153 research outputs found
Quantifying dynamical spillover in co-evolving multiplex networks
Multiplex networks (a system of multiple networks that have different types
of links but share a common set of nodes) arise naturally in a wide spectrum of
fields. Theoretical studies show that in such multiplex networks, correlated
edge dynamics between the layers can have a profound effect on dynamical
processes. However, how to extract the correlations from real-world systems is
an outstanding challenge. Here we provide a null model based on Markov chains
to quantify correlations in edge dynamics found in longitudinal data of
multiplex networks. We use this approach on two different data sets: the
network of trade and alliances between nation states, and the email and
co-commit networks between developers of open source software. We establish the
existence of "dynamical spillover" showing the correlated formation (or
deletion) of edges of different types as the system evolves. The details of the
dynamics over time provide insight into potential causal pathways
Dynamiques stochastiques sur réseaux complexes
Tableau d’honneur de la Faculté des études supérieures et postdoctorales, 2012-2013.Cette thèse a pour but d'élaborer et d'étudier des modèles mathématiques reproduisant le comportement de systèmes composés de plusieurs éléments dont les interactions forment un réseau complexe. Le corps du document est découpé en trois parties ; un chapitre introductif et une conclusion récapitulative complétent la thèse. La partie I s'intéresse à une dynamique spécifique (propagation de type susceptibleinfectieux- retiré, SIR) sur une classe de réseaux également spécifique (modèle de configuration). Ce problème a entre autres déjà été étudié comme un processus de branchement dans la limite où la taille du système est infinie, fournissant une solution probabiliste pour l'état final de ce processus stochastique. La principale contribution originale de la partie I consiste à modifier ce modèle afin d'introduire des éffets dûs à la taille finie du système et de permettre l'étude de son évolution temporelle (temps discret) tout en préservant la nature probabiliste de la solution. La partie II, contenant les principales contributions originales de cette thèse, s'intéresse aux processus stochastiques sur réseaux complexes en général. L'état du système (incluant la structure d'interaction) est partiellement représenté à l'aide de motifs, et l'évolution temporelle (temps continu) est étudiée à l'aide d'un processus de Markov. Malgré que l'état ne soit que partiellement représenté, des résultats satisfaisants sont souvent possibles. Dans le cas particulier du problème étudié en partie I, les résultats sont exacts. L'approche se révèle très générale, et de simples méthodes d'approximation permettent d'obtenir une solution pour des cas d'une complexité appréciable. La partie III cherche une solution analytique exacte sous forme fermée au modèle développé en partie II pour le problème initialement étudié en partie I. Le système est réexprimé en terme d'opérateurs et différentes relations sont utilisées afinn de tenter de le résoudre. Malgré l'échec de cette entreprise, certains résultats méritent mention, notamment une généralisation de la relation de Sack, un cas particulier de la relation de Zassenhaus.The goal of this thesis is to develop and study mathematical models reproducing the behaviour of systems composed of numerous elements whose interactions make a complex network structure. The body of the document is divided in three parts; an introductory chapter and a recapitulative conclusion complete the thesis. Part I pertains to a specific dynamics (susceptible-infectious-removed propagation, SIR) on a class of networks that is also specific (configuration model). This problem has already been studied, among other ways, as a branching process in the infinite system size limit, providing a probabilistic solution for the final state of this stochastic process. The principal original contribution of part I consists of modifying this model in order to introduce finite-size effects and to allow the study of its (discrete) time evolution while preserving the probabilistic nature of the solution. Part II, containing the principal contributions of this thesis, is interested in the general problem of stochastic processes on complex networks. The state of the system (including the interaction structure) is partially represented through motifs, then the (continuous) time evolution is studied with a Markov process. Although the state is only partially represented, satisfactory results are often possible. In the particular case of the problem studied in part I, the results are exact. The approach turns out to be very general, and simple approximation methods allow one to obtain a solution for cases of considerable complexity. Part III searches for a closed form exact analytical solution to the the model developed in part II for the problem initially studied in part I. The system is re-expressed in terms of operators and different relations are used in an attempt to solve it. Despite the failure of this enterprise, some results deserve mention, notably a generalization of Sack's relationship, a special case of the Zassenhaus relationship
Exact solution of bond percolation on small arbitrary graphs
We introduce a set of iterative equations that exactly solves the size
distribution of components on small arbitrary graphs after the random removal
of edges. We also demonstrate how these equations can be used to predict the
distribution of the node partitions (i.e., the constrained distribution of the
size of each component) in undirected graphs. Besides opening the way to the
theoretical prediction of percolation on arbitrary graphs of large but finite
size, we show how our results find application in graph theory, epidemiology,
percolation and fragmentation theory.Comment: 5 pages and 3 figure
Adaptive networks: coevolution of disease and topology
Adaptive networks have been recently introduced in the context of disease
propagation on complex networks. They account for the mutual interaction
between the network topology and the states of the nodes. Until now, existing
models have been analyzed using low-complexity analytic formalisms, revealing
nevertheless some novel dynamical features. However, current methods have
failed to reproduce with accuracy the simultaneous time evolution of the
disease and the underlying network topology. In the framework of the adaptive
SIS model of Gross et al. [Phys. Rev. Lett. 96, 208701 (2006)], we introduce an
improved compartmental formalism able to handle this coevolutionary task
successfully. With this approach, we analyze the interplay and outcomes of both
dynamical elements, process and structure, on adaptive networks featuring
different degree distributions at the initial stage.Comment: 11 pages, 8 figures, 1 appendix. To be published in Physical Review
Strategic tradeoffs in competitor dynamics on adaptive networks
Recent empirical work highlights the heterogeneity of social competitions
such as political campaigns: proponents of some ideologies seek debate and
conversation, others create echo chambers. While symmetric and static network
structure is typically used as a substrate to study such competitor dynamics,
network structure can instead be interpreted as a signature of the competitor
strategies, yielding competition dynamics on adaptive networks. Here we
demonstrate that tradeoffs between aggressiveness and defensiveness (i.e.,
targeting adversaries vs. targeting like-minded individuals) creates
paradoxical behaviour such as non-transitive dynamics. And while there is an
optimal strategy in a two competitor system, three competitor systems have no
such solution; the introduction of extreme strategies can easily affect the
outcome of a competition, even if the extreme strategies have no chance of
winning. Not only are these results reminiscent of classic paradoxical results
from evolutionary game theory, but the structure of social networks created by
our model can be mapped to particular forms of payoff matrices. Consequently,
social structure can act as a measurable metric for social games which in turn
allows us to provide a game theoretical perspective on online political
debates.Comment: 20 pages (11 pages for the main text and 9 pages of supplementary
material
Modeling the dynamical interaction between epidemics on overlay networks
Epidemics seldom occur as isolated phenomena. Typically, two or more viral
agents spread within the same host population and may interact dynamically with
each other. We present a general model where two viral agents interact via an
immunity mechanism as they propagate simultaneously on two networks connecting
the same set of nodes. Exploiting a correspondence between the propagation
dynamics and a dynamical process performing progressive network generation, we
develop an analytic approach that accurately captures the dynamical interaction
between epidemics on overlay networks. The formalism allows for overlay
networks with arbitrary joint degree distribution and overlap. To illustrate
the versatility of our approach, we consider a hypothetical delayed
intervention scenario in which an immunizing agent is disseminated in a host
population to hinder the propagation of an undesirable agent (e.g. the spread
of preventive information in the context of an emerging infectious disease).Comment: Accepted for publication in Phys. Rev. E. 15 pages, 7 figure
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