Exact coupling threshold for structural transition reveals diversified behaviors in interconnected networks

An interconnected network features a structural transition between two regimes [F. Radicchi and A. Arenas, Nat. Phys. 9, 717 (2013)]: one where the network components are structurally distinguishable and one where the interconnected network functions as a whole. Our exact solution for the coupling threshold uncovers network topologies with unexpected behaviors. Specifically, we show conditions that superdiffusion, introduced by Gómez et al. [Phys. Rev. Lett. 110, 028701 (2013)], can occur despite the network components functioning distinctly. Moreover, we find that components of certain interconnected network topologies are indistinguishable despite very weak coupling between them.

Figure 1

One-to-one interconnection of two networks $G_A$ and $G_B$, where the coupling weight is $p>0$.

Figure 2

Algebraic connectivity ${\lambda }_2\left(\mathit{L}\right)$ of an interconnected network with scale-free $G_A$ and random geometric $G_B$ as a function of the coupling weight $p$. For $p<p^{*}\simeq 0.27,$ algebraic connectivity is ${\lambda }_2\left(\mathit{L}\right)=2p$. For $p>p^{*}$, eigenvalue $\lambda =2p$ is no longer the algebraic connectivity of the interconnected network; thus, denoting a structural transition at $p=p^{*}$.

Figure 3

Bounds for the coupling threshold vs the exact values for a set of interconnected networks with identical averaged network. For each generated network, we compute different bounds for the coupling threshold and compare them with the exact value. The closer to the black dashed line $y=x$, the more accurate the bounds.