Burstiness and memory in complex systems

The dynamics of a wide range of real systems, from email patterns to earthquakes, display a bursty, intermittent nature, characterized by short timeframes of intense activity followed by long times of no or reduced activity. The understanding of the origin of such bursty patterns is hindered by the lack of tools to compare different systems using a common framework. Here we propose to characterize the bursty nature of real signals using orthogonal measures quantifying two distinct mechanisms leading to burstiness: the interevent time distribution and the memory. We find that while the burstiness of natural phenomena is rooted in both the interevent time distribution and memory, for human dynamics memory is weak, and the bursty character is due to the changes in the interevent time distribution. Finally, we show that current models lack in their ability to reproduce the activity pattern observed in real systems, opening up avenues for future work. Copyright c © EPLA, 2008 The dynamics of most complex systems is driven by the loosely coordinated activity of a large number of components, such as individuals in the society or molecules in the cell. While we witnessed much progress in the study of the networks behind these systems [1–4], advances towards understanding the system’s dynamics have been slower. With increasing potential to monitor the time-resolved activity of most components of selected complex systems, such as time-resolved email [5–7], web browsing [8], and gene expression [9,10] patterns, we have the opportunity to ask an important question: is the dynamics of complex systems governed by generic organizing principles, or each system has its own distinct dynamical features? While it is difficult to offer a definite answer to this question, a common feature across many systems is increasingly documented: the burstiness of the system’s activity patterns. Bursts, vaguely corresponding to significantly enhanced activity levels over short periods of time followed by long periods of inactivity, have been observed in a wide range of systems, from email patterns [6] to earthquakes [11,12] and gene expression [9]. Yet, often a burstiness is more of a metaphor than a quantitative feature, and opinions about its origin diverge. In human dynamics, burstiness has been reduced to the fat-tailed nature of the response time distribution [6,7], in contrast with earthquakes and weather patterns, where memory effects appear to play a key role as well [13,14]. Once present, burstiness can affect the spreading of viruses [15] or resource allocation [16,17]. Also, deviations towards a regular, “anti-bursty” behavior in heartbeat indicate disease progression [18]. Given the diversity of systems in which it emerges, there is a need to place burstiness on a firmer quantitative basis. Our goal in this letter is to make a step in this direction, by developing a diagnosis tool that can help quantify the magnitude and potential origin of the bursty patterns seen in different real systems. Such a tool may also lend insights into the analysis of fractal and self-similar bursty signals [19]. Let us consider a system whose components have a measurable activity pattern that can be mapped into a discrete signal, recording the moments when some events take place, like an email being sent, or a protein being translated. The activity pattern is random (Poisson process) if the probability of an event is time-independent. In this case the interevent time, τ , between two consecutive events follows an exponential distribution, PP(τ)∼ exp(−τ/τ0) (fig. 1a). An apparently bursty For systems with continuous signal, we may adopt a threshold method to transform it into a discrete one, and in many systems the statistical properties of the obtained signal are known to be threshold-independent [12,13].