Characterization of multiple topological scales in multiplex networks through supra-Laplacian eigengaps

Multilayer networks have been the subject of intense research during the past few years, as they represent better the interdependent nature of many real-world systems. Here, we address the question of describing the three different structural phases in which a multiplex network might exist. We show that each phase can be characterized by the presence of gaps in the spectrum of the supra-Laplacian of the multiplex network. We therefore unveil the existence of different topological scales in the system, whose relation characterizes each phase. Moreover, by capitalizing on the coarse-grained representation that is given in terms of quotient graphs, we explain the mechanisms that produce those gaps as well as their dynamical consequences.

Figure 1

Schematic representation of a multiplex network. Panel (a) shows an example of a multiplex network. The two other panels depict its corresponding coarse-grained reductions: the network of layers (b) and the aggregate network (c). Details of how these reductions are obtained can be found in Ref. [12].

Figure 2

Eigenvalues of a toy two-layer multiplex with four nodes per layer as a function of $p$. Continuous blue lines are the eigenvalues of the multiplex network, whereas the dashed red lines are the eigenvalues of the aggregate network.

Figure 3

Variation of the eigengap metric $g_n$ [Eq. (6)] with $p$. The figure represents the eigengap between the last bounded and the first unbounded eigenvalue for a node-aligned multiplex network made up by two Erdos-Renyi networks of 200 nodes and an average degree $\langle k\rangle =5$. The vertical continuous line is the analytical value of $p^{\diamond }$, while the dashed line is the bound provided by Eq. (12).

Figure 4

Variation of the entropic measures with $p$. The figure shows the behavior of the relative entropy ($\times 10$), main panel, and of the relative quantum entropy (inset) when $p$ is increased. The multiplex network is the same as described in the caption of Fig. 3 and the vertical line indicates the exact transition point $p^{\diamond }$.