Inter-event Time Distribution Research Articles

Regulation of burstiness by network-driven activation.

We prove that complex networks of interactions have the capacity to regulate and buffer unpredictable fluctuations in production events. We show that non-bursty network-driven activation dynamics can effectively regulate the level of burstiness in the production of nodes, which can be enhanced or reduced. Burstiness can be induced even when the endogenous inter-event time distribution of nodes' production is non-bursty. We find that hubs tend to be less susceptible to the networked regulatory effects than low degree nodes. Our results have important implications for the analysis and engineering of bursty activity in a range of systems, from communication networks to transcription and translation of genes into proteins in cells.

Improve the algorithmic performance of collaborative filtering by using the interevent time distribution of human behaviors

Recently, many scaling laws of interevent time distribution of human behaviors are observed and some quantitative understanding of human behaviors are also provided by researchers. In this paper, we propose a modified collaborative filtering algorithm by making use the scaling law of human behaviors for information filtering. Extensive experimental analyses demonstrate that the accuracies on MovieLensand Last.fm datasets could be improved greatly, compared with the standard collaborative filtering. Surprisingly, further statistical analyses suggest that the present algorithm could simultaneously improve the novelty and diversity of recommendations. This work provides a creditable way for highly efficient information filtering.

A new expression for the earthquake interevent time distribution

SUMMARY The distribution of interevent times between two successive earthquakes is generally modelled as Gamma distribution. However, significant deviations from Gamma distribution have been observed at small interevent times. These deviations can be easily modelled through the introduction of a new parameter in the interevent time distribution. The log-likelihood estimations of the model parameters performed for the data from four earthquake catalogues reveal that the proposed model fits the data well. Finally, the new parameter is interpreted in terms of the c parameter of the Omori law.

Universal and Distinct Properties of Communication Dynamics

With the advancement of information systems, means of communications are becoming cheaper, faster, and more available. Today, millions of people carrying smartphones or tablets are able to communicate practically any time and anywhere they want. They can access their e-mails, comment on weblogs, watch and post videos and photos (as well as comment on them), and make phone calls or text messages almost ubiquitously. Given this scenario, in this article, we tackle a fundamental aspect of this new era of communication: How the time intervals between communication events behave for different technologies and means of communications. Are there universal patterns for the Inter-Event Time Distribution (IED)? How do inter-event times behave differently among particular technologies? To answer these questions, we analyzed eight different datasets from real and modern communication data and found four well-defined patterns seen in all the eight datasets. Moreover, we propose the use of the Self-Feeding Process (SFP) to generate inter-event times between communications. The SFP is an extremely parsimonious point process that requires at most two parameters and is able to generate inter-event times with all the universal properties we observed in the data. We also show three potential applications of the SFP: as a framework to generate a synthetic dataset containing realistic communication events of any one of the analyzed means of communications, as a technique to detect anomalies, and as a building block for more specific models that aim to encompass the particularities seen in each of the analyzed systems.

Intermediate‐depth icequakes and harmonic tremor in an Alpine glacier (Glacier d'Argentière, France): Evidence for hydraulic fracturing?

