Supersymmetric multiplex networks described by coupled Bose and Fermi statistics

Until now, no simple symmetries have been detected in complex networks. Here we show that in growing multiplex networks the symmetries of multilayer structures can be exploited by their dynamical rules, forming supersymmetric multiplex networks described by coupled Bose-Einstein and Fermi-Dirac quantum statistics. The supersymmetric multiplex is formed by layers which are scale-free networks and can display a Bose-Einstein condensation of the links. To characterize the complexity of the supersymmetric multiplex using quantum information tools, we extend the definition of the network entanglement entropy to the layers of any multiplex network. Interestingly, we observe a very simple relation between the entanglement entropies of the layers of the supersymmetric multiplex network and the entropy rate of the same multiplex network. This relation therefore connects the classical nonequilibrium growing dynamics of the supersymmetric multiplex network with its quantum information static characteristics.

Figure 1

The degree distributions $P^{\left[1\right]}\left(k\right)$ and $P^{\left[2\right]}\left(k\right)$ in the two layers of the supersymmetric network. The data are shown for networks of $N={10}^4$ nodes with energy distribution $g\left(\epsilon \right)=(1+\theta ){\epsilon }^{\theta }$ and $\theta =0.5,\alpha =0.3,\beta =0,1,5$. The curves are averaged over 100 multiplex networks realizations. The lines indicate the mean-field expectation for $\beta =0$ and $\beta =1$ and are in very good agreement with the simulation results. The degree distribution for $\beta =5$ is a typical degree distribution below the condensation transition where very big hubs emerge in the network.

Figure 2

The average degree in layer 2 conditioned on the degree in layer 1, $〈k^{\left[2\right]}|k^{\left[1\right]}〉$ is shown and compared with the mean-field theoretical expectations finding very good agreement. The data are shown for networks of $N={10}^4$ nodes, energy distribution $g\left(\epsilon \right)=(1+\theta ){\epsilon }^{\theta }$ with $\theta =0.5,\alpha =0.3$, and $\beta =0,1,5$. The curves are averaged over 100 multiplex networks realizations.

Figure 3

The fraction of links $k_{\max }^{\left[1\right]}/\left(mN\right)$ and $k_{\max }^{\left[2\right]}/\left(mN\right)$ linked to the most connected node in layer 1 and in layer 2 are shown together with the total overlap of the links ${\mathcal{O}}^{[1,2]}$ versus $T=1/\beta$. The data are shown for networks of $N={10}^3,{10}^4,{10}^5$ nodes with energy distribution $g\left(\epsilon \right)=(1+\theta ){\epsilon }^{\theta }$ and $\theta =0.5,\alpha =0.3$. The curves are averaged over 100 multiplex networks realizations for $N={10}^3$ and $N={10}^4$ and over 30 multiplex network realization for $N={10}^5$.