We study the time scales associated with diffusion processes that take place on multiplex networks, i.e., on a set of networks linked through interconnected layers. To this end, we propose the construction of a supra-laplacian matrix, which consists of a dimensional lifting of the laplacian matrix of each layer of the multiplex network. We use perturbative analysis to reveal analytically the structure of eigenvectors and eigenvalues of the complete network in terms of the spectral properties of the individual layers. The spectrum of the supra-laplacian allows us to understand the physics of diffusionlike processes on top of multiplex networks.

We study the relationship between topological scales and dynamic time scales in complex networks. The analysis is based on the full dynamics towards synchronization of a system of coupled oscillators. In the synchronization process, modular structures corresponding to well-defined communities of nodes emerge in different time scales, ordered in a hierarchical way. The analysis also provides a useful connection between synchronization dynamics, complex networks topology, and spectral graph analysis.

Multifractal formalism in the microcanonical framework has proved to be a valuable approach to understand and analyze complex signals, typically associated with natural phenomena in scale invariant systems. In this paper, we discuss the multifractal microcanonical formalism in a comprehensive, unified way, including new theoretical proofs and validation tests on real signals, so completing some known gaps in the foundations of this theory. We also review the latest advances and describe the present perspectives in this field. Some technical details on the implementation of involved algorithms and relevant open issues are also discussed.

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We present an extended analysis, based on the dynamics towards synchronization of a system of coupled oscillators, of the hierarchy of communities in complex networks. In the synchronization process, different structures corresponding to well defined communities of nodes appear in a hierarchical way. The analysis also provides a useful connection between synchronization dynamics, complex networks topology and spectral graph analysis.