Source-linked AI summary
Universal features of correlated bursty behaviour
Márton Karsai, Kimmo Kaski, Albert-László Barabási, János Kertész
TL;DR
Fat-tailed inter-event times and autocorrelation do not fully distinguish heterogeneity from true temporal correlations. The paper introduces burst-size distributions and a memory-based phenomenological model, finding robust scale-invariant burst lengths across human, seismic, and neuronal sequences.
Problem
Inter-event-time distributions and autocorrelation functions cannot fully characterize temporally heterogeneous signals or reliably identify true correlations.
Method
The paper analyzes burst-size distributions, interprets correlated bursts through memory effects, and develops a two-state phenomenological model with reinforcement dynamics.
Results
Scale-invariant burst-length distributions remain similar after activity grouping and daily-fluctuation removal, with long bursts retained at β ≃4.1 in mobile-call data.
Takeaways & Limitations
The number of correlated events in bursty cascades provides a cross-system measure of temporal correlations and heterogeneity.
Abstract
from arXiv · showhide
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.
A. Correlated events
The paper identifies bursty clusters by linking consecutive events separated by at most Δt and counting events within each cluster. For independent events, the resulting P(E) decays exponentially, so deviations indicate temporal correlations.
- A. Correlated events: Bursty periods contain consecutive events whose inter-event gaps do not exceed Δt.The number of events in each period is denoted E, and their distribution is P(E).
- A. Correlated events: For independent events, P(E) is determined by the inter-event-time distribution P(tie).
- A. Correlated events: Deviations from exponential P(E) decay indicate correlations between consecutive event times, even when P(tie) is fat-tailed.
1. Bursty sequences in human communication
Human communication sequences show correlated, scale-invariant bursty behavior across calls, messages, and emails. The P(E) distribution distinguishes these correlations from independent-event behavior and remains robust across activity groups and after removing daily fluctuations.
- 1. Bursty sequences in human communication: Emails show long-term correlations up to 8 hours, consistent with the office-hour rhythm of the university-staff dataset.
- 1. Bursty sequences in human communication: P(E) reveals strong temporal correlations in mobile-call, text-message, and email sequences, unlike the exponential P(E) of independent events.
- 1. Bursty sequences in human communication: P(E) remains fat-tailed for users grouped by activity and retains similar scaling after daily fluctuations are removed.
- 1. Bursty sequences in human communication: Mobile calls show exponents α ≃0.5, β ≃4.1 and γ ≃0.7; text messages show α ≃0.6, β ≃3.9 and γ ≃0.7.
2. Bursty periods in natural phenomena
Earthquake and single-neuron sequences both exhibit heterogeneous, correlated bursty behavior. Their burst-size distributions are broad, with system-specific scaling exponents.
- 2. Bursty periods in natural phenomena: Earthquake sequences at one Japanese station show broad P(E) distributions for Δt windows of 2–32 hours, with exponent β different from communication data.
- 2. Bursty periods in natural phenomena: Single-neuron spike trains display fat-tailed train lengths, indicating correlations between consecutive bursty spikes.
3. Memory process
The paper interprets correlated bursty events as a memory process and tests a phenomenological model against empirical sequences. The model captures common dynamics across social, physical, and biological systems, while finite data impose a cutoff on observed memory.
- 3. Memory process: Memory effects explain bursty trains across human communication, earthquakes, and neuron spikes, while the model captures their shared temporal features.
- 3. Memory process: A memory function p(n) gives the probability of another event within Δt after n prior events in the same cascade.
- 3. Memory process: Finite sequence length limits the longest bursty train, producing a finite cutoff in p(n).
- 3. Memory process: The empirical mobile-call memory function fits the theoretical form with ν = 2.971±0.072, implying β ≃3.971, close to the directly fitted β ≃4.1.
- 3. Memory process: The phenomenological model uses normal and excited states, with correlated high-frequency events generated through reinforcement and memory.
4. Reinforcement dynamics with memory
The model combines two activity states with reinforcement-based waiting times and memory-driven transitions to reproduce heterogeneous, correlated bursts. Simulations yield scale-free inter-event times and autocorrelations with linked exponents.
- Model definition: The two-state model separates normal state A, with independent longer waits, from excited state B, with correlated higher-frequency events.State B represents burst activity.
- Model definition: Reinforcement makes continued waiting more likely after longer elapsed times, while transition probabilities govern switching between states.The model uses π for switching from A to B and 1−π for remaining in A.
- Numerical results: β = 3.0 is the fitted exponent of the synthetic burst-event distribution P(E).Results were averaged over 1000 independent realizations.
- Numerical results: γ = 1.3 characterizes the simulated scale-free inter-event time distribution, satisfying γ = µA + 1.The simulation uses a maximum inter-event time tmax ie = 10^6.
- Numerical results: α = 0.7 characterizes the power-law decay of the simulated autocorrelation function, satisfying α + γ = 2.The autocorrelation calculation uses a maximum lag τ max = 10^4.
II. DISCUSSION
The discussion presents correlated burst lengths as a scale-free, system-dependent signature across heterogeneous temporal systems and interprets the dynamics through memory effects.