AbstractWe detected several thousand deep englacial icequakes on Glacier d'Argentière (Mont‐Blanc massif) between 30 March and 3 May 2012. These events have been classified in eight clusters. Inside each cluster, the waveforms are similar for P waves and S waves, although the time delay between the P waves and the S waves vary by up to 0.03 s, indicating an extended source area. Although these events were recorded by a single accelerometer, they were roughly located using a polarization analysis. The deepest events were located at a depth of 130 m, 60 m above the ice/bed interface. The clusters are separated in space. The largest cluster extends over about 100 m. For this cluster, the strike of the rupture plane is nearly parallel to the direction of the open crevasses, and the dip angle is 56°. Deep icequakes occur in bursts of activity that last for a few hours and are separated by quiet periods. Many events occurred on 28 and 29 April 2012, during the warmest days, when snowmelting was likely important. The distributions of interevent times and peak amplitudes obey power laws as also observed for earthquakes, but with larger exponents. The polarity of the P waves for all of the events is consistent with tensile faulting. Finally, between 25 April and 3 May, we observed a gliding harmonic tremor with a fundamental resonance frequency that varied between 30 Hz and 38 Hz, with additional higher‐frequency harmonics. During this time we also observed shallow hybrid events with high‐frequency onsets and a monochromatic coda. These events might be produced by the propagation of fractures and the subsequent flow of water into the fracture. The strongest resonance was observed just after a strong burst of deep icequakes and during an unusually warm period when the snow height decreased by 60 cm in 1 week. The resonance frequency shows a succession of several sharp decreases and phases of progressive increases. One of the strongest negative steps of the resonance frequency on 28 April coincides with a burst of deep icequakes. These events appear to be associated with the propagation of fractures, which can explain the decrease in the resonance frequency. Finally, we observed an acceleration of glacier flow on 29 April, suggesting that meltwater had reached the ice/bed interface. These observations suggest that deep icequakes are due to hydraulic fracturing and that they can be used to track fluid flow inside glaciers.

Steady state and mean recurrence time for random walks on stochastic temporal networks.

Random walks are basic diffusion processes on networks and have applications in, for example, searching, navigation, ranking, and community detection. Recent recognition of the importance of temporal aspects on networks spurred studies of random walks on temporal networks. Here we theoretically study two types of event-driven random walks on a stochastic temporal network model that produces arbitrary distributions of interevent times. In the so-called active random walk, the interevent time is reinitialized on all links upon each movement of the walker. In the so-called passive random walk, the interevent time is reinitialized only on the link that has been used the last time, and it is a type of correlated random walk. We find that the steady state is always the uniform density for the passive random walk. In contrast, for the active random walk, it increases or decreases with the node's degree depending on the distribution of interevent times. The mean recurrence time of a node is inversely proportional to the degree for both active and passive random walks. Furthermore, the mean recurrence time does or does not depend on the distribution of interevent times for the active and passive random walks, respectively.

Building Time‐Dependent Earthquake Recurrence Models for Probabilistic Risk Computations

Abstract We present a risk assessment perspective on earthquake recurrence on mature faults and the ways that it can be modeled. Specific needs unique to risk models compared with probabilistic seismic‐hazard assessment include (1) the determination of loss per event to assess the correlation of risk between sites for portfolio management, (2) loss at all return periods, and sometimes (3) realizations of the order in which events occur. We use three mature faults in New Zealand as a case study and review the impact of parameter and model uncertainty on risk metrics. We review the physical characteristics of several faulting environments, contrasting them against the properties of three probability density functions (PDFs) widely used to characterize such interevent time distributions. We review the data available to constrain both the priors and the recurrence process and conclude that to use them to quantify the recurrence of large events on mature faults, it is best to compute a Bayesian combination of models (i.e., the weighted average of individual PDFs using their Bayes factors). Finally, we propose the following to the community: (1) the initiation of a general discussion on how best to incorporate our knowledge (e.g., from earthquake geology) on plausible models and model parameters, while preserving the information on what we do not know and (2) the creation and maintenance of a global database of priors, data, and model evidence, classified by tectonic region, fluid characteristic, fault geometry, and other relevant properties so that we can monitor whether the time‐dependent recurrence behavior of certain types of faults or faulting environments can be better described by one (or a combination of) recurrence model(s).

Evidence of non extensivity in the evolution of seismicity along the San Andreas Fault, California, USA: An approach based on Tsallis statistical physics

