Optimizing diffusion in multiplexes by maximizing layer dissimilarity

Diffusion in a multiplex depends on the specific link distribution between the nodes in each layer, but also on the set of the intralayer and interlayer diffusion coefficients. In this work we investigate, in a quantitative way, the efficiency of multiplex diffusion as a function of the topological similarity among multiplex layers. This similarity is measured by the distance between layers, taken among the pairs of layers. Results are presented for a simple two-layer multiplex, where one of the layers is held fixed, while the other one can be rewired in a controlled way in order to increase or decrease the interlayer distance. The results indicate that, for fixed values of all intra- and interlayer diffusion coefficients, a large interlayer distance generally enhances the global multiplex diffusion, providing a topological mechanism to control the global diffusive process. For some sets of networks, we develop an algorithm to identify the most sensitive nodes in the rewirable layer, so that changes in a small set of connections produce a drastic enhancement of the global diffusion of the whole multiplex system.

Figure 1

Illustration of a simple multiplex network composed of $M=2$ layers and $N=7$ nodes. Note that each node is represented once in each of the two interaction layers, and that an interlayer connection (dotted lines) is established between its two representations.

Figure 2

Illustration of the rewiring process in the layer ${\mathbf{L}}_{\mathbf{2}}$ associated to switching the labels of nodes 5 and 6. In (a), the two layers are identical. In (b), the labels $5^{\prime }$ and $6^{\prime }$ have been switched, but they are still connected to nodes 5 and 6 by tilted lines. In (c), the new nodes $5^{\prime }$ and $6^{\prime }$ are moved back to the positions vertically below the nodes 5 and 6, resulting in a displacement of the connections in ${\mathbf{L}}_{\mathbf{2}}$.

Figure 3

Typical results for the behavior of $\delta \left(s\right)$ as a function of $s$ resulting from the rewiring procedure. (a) Multiplex assembled at $s=0$ by one single $N=1000$, $p=0.01$ ER network. The solid circles indicate the values $s=434$ and $s=5\times {10}^6$. (b) Multiplexes assembled at $s=0$ by two distinct samples of $N=500$, $p=0.01$ ER networks. (c) Multiplexes assembled at $s=0$ by two distinct samples of $N=1000$ BA networks. In (b) and (c), the two branches indicated by ${\delta }_{+}$ and ${\delta }_{-}$ are obtained by tuning the MC procedure to increase and decrease the value of $\delta \left(s\right)$.

Figure 4

Typical results produced within the adopted framework for multiplexes assembled from one single $N=500$, $p=0.01$ ER network. (a) Dependency of ${\lambda }_2$ on $D_x$: ${\lambda }_2=2D_x$ (cyan short-dotted line), ${\lambda }_2\left(\alpha \right)={\lambda }_2\left(\beta \right)$ (black solid line), $\overline{{\lambda }_2}(s={10}^7)$ (red dash-dotted line), ${\mathrm{\Lambda }}_2(D_x;s=434)$ (dark yellow dashed line), ${\mathrm{\Lambda }}_2(D_x;s={10}^7)$ (dark cyan short-dashed line). (b) Dependency of $\overline{{\lambda }_2}\left(s\right)$ on $s$. The red circles around squares indicate the values $s=433$ and 434, where the value of $\overline{{\lambda }_2}\left(s\right)$ jumps by a large amount.

Figure 5

Dependency of $\overline{{\lambda }_2}\left(s\right)$ on $s$ by rewiring targeted pairs of nodes in $\beta$ for an $N=500$ ER multiplex with $p=0.01$ (a), and an $N=750$ ER multiplex with $p=0.008$ (b).

Figure 6

Typical results produced within the adopted framework for multiplexes assembled from one single $N=1000$, $\langle k\rangle =4$ BA network. (a) Dependency of ${\lambda }_2$ on $D_x$ for random rewiring: ${\lambda }_2=2D_x$ (cyan short-dotted line), ${\lambda }_2\left(\alpha \right)={\lambda }_2\left(\beta \right)$ (black solid line), $\overline{{\lambda }_2}(s={10}^7)$ (red dash-dotted line), ${\mathrm{\Lambda }}_2(D_x;s=500)$ (dark yellow dashed line), ${\mathrm{\Lambda }}_2(D_x;s={10}^7)$ (dark cyan short-dashed line). (b) Dependency of ${\lambda }_2$ on $D_x$ for the optimal BA rewiring scheme: ${\lambda }_2=2D_x$ (cyan short-dotted line), ${\lambda }_2\left(\alpha \right)={\lambda }_2\left(\beta \right)$ (black solid line), $\overline{{\lambda }_2}(s=499)$ (red dash-dotted line), ${\mathrm{\Lambda }}_2(D_x;s=256)$ (dark yellow dashed line), ${\mathrm{\Lambda }}_2(D_x;s=499)$ (dark cyan short-dashed line).

