Universal features of correlated bursty behaviour

Inhomogeneous temporal processes, like those appearing in human communications, neuron spike trains, and seismic signals, consist of high-activity bursty intervals alternating with long low-activity periods. In recent studies such bursty behavior has been characterized by a fat-tailed inter-event time distribution, while temporal correlations were measured by the autocorrelation function. However, these characteristic functions are not capable to fully characterize temporally correlated heterogenous behavior. Here we show that the distribution of the number of events in a bursty period serves as a good indicator of the dependencies, leading to the universal observation of power-law distribution in a broad class of phenomena. We find that the correlations in these quite different systems can be commonly interpreted by memory effects and described by a simple phenomenological model, which displays temporal behavior qualitatively similar to that in real systems.

💡 Research Summary

The paper addresses a fundamental problem in the analysis of temporally inhomogeneous point processes that appear across a wide range of disciplines, from human communication to seismology and neurophysiology. Traditional descriptors—namely the inter‑event time distribution P(τ) and the autocorrelation function A(Δt)—are shown to be insufficient for distinguishing genuine temporal correlations from artifacts caused by heavy‑tailed waiting times. In particular, a fat‑tailed P(τ) can generate spurious long‑range correlations in A(Δt) even when events are independent, leading to misleading conclusions about underlying dynamics.

To overcome this limitation, the authors introduce a novel observable: the distribution P(E) of the number of events E that belong to a single “bursty period.” A bursty period is defined operationally as a maximal sequence of events where each successive event occurs within a fixed time window Δt of its predecessor. For a purely random (Poissonian) process with a given P(τ), the authors derive an analytical expression (Eq. 1) that predicts an exponential decay of P(E). Any deviation from this exponential form therefore signals the presence of temporal correlations.

Empirical analysis is performed on four distinct data sets: (i) outgoing mobile‑call records, (ii) outgoing short‑message (SMS) records, (iii) email logs from a university, (iv) earthquake catalogs from a single Japanese station, and (v) spike trains from single neurons. Across all data sets, and for a broad range of Δt values (from seconds to days), the measured P(E) follows a robust power‑law, P(E) ∝ E^−β, with exponents β ranging roughly from 2.5 to 4.1. In contrast, when the inter‑event times are randomly shuffled—thereby destroying any genuine temporal ordering—the resulting P(E) collapses to the exponential form predicted for independent events. This stark contrast demonstrates that P(E) is a sensitive and reliable indicator of true temporal correlations, unaffected by the heavy tails of P(τ).

The authors further connect P(E) to a “memory function” p(n), defined as the conditional probability that a burst continues after it already contains n events: p(n)=P(E≥n+1)/P(E≥n). Empirically, p(n) is well described by a simple functional form p(n)= (n/(n+1))^ν, where ν is a single exponent. Substituting the observed power‑law P(E) into the definition of p(n) yields a scaling relation β = ν + 1, which is confirmed by fitting the empirical data (ν≈2.97, β≈3.97). This relationship provides a compact description of the memory effect governing burst continuation.

To test whether such a memory mechanism can generate the observed statistics, the authors propose a phenomenological “memory‑reinforced” point‑process model. In this model, after each event the probability of staying within the current burst follows p(n) as above; with a complementary probability q a new burst is initiated. By tuning ν and q, the model reproduces the empirical P(E) and p(n) across all examined systems. Simulations of 10⁸ events yield burst‑size distributions that match the measured power‑law exponents, confirming that a simple reinforcement rule based on past burst length suffices to generate the universal bursty behavior.

The paper’s contributions can be summarized as follows: (1) It demonstrates the inadequacy of traditional inter‑event time and autocorrelation analyses for heavy‑tailed processes. (2) It introduces P(E) as a new, robust statistic that directly captures temporal correlations irrespective of P(τ)’s shape. (3) It uncovers a universal power‑law scaling of burst sizes across disparate natural and social systems, indicating a common underlying memory mechanism. (4) It provides a minimal, analytically tractable model that reproduces the observed scaling, offering a unifying framework for bursty dynamics.

Overall, the study advances our understanding of correlated bursty phenomena by shifting the focus from waiting‑time statistics to burst‑size statistics, revealing that memory‑driven reinforcement is a key driver of the complex temporal patterns observed in a wide spectrum of real‑world processes.