We examine the nature of the seismogenetic system along the San Andreas Fault (SAF), California, USA, by searching for evidence of complexity and non-extensivity in the earthquake record. We use accurate, complete and homogeneous earthquake catalogues in which aftershocks are included (raw catalogues), or have been removed by a stochastic declustering procedure (declustered catalogues). On the basis of Non-Extensive Statistical Physics (NESP), which generalizes the Boltzmann–Gibbs formalism to non-equilibrating (complex) systems, we investigate whether earthquakes are generated by an extensive self-excited Poisson process or by a non-extensive complex process. We examine bivariate cumulative frequency distributions of earthquake magnitudes and interevent times and determine the size and time dependence of the respective magnitude and temporal entropic indices, which indicate the level on non-equilibrium (correlation). It is shown that the magnitude entropic index is very stable and corresponds to proxy b-values that are remarkably consistent with the b-values computed by conventional means. The temporal entropic index computed from the raw catalogues indicate moderately to highly correlated states during the aftershock sequences of large earthquakes, progressing to quasi-uncorrelated states as these die out and before the next large event. Conversely, the analysis of the declustered catalogues shows that background seismicity exhibits moderate to high correlation that varies significantly albeit smoothly with time. This indicates a persistent sub-extensive seismogenetic system. The degree of correlation is generally higher in the southern SAF segment, which is consistent with the observation of shorter return periods for large earthquakes. A plausible explanation is that because aftershock sequences are localized in space and time, their efficient removal unveils long-range background interactions which are obscured by their presence! Our results indicate complexity in the expression of background seismicity along the San Andreas Fault, with criticality being a very likely mechanism as a consequence of the persistent non-equilibrium inferred from the temporal entropic index. However, definite conclusions cannot be drawn until the earthquake record is exhaustively studied in all its forms.

A comparative analysis of intra-city human mobility by taxi

Quantitative understanding of human movement behaviors would provide helpful insights into the mechanisms of many socioeconomic phenomena. In this paper, we investigate human mobility patterns through analyzing taxi-trace datasets collected from five metropolitan cities in two countries. We focus on three statistics for each dataset: the displacement of each occupied trip, the duration of each occupied trip, and the time interval between successive occupied trips by the same taxi (interevent time). The results indicate that the displacement distributions of human travel by taxi tend to follow exponential laws in two displacement ranges rather than power laws; the trip duration distributions can be approximated by log-normal distributions; the interevent time distributions can be well characterized by log-normal bodies followed by power law tails. For each considered measure, the rescaled distributions of all cities collapsed into a master curve. These results provide empirical evidence supporting the common regularity of intra-city human mobility. Moreover, we show that airport locations could play a role in explaining the spikes of displacement distributions of taxi trips in certain cities.

Interevent time distributions of human multi-level activity in a virtual world

Studying human behavior in virtual environments provides extraordinary opportunities for a quantitative analysis of social phenomena with levels of accuracy that approach those of the natural sciences. In this paper we use records of player activities in the massive multiplayer online game Pardus over 1238 consecutive days, and analyze dynamical features of sequences of actions of players. We build on previous work where temporal structures of human actions of the same type were quantified, and provide an empirical understanding of human actions of different types. This study of multi-level human activity can be seen as a dynamic counterpart of static multiplex network analysis. We show that the interevent time distributions of actions in the Pardus universe follow highly non-trivial distribution functions, from which we extract action-type specific characteristic “decay constants”. We discuss characteristic features of interevent time distributions, including periodic patterns on different time scales, bursty dynamics, and various functional forms on different time scales. We comment on gender differences of players in emotional actions, and find that while males and females act similarly when performing some positive actions, females are slightly faster for negative actions. We also observe effects on the age of players: more experienced players are generally faster in making decisions about engaging in and terminating enmity and friendship, respectively.

Emergent spike patterns in neuronal populations.

This numerical study documents and analyzes emergent spiking behavior in local neuronal populations. Emphasis is given to a phenomenon we call clustering, by which we refer to a tendency of random groups of neurons large and small to spontaneously coordinate their spiking activity in some fashion. Using a sparsely connected network of integrate-and-fire neurons, we demonstrate that spike clustering occurs ubiquitously in both high firing and low firing regimes. As a practical tool for quantifying such spike patterns, we propose a simple scheme with two parameters, one setting the temporal scale and the other the amount of deviation from the mean to be regarded as significant. Viewing population activity as a sequence of events, meaning relatively brief durations of elevated spiking, separated by inter-event times, we observe that background activity tends to give rise to extremely broad distributions of event sizes and inter-event times, while driving a system imposes a certain regularity on its inter-event times, producing a rhythm consistent with broad-band gamma oscillations. We note also that event sizes and inter-event times decorrelate very quickly. Dynamical analyses supported by numerical evidence are offered.