Figure 7

Typical results produced within the adopted framework for multiplexes assembled from one single $N=500$, $\rho =0.05$ WS network. (a) Dependency of $\overline{{\lambda }_2}\left(s\right)$ (black squares) and $D_m\left(s\right)$ (blue diamonds) on $s$. The symbols surrounded by red circles on the left vertical axis, which correspond to the values at $s=0$, have been inserted for the purpose of comparison. (b) Dependency of ${\lambda }_2$ on $D_x$ for random rewiring: ${\lambda }_2=2D_x$ (cyan short-dotted line), ${\lambda }_2\left(\alpha \right)={\lambda }_2\left(\beta \right)$ (black solid line), $\overline{{\lambda }_2}(s={10}^3)$ (red dash-dotted line), ${\mathrm{\Lambda }}_2(D_x;s={10}^3)$ (dark cyan short-dashed line), ${\mathrm{\Lambda }}_2(D_x;s=3000)$ (dark yellow dashed line), ${\mathrm{\Lambda }}_2(D_x;s={10}^4)$ (gray dots). (c) Dependency of ${\lambda }_2$ on $D_x$ for random rewiring: ${\lambda }_2=2D_x$ (cyan short-dotted line), ${\lambda }_2\left(\alpha \right)={\lambda }_2\left(\beta \right)$ (black solid line), $\overline{{\lambda }_2}(s={10}^3)$ (red dash-dotted line), ${\mathrm{\Lambda }}_2(D_x;s=100)$ (dark cyan short-dashed line), ${\mathrm{\Lambda }}_2(D_x;s=200)$ (dark yellow dashed line), ${\mathrm{\Lambda }}_2(D_x;s=500)$ (gray dots).

Figure 8

Typical results for multiplexes assembled by two realizations of $N=500$, $p=0.01$ ER networks. (a) Dependency of ${\lambda }_2$ on $D_x$ for ${\delta }_{+}$ multiplexes: ${\lambda }_2=2D_x$ (cyan short-dotted line), ${\lambda }_2\left(\alpha \right)$ (black solid line), ${\lambda }_2\left(\beta \right)$ (gray dots), $\overline{{\lambda }_2}(s={10}^7)$ (red dash-dotted line), ${\mathrm{\Lambda }}_2(D_x;s=0)$ (dark yellow dashed line), ${\mathrm{\Lambda }}_2(D_x;s={10}^7)$ (dark cyan short-dashed line). (b) Dependency of ${\lambda }_2$ on $D_x$ for ${\delta }_{-}$ multiplexes: ${\lambda }_2=2D_x$ (cyan short-dotted line), ${\lambda }_2\left(\alpha \right)$ (black solid line), ${\lambda }_2\left(\beta \right)$ (gray dots), $\overline{{\lambda }_2}(s=0)$ (red dash-dotted line), ${\mathrm{\Lambda }}_2(D_x;s=0)$ (dark yellow dashed line), ${\mathrm{\Lambda }}_2(D_x;s={10}^7)$ (dark cyan short-dashed line).

Figure 9

Dependency of $\overline{{\lambda }_2}\left(s\right)$ as a function of $s$ for multiplexes assembled by two realizations of $N=500$, $p=0.01$ ER networks. Panels (a) and (b) correspond to the situations in which the network distance is given by ${\delta }_{+}$ and ${\delta }_{-}$, respectively.

Figure 10

Typical results for multiplexes assembled by two realizations of $N=1000$, BA networks. (a) Dependency of ${\lambda }_2$ on $D_x$ for ${\delta }_{+}$ multiplexes: ${\lambda }_2=2D_x$ (cyan short-dotted line), ${\lambda }_2\left(\alpha \right)$ (black solid line), $\overline{{\lambda }_2}(s={10}^7)$ (red dash-dotted line), ${\mathrm{\Lambda }}_2(D_x;s=0)$ (dark yellow dashed line), ${\mathrm{\Lambda }}_2(D_x;s={10}^7)$ (dark cyan short-dashed line). (c) Dependency of ${\lambda }_2$ on $D_x$ for ${\delta }_{-}$ multiplexes: ${\lambda }_2=2D_x$ (cyan short-dotted line), ${\lambda }_2\left(\alpha \right)$ (black solid line), $\overline{{\lambda }_2}(s=0)$ (red dash-dotted line), ${\mathrm{\Lambda }}_2(D_x;s=0)$ (dark yellow dashed line), ${\mathrm{\Lambda }}_2(D_x;s={10}^7)$ (dark cyan short-dashed line). For the sake of clearer pictures, the horizontal line ${\lambda }_2\left(\beta \right)$ is not shown, once it is different but very close to ${\lambda }_2\left(\alpha \right)$.