlated [(red) circles], and an uncorrelated [(green) triangles] network consisting of two ER or BA networks of size $N={10}^4$ and mean degree 6. Results obtained by averaging all the graphs. (a) Original graphs of coupled ER networks. (b) Original graphs of coupled BA networks. (c) Interaction graphs with interaction limit $m=12$ obtained from coupled ER networks. (d) Interaction graphs with interaction limit $m=12$ obtained from coupled BA networks.

Figure 4

Nearest-neighbors average connectivity (NNAC) as a function of node degree for a single BA network of size $N={10}^4$ and mean degree 6. Results obtained by averaging all the graphs.

Figure 5

Fraction of infected nodes at the end of the epidemic outbreaks $R_{\infty }$ as a function of the interaction limit $m$ for constrained epidemic spreading at infection rate $\lambda =0.1$, running in positively [(black) squares] and negatively [(red) circles] correlated networks consisting of two ER or BA networks of size $N={10}^4$ and mean degree 6. Symbols and error bars (if larger than the symbol size) correspond to the mean and standard deviation computed over 1000 simulation runs. (a) Coupled ER networks; (b) Coupled BA networks.