Impact of the time pattern of human behaviors on information spreading

There is sufficient evidence to support that the inter-event time distribution of human behaviors is heavy tailed. This fact plays an important role in human dynamics at collective levels. Recently, researchers found that the heavy-tailed inter-activity patterns of human behaviors slow down the spreading significantly. In this paper, we investigate the influence of people's interest time length on information spreading. Considering that information has its own time period of being interested in, we find that for small interest time period, heavy-tailed human behaviors can enhance the information spreading. While for large interest time period, the heavy-tailed human behaviors will suppress the spreading. These results can help us understanding the rule of information spreading, controlling and limiting the outbreak of rumors.

Simulating non-Markovian stochastic processes.

We present a simple and general framework to simulate statistically correct realizations of a system of non-Markovian discrete stochastic processes. We give the exact analytical solution and a practical and efficient algorithm like the Gillespie algorithm for Markovian processes, with the difference being that now the occurrence rates of the events depend on the time elapsed since the event last took place. We use our non-Markovian generalized Gillespie stochastic simulation methodology to investigate the effects of nonexponential interevent time distributions in the susceptible-infected-susceptible model of epidemic spreading. Strikingly, our results unveil the drastic effects that very subtle differences in the modeling of non-Markovian processes have on the global behavior of complex systems, with important implications for their understanding and prediction. We also assess our generalized Gillespie algorithm on a system of biochemical reactions with time delays. As compared to other existing methods, we find that the generalized Gillespie algorithm is the most general because it can be implemented very easily in cases (such as for delays coupled to the evolution of the system) in which other algorithms do not work or need adapted versions that are less efficient in computational terms.

Dynamic features analysis for the large-scale logistics system warehouse-out operation

In the paper, we research on the behavior dynamics for the large-scale logistics system warehouse-out operation systematically. First, we discover that steel products warehouse-out of different warehouses in a large-scale logistics system can be characterized by burst, and the warehouse-out inter-event time follows the power-law distribution with exponents close to α=2.5, which differs from the two classical models proposed by Barabasi (2005) and Vazquez (2005) respectively. By analyzing the warehouse-out inter-event time distribution of the products in one certain large-scale logistics system, we further discuss burst features and mechanisms of logistics system. Additionally, we find that in population behaviors, burst features can be explained by the priority that rooted in holidays and interior task scheduling. However, warehouse-out behaviors of active individuals do not show any features of burst.Further, we find that warehouse-out quantity of steel products follows Fractal Brownian motion with the HURST exponent higher than 0.5 by means of R/S, which infers that the quantity of products in a logistics system is not only guided by prices in the present market, but also related closely to the previous quantity of warehouse-out. Based on V statistic, we compare memory length of different products in warehouses.Finally, we apply complex networks visibility graphs for further validation of fractal features in a logistics system and find that almost every visibility graph exhibits small-world and scale-free features. Both R/S and complex networks visibility graphs reinforce that the warehouse-out quantity of products in a logistics system is not a random walk process, but contains intrinsic regularities and long-term correlation between present and previous warehouse-out quantity.

Structural differences between open and direct communication in an online community

Most research of online communication focuses on modes of communication that are either open (like forums, bulletin boards, Twitter, etc.) or direct (like e-mails). In this work, we study a dataset that has both types of communication channels. We relate our findings to theories of social organization and human dynamics. The data comprises 36,492 users of a movie discussion community. Our results show that there are differences in the way users communicate in the two channels that are reflected in the shape of degree- and interevent time distributions. The open communication that is designed to facilitate conversations with any member shows a broader degree distribution and more of the triangles in the network are primarily formed in this mode of communication. The direct channel is presumably preferred by closer communication and the response time in dialogs is shorter. On a more coarse-grained level, there are common patterns in the two networks. The differences and overlaps between communication networks, thus, provide a unique window into how social and structural aspects of communication establish and evolve.

