Conrad J. Pérez-Vicente

Synchronization Reveals Topological Scales in Complex 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.

Diffusion dynamics on multiplex networks.

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.

Synchronization processes in complex networks

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.

Numerical methods for the estimation of multifractal singularity spectra on sampled data

Physical variables in scale invariant systems often show chaotic, turbulent-like behavior, commonly associated to the existence of an underlying fractal or multifractal structure. However, the assessment of multifractality over experimental, discretized data requires of appropriate methods and to establish criteria to measure the confidence degree on the estimates. In this paper we have evaluated the quality of different techniques used for multifractal analysis. We have tested four different techniques: the moment (M) method, the wavelet transform modulus maxima (WTMM) method, the gradient modulus wavelet projection (GMWP) method and the gradient histogram (GH) method, which are used to estimate the singularity spectra of multifractal signals. The test consists in analyzing synthetic multifractal 1D signals with given multifractal spectrum. We have compared the results, studying the sensibility of each method to the length of the series, size of the ensemble and type of spectrum. Our results show that GMWP method is the one attaining the best performance, providing reliable estimates which can be improved when the statistics is increased. All the other methods are affected by problems such as the linearization of the right tail of the spectrum, and some of them are very demanding in data.

Microcanonical multifractal formalism—a geometrical approach to multifractal systems: Part I. Singularity 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.

Multifractal geometry in stock market time series

It has been recently noticed that time series of returns in stock markets are of multifractal (multiscaling) character. In that context, multifractality has been always evidenced by its statistical signature (i.e., the scaling exponents associated to a related variable). However, a direct geometrical framework, much more revealing about the underlying dynamics, is possible. In this paper, we present the techniques allowing the multifractal decomposition. We will show that there exists a particular fractal component, the most singular manifold (MSM), which contains the relevant information about the dynamics of the series: it is possible to reconstruct the series (at a given precision) from the MSM. We analyze the dynamics of the MSM, which shows revealing features about the evolution of this type of series.

On optimal wavelet bases for the realization of microcanonical cascade processes

Multiplicative cascades are often used to represent the structure of turbulence. Under the action of a multiplicative cascade, the relevant variables of the system can be understood as the result of a successive transfer of information in cascade from large to small scales. However, to make this cascade transfer explicit (i.e, being able to decompose each variable as the product of larger scale contributions) is only achieved when signals are represented in an optimal wavelet basis. Finding such a basis is a data-demanding, highly-complex task. In this paper we propose a formalism that allows to find the optimal wavelet of a signal in an efficient, little data-demanding way. We confirm the appropriateness of this approach by analyzing the results on synthetic signals constructed with prescribed optimal bases. We show the validity of our approach constrained to given families of wavelets, though it can be generalized for a continuous unconstrained search scheme.

Partially and fully frustrated coupled oscillators with random pinning fields

We have studied two specific models of frustrated and disordered coupled Kuramoto oscillators, all driven with the same natural frequency, in the presence of random external pinning fields. Our models are structurally similar, but differ in their degree of bond frustration and in their finite size ground state properties (one has random ferro- and anti-ferromagnetic interactions and the other has random chiral interactions). We have calculated the equilibrium properties of both models in the thermodynamic limit using the replica method, with emphasis on the role played by symmetries of the pinning field distribution, leading to explicit predictions for observables, transitions and phase diagrams. For absent pinning fields our two models are found to behave identically, but pinning fields (provided with appropriate statistical properties) break this symmetry. Simulation data lend satisfactory support to our theoretical predictions.

The configuration multi-edge model: Assessing the effect of fixing node strengths on weighted network magnitudes

Complex networks grow subject to structural constraints which affect their measurable properties. Assessing the effect that such constraints impose on their observables is thus a crucial aspect to be taken into account in their analysis. To this end, we examine the effect of fixing the strength sequence in multi-edge networks on several network observables such as degrees, disparity, average neighbor properties and weight distribution using an ensemble approach. We provide a general method to calculate any desired weighted network metric and we show that several features detected in real data could be explained solely by structural constraints. We thus justify the need of analytical null models to be used as basis to assess the relevance of features found in real data represented in weighted network form.

A solvable model of the genesis of amino-acid sequences via coupled dynamics of folding and slow-genetic variation

We study the coupled dynamics of primary and secondary structures formation (i.e. slow-genetic sequence selection and fast folding) in the context of a solvable microscopic model that includes both short-range steric forces and long-range polarity-driven forces. Our solution is based on the diagonalization of replicated transfer matrices, and leads in the thermodynamic limit to explicit predictions regarding phase transitions and phase diagrams at genetic equilibrium. The predicted phenomenology allows for natural physical interpretations, and finds satisfactory support in numerical simulations.