Microdynamics in stationary complex networks

arXiv:0811.1051v1 [physics.data-an] 6 Nov 2008 Microdynamics in stationary complex networks Aurelien Gautreau,1 Alain Barrat,1, 2, 3 and Marc Barth´elemy4, 5 1Laboratoire de Physique Th´eorique (CNRS UMR 8627), Universit´e de Paris-Sud, 91405 Orsay, France 2Centre de Physique Th´eorique (CNRS UMR 6207), Luminy Case 907, 13288 Marseille Cedex 9, France 3Complex Networks Lagrange Laboratory, ISI Foundation, Torino, Italy 4CEA-D´epartement de Physique Th´eorique et Appliqu´ee, 91680 Bruyeres-Le-Chatel, France 5Centre d’Analyse et de Math´ematique Sociales (CAMS, UMR 8557 CNRS-EHESS), Ecole des Hautes Etudes en Sciences Sociales, 54 bd. Raspail, F-75270 Paris Cedex 06, France (Dated: November 24, 2021) Many complex systems, including networks, are not static but can display strong fluctuations at various time scales. Characterizing the dynamics in complex networks is thus of the utmost importance in the understanding of these networks and of the dynamical processes taking place on them. In this article, we study the example of the US airport network in the time period 1990−2000. We show that even if the statistical distributions of most indicators are stationary, an intense activity takes place at the local (‘microscopic’) level, with many disappearing/appearing connections (links) between airports. We find that connections have a very broad distribution of lifetimes, and we introduce a set of metrics to characterize the links’ dynamics. We observe in particular that the links which disappear have essentially the same properties as the ones which appear, and that links which connect airports with very different traffic are very volatile. Motivated by this empirical study, we propose a model of dynamical networks, inspired from previous studies on firm growth, which reproduces most of the empirical observations both for the stationary statistical distributions and for the dynamical properties. PACS numbers: INTRODUCTION Despite the presence of stable statistical regularities at the global level, many systems exhibit an intense activity at the level of individual components, i.e. at the ‘micro- scopic’ level. An important illustration of this fact was recently put forward by Batty [1] in the case of city popu- lations. Indeed, even if the population Zipf plots display negligible changes in time, the same city can have very different ranks in the course of history. Similarly, many other systems, in particular occurring in human dynam- ics studies, present simultaneously stationary statistical distributions and strong time fluctuations at the micro- scopic level, with activity bursts separated by very het- erogeneous time intervals [2, 3, 4, 5]. These systems thus challenge us with the fundamental puzzle which consists in reconciling an important dynamical activity occurring at the local level on many timescales and the emergence of stable distributions at a macroscopic level which can be maintained even when the external conditions are highly non-stationary [6]. For instance, the dynamics of the rank is not consistent with processes such as preferential attachment [7] where the rank is essentially constant in time. These issues naturally apply to the case where complex systems are structured under the form of large networks. In most recent studies, these networks have been consid- ered as static objects with a fixed topology. However, their structure may in principle evolve, links may ap- pear and disappear. Such topological fluctuations have important consequences: many dynamical processes take place on complex networks [8, 9, 10, 11], and a non trivial interplay can occur between the evolutions of the topol- ogy and of these dynamical processes. The structure of the network strongly influences the characteristics of the dynamical processes [11], and the topology of the net- work can simultaneously be modified as a consequence of the process itself. In this framework, recent studies have been devoted to simple models of coevolution and adap- tive networks [12, 13, 14, 15, 16]. Another illustration of the importance of taking into account the dynamics of the network is given by concurrency effects in epidemi- ology [17]. Indeed, while a contact network is usually measured at a certain instant or aggregated over a cer- tain period, the actual spread of epidemics depends on the instantaneous contacts. In such contexts, it is thus crucial to gain insights into the dynamics of the network, possibly by putting forward convenient new measures and to propose possible models for it. These considerations emphasize the need for empiri- cal observations and models for the dynamics of complex networks, which are up to now quite scarce. In this pa- per, we study the case of the US airport network (USAN) where nodes are airports and links represent direct con- nections between them. It is indeed possible to gather data on the time evolution of this network [18] (see also [19] for a study of the yearly evolution of the Brazilian airport ne

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