68 research outputs found
On degree-degree correlations in multilayer networks
We propose a generalization of the concept of assortativity based on the
tensorial representation of multilayer networks, covering the definitions given
in terms of Pearson and Spearman coefficients. Our approach can also be applied
to weighted networks and provides information about correlations considering
pairs of layers. By analyzing the multilayer representation of the airport
transportation network, we show that contrasting results are obtained when the
layers are analyzed independently or as an interconnected system. Finally, we
study the impact of the level of assortativity and heterogeneity between layers
on the spreading of diseases. Our results highlight the need of studying
degree-degree correlations on multilayer systems, instead of on aggregated
networks.Comment: 8 pages, 3 figure
A polynomial eigenvalue approach for multiplex networks
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. More specifically, we reduce the dimensionality of our
matrices but increase the power of the characteristic polynomial, i.e, a
polynomial eigenvalue problem. Such an approach may sound counterintuitive at
first glance, but it allows us to relate the quadratic problem for a 2-Layer
multiplex system with the spectra of the aggregated network and to derive
bounds for the spectra, among many other interesting analytical insights.
Furthermore, it also permits to directly obtain analytical and numerical
insights on the eigenvalue behavior as a function of the coupling between
layers. Our study includes the supra-adjacency, supra-Laplacian, and the
probability transition matrices, which enable 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.Comment: 15 pages including figures. Submitted for publicatio
Layer degradation triggers an abrupt structural transition in multiplex networks
Network robustness is a central point in network science, both from a
theoretical and a practical point of view. In this paper, we show that layer
degradation, understood as the continuous or discrete loss of links' weight,
triggers a structural transition revealed by an abrupt change in the algebraic
connectivity of the graph. Unlike traditional single layer networks, multiplex
networks exist in two phases, one in which the system is protected from link
failures in some of its layers and one in which all the system senses the
failure happening in one single layer. We also give the exact critical value of
the weight of the intra-layer links at which the transition occurs for
continuous layer degradation and its relation to the value of the coupling
between layers. This relation allows us to reveal the connection between the
transition observed under layer degradation and the one observed under the
variation of the coupling between layers.Comment: 8 pages, and 8 figures in Revtex style. Submitted for publicatio
A process of rumor scotching on finite populations
Rumor spreading is a ubiquitous phenomenon in social and technological
networks. Traditional models consider that the rumor is propagated by pairwise
interactions between spreaders and ignorants. Spreaders can become stiflers
only after contacting spreaders or stiflers. Here we propose a model that
considers the traditional assumptions, but stiflers are active and try to
scotch the rumor to the spreaders. An analytical treatment based on the theory
of convergence of density dependent Markov chains is developed to analyze 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 be applied to study systems in which informed agents try
to stop the rumor propagation. In addition, our results can be considered to
develop optimal information dissemination strategies and approaches to control
rumor propagation.Comment: 13 pages, 11 figure
Data-driven contact structures: from homogeneous mixing to multilayer networks
The modeling of the spreading of communicable diseases has experienced
significant advances in the last two decades or so. This has been possible due
to the proliferation of data and the development of new methods to gather, mine
and analyze it. A key role has also been played by the latest advances in new
disciplines like network science. Nonetheless, current models still lack a
faithful representation of all possible heterogeneities and features that can
be extracted from data. Here, we bridge a current gap in the mathematical
modeling of infectious diseases and develop a framework that allows to account
simultaneously for both the connectivity of individuals and the age-structure
of the population. We compare different scenarios, namely, i) the homogeneous
mixing setting, ii) one in which only the social mixing is taken into account,
iii) a setting that considers the connectivity of individuals alone, and
finally, iv) a multilayer representation in which both the social mixing and
the number of contacts are included in the model. We analytically show that the
thresholds obtained for these four scenarios are different. In addition, we
conduct extensive numerical simulations and conclude that heterogeneities in
the contact network are important for a proper determination of the epidemic
threshold, whereas the age-structure plays a bigger role beyond the onset of
the outbreak. Altogether, when it comes to evaluate interventions such as
vaccination, both sources of individual heterogeneity are important and should
be concurrently considered. Our results also provide an indication of the
errors incurred in situations in which one cannot access all needed information
in terms of connectivity and age of the population.Comment: 11 pages, 4 figure
Social contagion models on hypergraphs
Our understanding of the dynamics of complex networked systems has increased
significantly in the last two decades. However, most of our knowledge is built
upon assuming pairwise relations among the system's components. This is often
an oversimplification, for instance, in social interactions that occur
frequently within groups. To overcome this limitation, here we study the
dynamics of social contagion on hypergraphs. We develop an analytical framework
and provide numerical results for arbitrary hypergraphs, which we also support
with Monte Carlo simulations. Our analyses show that the model has a vast
parameter space, with first and second-order transitions, bi-stability, and
hysteresis. Phenomenologically, we also extend the concept of latent heat to
social contexts, which might help understanding oscillatory social behaviors.
Our work unfolds the research line of higher-order models and the analytical
treatment of hypergraphs, posing new questions and paving the way for modeling
dynamical processes on these networks.Comment: 17 pages, including 14 figure
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.Comment: Review article. 73 pages, including 24 figure
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 SIS and SIR dynamics, as well as upper and lower bounds for
the disease prevalence in the steady state for the SIS scenario. Using the
quasi-stationary 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. 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 verified the barrier effect, i.e., for
three-layer configuration, when the layer with the largest 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.Comment: Revised version. 25 pages and 18 figure
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