A polynomial eigenvalue approach for multiplex networks

Abstract

We explore the block nature of the matrix representation of multiplex networks, introducing a new formalism to deal with its spectral properties as a function of the inter-layer coupling parameter. This approach allows us to derive interesting results based on an interpretation of the traditional eigenvalue problem. Specifically, our formalism is based on the reduction of the dimensionality of a matrix of interest but increasing the power of the characteristic polynomial, i.e, a polynomial eigenvalue problem. This approach may sound counterintuitive at first, but it enable us to relate the quadratic eigenvalue problem for a 2-Layer multiplex network with the spectra of its respective aggregated network. Additionally, it also allows us to derive bounds for the spectra, among many other interesting analytical insights. Furthermore, it also permits us to directly obtain analytical and numerical insights on the eigenvalue behavior as a function of the coupling between layers. Our study includes the supraadjacency, supra-Laplacian and the probability transition matrices, which enables us to put our results under the perspective of structural phases in multiplex networks. We believe that this formalism and the results reported will make it possible to derive new results for multiplex networks in the future. phase, for small values of p, where the layers are virtually decoupled and act by themselves, with a negligible interaction between layers, (ii) multiplex/multilayer phase, where the system is coupled and the intra-layer edges play an important role and (iii) the aggregate network phase, where the system behaves as the superposition of all layers. It is clear that a good understanding of the eigenvalues' behavior might be useful since we could move our system into different structural regimes, aiming at different goals such as improved robustness, better performance regarding diffusion or spreading, among many other possible applications.The different structural phases above are related through the interlacing properties of quotient graphs [10,11]. More specifically, in [10] the authors showed that the spectra of different scales of a multiplex network (aggregated network, the network of layers and individual layers) characterize the three phases. In practical terms, the interlacing provides us bounds for the spectra [10, 11], but also emphasizes that the different scales are intrinsically connected. Indeed, it is impossible to tune the leading eigenvalue of the network of layers without also increasing the leading eigenvalue of the whole multiplex. Furthermore, in [12] the authors characterized multiple topological scales using the supra-Laplacian matrix. More specifically, they analyzed eigengaps to characterize them.Following a different approach, in [13], the author evaluated the normalized Laplacian matrix, which, in fact, shares the same set of eigenvalues of the probability transition matrix (see section 4.3), and proposed a similar classification. However, it is worth mentioning that in [13], a different nom...

Social contagion models on hypergraphs

“…A particularly relevant development is the extension of contagion processes to multilayer networks, which in turn paved the way to combinatorial higher-order models. Indeed, multilayer's structural [13][14][15][16][17] and spreading and diffusion properties [5,13,14,18] have a new and richer phenomenology. Nevertheless, as recently argued in [19], real data is revealing that pairwise relationships -the fundamental interaction units of networks -do not capture complex dependencies.…”

Diffusive behavior of multiplex networks

“…Interestingly [15], showed that multiplex networks can have super-diffusive behavior, meaning that their time scale to relax to the steady state is smaller than that of any layer taken in isolation. Since then, the spectral properties of multiplex networks have been widely investigated [19][20][21][22][23], and the formalism has been extended to describe more complex phenomena, such as reaction-diffusion [24][25][26] and synchronization processes [27]. Nevertheless, while it was suggested that low correlation in the structure of the layers can enhance diffusion in a multiplex [28], a rigorous theory for the emergence of super-diffusion is currently lacking.…”

Layer degradation triggers an abrupt structural transition in multiplex networks

“…t α = t, ∀α. In this scenario, we can obtain the exact transition point, t * , by reformulating the eigenvalue problem for the supra-Laplacian in terms of a polynomial eigenvalue problem [26]. In general, a polynomial eigenvalue problem is formulated as an equation of the form…”