- II. DISCUSSION: The number of correlated events in bursty cascades detects both correlations and heterogeneity in temporal sequences.It captures behavior not observed from inter-event distributions and autocorrelation functions alone.
- II. DISCUSSION: Scale-free burst-length distributions appear across the studied systems, with exponents that vary by system.The paper identifies this as a universal property of correlated temporal patterns.
- II. DISCUSSION: Memory effects account for heterogeneous temporal behavior, while a phenomenological model captures common features of the empirical systems.The model is intended to clarify the role of memory during temporal evolution.
III. MATERIALS AND METHODS
The materials and methods describe extracting individual communication-event sequences and shuffling inter-event times to construct independent-event comparisons.
- III. MATERIALS AND METHODS: The study uses time-stamped communication records from three datasets and extracts outgoing-event sequences for each individual.The cited passage specifies a mobile-call dataset covering approximately 325 × 10^6 records and 6.5 × 10^6 users over 120 days.
- III. MATERIALS AND METHODS: Shuffling inter-event times across users preserves each individual’s event count while disrupting the original temporal ordering.This procedure constructs independent-event sequences for comparison.
- III. MATERIALS AND METHODS: For the shuffled independent sequences, P(E) is calculated with one Δt window size to demonstrate exponential behavior.
Supplementary Informations
The supplied passage identifies the paper’s authors: M. Karsai, K. Kaski, A.-L. Barabási, and J. Kertész.
- Supplementary Informations: M. Karsai is listed among the paper’s authors.
- Supplementary Informations: K. Kaski is listed among the paper’s authors.
- Supplementary Informations: A.-L. Barabási and J. Kertész are listed among the paper’s authors.
IV. SCALING OF THE AUTOCORRELATION FUNCTION IN HETEROGENEOUS INDEPENDENT PROCESSES
Independent heterogeneous event sequences can produce apparent power-law autocorrelations, with the effective exponents linked to the inter-event-time exponent. Numerical simulations verify this relationship for γ = 1.5.
- For independent sequences with P(t_ie) ∼ t^-γ, the autocorrelation can scale as A(τ) ∼ τ^-α, creating apparent correlations.This scaling applies for 1 ≤ γ ≤ 2.
- γ = 1.5 yields an effective autocorrelation exponent α = 0.5 in the numerical simulation.The simulated result follows the α + γ = 2 relation.
V. P(E) DISTRIBUTON IN INDEPENDENT MODELS
For independent event processes, the burst-size distribution P(E) decays exponentially, even when inter-event times are fat-tailed. In empirical call data, power-law P(E) scaling instead reveals temporal correlations and remains robust across time windows and activity groups.
- Independent models: Independent events produce exponentially decaying P(E), regardless of whether inter-event times follow power-law or exponential distributions.The analytical result assumes a finite time window and independently sampled inter-event times; simulations confirm the predicted decay.
- Time-window regimes: For ∆t < tl_c, P(E) is concave; for ∆t > th_c, uncorrelated periods merge and P(E) approaches the strength distribution.The intermediate regime is characterized by scale-free P(E) with an exponent insensitive to ∆t.
- Empirical call data: β = 4.1, tl_c = 20 seconds, and th_c = 12 hours characterize the mobile-call P(E) scaling and its two characteristic time scales.The shorter scale corresponds to typical reaction time, while the longer reflects correlated periods of daily human activity.
- Robustness checks: P(E) retains scale-free behavior with β ≃ 4.1 after combining call sequences from 106 users, ruling out broad strength distribution as its sole source.The critical regime persists as ∆t increases from its minimum to maximum.
- Strength decomposition: Across strength groups, P(t_ie) collapses onto one master curve after average-time scaling, while P(E) remains fat-tailed with β = 2.45...4.3.Long correlated bursty periods occur even for users with few calls, but less frequently than for highly active users.
VIII. DE-SEASONED RESULTS
De-seasoning reduces daily-cycle effects while preserving the broad inter-event-time and burst-size scaling observed in outgoing call sequences. Earthquake analyses likewise retain scaling across area-bin sizes, supporting genuine temporal correlations rather than collection artifacts.
- De-seasoning: Rescaling event times dilates high-frequency periods and contracts low-frequency periods, reducing cyclic fluctuations before analyzing call sequences.The procedure normalizes the rescaled event rate and preserves the event-time density relation.
- De-seasoning: Autocorrelation weakens after de-seasoning but retains long temporal correlations, with α ≃ 0.75 versus α = 0.5 before rescaling.The inter-event-time distribution remains similar, with γ ≃ 0.7.
- De-seasoning: β ≃ 4.1: outgoing-call burst-size distributions retain power-law-like decay across different Δt values after daily patterns are removed.Long bursty periods therefore remain visible in the de-seasoned sequences.
- Earthquake robustness: Scaling behavior of earthquake characteristic functions is recovered with equal-size area bins, indicating that temporal correlations characterize the sequences rather than the collection method.The analysis compares area-bin sizes of 100 km^2, 10^2 km^2, and 10^4 km^2.
- Memory-function estimation: Empirical complement memory functions exhibit finite-size effects, while logarithmic binning and asymptotic fitting provide the comparison with the analytical memory form.The analytical and empirical functions fit well asymptotically for larger n, with deviations at small n because empirical P(E) distributions are not perfect power laws.