Spreading dynamics on networks: the role of burstiness, topology and non-stationarity

Spreading on networks is influenced by a number of factors, including different parts of the inter-event time distribution (IETD), the topology of the network and non-stationarity. In order to understand the role of these factors we study the SI model on temporal networks with different aggregated topologies and different IETDs. Based on analytic calculations and numerical simulations, we show that if the stationary bursty process is governed by power-law IETD, the spreading can be slowed down or accelerated as compared to a Poisson process; the speed is determined by the short time behaviour, which in our model is controlled by the exponent. We demonstrate that finite, so called ‘locally tree-like’ networks, like the Barabási–Albert networks behave very differently from real tree graphs if the IETD is strongly fat-tailed, as the lack or presence of rare alternative paths modifies the spreading. A further important result is that the non-stationarity of the dynamics has a significant effect on the spreading speed for strongly fat-tailed power-law IETDs, thus bursty processes characterized by small power-law exponents can cause slow spreading in the stationary state but also very rapid spreading, with this heavily dependent on the age of the processes.

Empirical Analysis on Human Dynamics of QQ Group Communication

The understanding of the temporal patterns of individual human interactions is essential in explaining many characteristics of human behavior. The instant communication tool QQ is developing rapidly on the network in recent years. Based on the QQ group communication records provided by some volunteers, this paper investigates the statistics of the inter-event intervals between two consecutive messages. The result shows that they follow power-law distribution. And there are obvious positive correlation between the activity of QQ group and the power-law exponent of inter-event time distribution. In further study, we find that the distribution displays a strong bursty property yet weak memory effects, then we analysis the distribution of the number of events in a bursty period.

Evidence of [formula omitted]-exponential statistics in Greek seismicity

We study the seismicity (global seismic activity) that occurred in Greece between 1976 and 2009 based on the dataset reported in Makropoulos et al. (2012), using concepts of non-extensive Statistical Physics. By considering the entire and declustered datasets, for which the aftershocks have been removed, we initially investigate the frequency–magnitude distribution and find that both datasets are well approximated by a physical model derived in the framework of non-extensive Statistical Physics. We then carry out a study of the distribution of interevent times of seismic events for different magnitude thresholds and discover that the data are well approximated by a statistical distribution of the q-exponential type that allows us to compute analytically the risk function of earthquake production. Our analysis thus reveals further evidence that the underlying dynamical process of earthquake birth reflects a kind of nonlinear memory due to long-term persistence of seismic events.

Universal bursty behaviour in human violent conflicts.

Understanding the mechanisms and processes underlying the dynamics of collective violence is of considerable current interest. Recent studies indicated the presence of robust patterns characterizing the size and timing of violent events in human conflicts. Since the size and timing of violent events arises as the result of a dynamical process, we explore the possibility of unifying these observations. By analyzing available catalogs on violent events in Iraq (2003–2005), Afghanistan (2008–2010) and Northern Ireland (1969–2001), we show that the inter-event time distributions (calculated for a range of minimum sizes) obeys approximately a simple scaling law which holds for more than three orders of magnitude. This robust pattern suggests a hierarchical organization in size and time providing a unified picture of the dynamics of violent conflicts.

Analytically Solvable Model of Spreading Dynamics with Non-Poissonian Processes

Non-Poissonian bursty processes are ubiquitous in natural and social phenomena, yet little is known about their effects on the large-scale spreading dynamics. In order to characterize these effects we devise an analytically solvable model of Susceptible-Infected (SI) spreading dynamics in infinite systems for arbitrary inter-event time distributions and for the whole time range. Our model is stationary from the beginning, and the role of lower bound of inter-event times is explicitly considered. The exact solution shows that for early and intermediate times the burstiness accelerates the spreading as compared to a Poisson-like process with the same mean and same lower bound of inter-event times. Such behavior is opposite for late time dynamics in finite systems, where the power-law distribution of inter-event times results in a slower and algebraic convergence to fully infected state in contrast to the exponential decay of the Poisson-like process. We also provide an intuitive argument for the exponent characterizing algebraic convergence.