Separating Internal and External Dynamics of Complex Systems

Abstract

The observable behavior of a complex system reflects the mechanisms governing the internal interactions between the system's components and the effect of external perturbations. Here we show that by capturing the simultaneous activity of several of the system's components we can separate the internal dynamics from the external fluctuations. The method allows us to systematically determine the origin of fluctuations in various real systems, finding that while the Internet and the computer chip have robust internal dynamics, highway and Web traffic are driven by external demand. As multichannel measurements are becoming the norm in most fields, the method could help uncover the collective dynamics of a wide array of complex systems.Decades of research has lead to the development of sophisticated tools to analyze time series generated by various dynamical systems, allowing us to extract short and long range temporal correlations, periodic patterns and stationarity information [1,2,3,4]. We lack, however, systematic methods to extract from multiple datasets information not already provided by a single time series. Indeed, advances in computer aided measurement techniques increasingly offer the possibility to separately but simultaneously record the time dependent activity of a system's many components, such as information flow on thousands of Internet routers or highway traffic on numerous highways. As the time dependent activity of each component (router or highway) captures the system's dynamics from a different angle, these parallel time series offer us increasingly complete information about the system's collective behavior. Yet, we have difficulty answering a simple question: How can we uncover from multiple time series a system's internal dynamics?Multiple time series are typically available for complex systems whose dynamics is determined by the interaction of a large number of components that communicate with each other through some complex network [5]. The dynamics of each component is determined by two factors: (1) interactions between the components, governed by some internal dynamical rules that distribute the activity between the various parts of the system and (2) global variations in the overall activity of the system. For example, the traffic increase on highways during peak hours and surges in the number of Internet users during working hours represent global activity changes that have a strong impact on the local activity of each component (highway or router) as well. Different components are influenced to a different degree by these global changes, making impossible for an observer that has access only to a single component to separate the internal dynamics from the externally imposed fluctuations. Most important, the inevitable fluctuations in the external conditions systematically obscure the mechanisms that govern the system's internal dynamics.Here we propose a method to separate in a systematic manner for each time series the external from the internal contributions, and validate it on model ...

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“…The resulting dependence turns out to be proportional, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hN 2 i À hNi 2 p % 0:1hNi, over a broad region of values hNi, which crucially differs from the square root behavior of standard statistical fluctuations [19]. The usual reason for such a strong difference is that the fluctuations of the quantities under study are not statistically independent [20,21]. In this respect, there is only one factor in the evolution of the array of numbers which can break the statistical independence of fluctuations, namely, the variation of the influx of numbers.…”

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“…Another striking example closely related to the problem here addressed in which the degree of the nodes determines the dynamical properties is the scaling law characterizing flow fluctuations in complex networks [16][17][18][19]. Admittedly, the mean traffic f and its standard deviation σ can be related through the simple scaling form σ ∼ f α [16][17][18]. However, this relation, which was previously thought to be universal with α being between 1/2 and 1, is not satisfied for all values of k (i.e., the exponent is not universal and depends, among other factors, on the degree of the nodes [19]).…”