Guilherme Ferraz de Arruda

Disease Localization in Multilayer Networks

We present a continuous formulation of epidemic spreading on multilayer networks using a tensorial representation, extending the models of monoplex networks to this context. We derive analytical expressions for the epidemic threshold of the susceptible-infected-susceptible (SIS) and susceptible-infected-recovered dynamics, as well as upper and lower bounds for the disease prevalence in the steady state for the SIS scenario. Using the quasistationary state method, we numerically show the existence of disease localization and the emergence of two or more susceptibility peaks, which are characterized analytically and numerically through the inverse participation ratio. At variance with what is observed in single-layer networks, we show that disease localization takes place on the layers and not on the nodes of a given layer. Furthermore, when mapping the critical dynamics to an eigenvalue problem, we observe a characteristic transition in the eigenvalue spectra of the supra-contact tensor as a function of the ratio of two spreading rates: If the rate at which the disease spreads within a layer is comparable to the spreading rate across layers, the individual spectra of each layer merge with the coupling between layers. Finally, we report on an interesting phenomenon, the barrier effect; i.e., for a three-layer configuration, when the layer with the lowest eigenvalue is located at the center of the line, it can effectively act as a barrier to the disease. The formalism introduced here provides a unifying mathematical approach to disease contagion in multiplex systems, opening new possibilities for the study of spreading processes.

Multilayer Networks: Metrics and Spectral Properties

Multilayer networks represent systems in which there are several topological levels each one representing one kind of interaction or interdependency between the systems’ elements. These networks have attracted a lot of attention recently because their study allows considering different dynamical modes concurrently. Here, we revise the main concepts and tools developed up to date. Specifically, we focus on several metrics for multilayer network characterization as well as on the spectral properties of the system, which ultimately enable for the dynamical characterization of several critical phenomena. The theoretical framework is also applied for description of real-world multilayer systems.

Structure and dynamics of functional networks in child-onset schizophrenia

OBJECTIVE Schizophrenia is a neuropsychiatric disorder characterized by cognitive and emotional deficits and associated with various abnormalities in the organization of neural circuits. It is currently unclear how and to which extend the global network organization is changed due to such disorder. In this work, we analyzed cortical networks of healthy subjects and patients with child-onset schizophrenia to address this issue. METHODS We performed a comparison of cortical networks extracted from functional MRI data of patients with schizophrenia and healthy subjects considering their topological and dynamical properties. RESULTS Among 54 network measures tested, only four contributed substantially to a discrimination between the classes of healthy and schizophrenic subjects, with a sensitivity of 90% and specificity of 74%. However, such classes of networks did not differ significantly with respect to the level of network resilience and synchronization. CONCLUSIONS Schizophrenic subjects have cortical regions with higher variance of network centrality, but less modular structure. SIGNIFICANCE Our findings suggest that it is possible to establish data analysis routines that allow automatic diagnosis of a multifaceted disease like child-onset schizophrenia based on fMRI data of individual subjects and extracted network properties.

Scaling properties of multilayer random networks

Multilayer networks are widespread in natural and manmade systems. Key properties of these networks are their spectral and eigenfunction characteristics, as they determine the critical properties of many dynamics occurring on top of them. Here, we numerically demonstrate that the normalized localization length β of the eigenfunctions of multilayer random networks follows a simple scaling law given by β=x^{*}/(1+x^{*}), with x^{*}=γ(b_{eff}^{2}/L)^{δ}, δ∼1, and b_{eff} being the effective bandwidth of the adjacency matrix of the network, whose size is L. The scaling law for β, that we validate on real-world networks, might help to better understand criticality in multilayer networks and to predict the eigenfunction localization properties of them.

Diluted banded random matrices: scaling behavior of eigenfunction and spectral properties

We demonstrate that the normalized localization length β of the eigenfunctions of diluted (sparse) banded random matrices follows the scaling law . The scaling parameter of the model is defined as , where is the average number of non-zero elements per matrix row, N is the matrix size, and . Additionally, we show that also scales the spectral properties of the model (up to certain sparsity) characterized by the spacing distribution of eigenvalues.

A process of rumour scotching on finite populations

Rumour spreading is a ubiquitous phenomenon in social and technological networks. Traditional models consider that the rumour is propagated by pairwise interactions between spreaders and ignorants. Only spreaders are active and may become stiflers after contacting spreaders or stiflers. Here we propose a competition-like model in which spreaders try to transmit an information, while stiflers are also active and try to scotch it. We study the influence of transmission/scotching rates and initial conditions on the qualitative behaviour of the process. An analytical treatment based on the theory of convergence of density-dependent Markov chains is developed to analyse how the final proportion of ignorants behaves asymptotically in a finite homogeneously mixing population. We perform Monte Carlo simulations in random graphs and scale-free networks and verify that the results obtained for homogeneously mixing populations can be approximated for random graphs, but are not suitable for scale-free networks. Furthermore, regarding the process on a heterogeneous mixing population, we obtain a set of differential equations that describes the time evolution of the probability that an individual is in each state. Our model can also be applied for studying systems in which informed agents try to stop the rumour propagation, or for describing related susceptible–infected–recovered systems. In addition, our results can be considered to develop optimal information dissemination strategies and approaches to control rumour propagation.

Fundamentals of spreading processes in single and multilayer complex networks

Spreading processes have been largely studied in the literature, both analytically and by means of large-scale numerical simulations. These processes mainly include the propagation of diseases, rumors and information on top of a given population. In the last two decades, with the advent of modern network science, we have witnessed significant advances in this field of research. Here we review the main theoretical and numerical methods developed for the study of spreading processes on complex networked systems. Specifically, we formally define epidemic processes on single and multilayer networks and discuss in detail the main methods used to perform numerical simulations. Throughout the review, we classify spreading processes (disease and rumor models) into two classes according to the nature of time: (i) continuous-time and (ii) cellular automata approach, where the second one can be further divided into synchronous and asynchronous updating schemes. Our revision includes the heterogeneous mean-field, the quenched-mean field, and the pair quenched mean field approaches, as well as their respective simulation techniques, emphasizing similarities and differences among the different techniques. The content presented here offers a whole suite of methods to study epidemic-like processes in complex networks, both for researchers without previous experience in the subject and for experts.

Multiplex Networks: Basic Definition and Formalism

In this chapter, we present and define multiplex networks as they will be used in this book.

Polynomial Eigenvalue Formulation

In the previous chapters, we have dealt with the matricial representation for multiplex systems. Here, we focus on the matricial representation and explore the block nature of such representation.

Structural Organization and Transitions

Complex networks show nontraditional critical properties due to their extreme compactness (small-world property) together with their complex organization [28].