Heavy-tailed Inter-event Time Distributions Research Articles

Non-Markovian epidemic spreading on temporal networks

Many empirical studies have revealed that the occurrences of contacts associated with human activities are non-Markovian temporal processes with a heavy tailed inter-event time distribution. Besides, there has been increasing empirical evidence that the infection and recovery rates are time-dependent. However, we lack a comprehensive framework to analyze and understand non-Markovian contact and spreading processes on temporal networks. In this paper, we propose a general formalism to study non-Markovian dynamics on non-Markovian temporal networks. We find that, under certain conditions, non-Markovian dynamics on temporal networks are equivalent to Markovian dynamics on static networks. Interestingly, this result is independent of the underlying network topology.

Modeling temporal networks with bursty activity patterns of nodes and links

The concept of temporal networks provides a framework to understand how the interaction between system components changes over time. In empirical communication data, we often detect non-Poissonian, so-called bursty behavior in the activity of nodes as well as in the interaction between nodes. However, such reconciliation between node burstiness and link burstiness cannot be explained if the interaction processes on different links are independent of each other. This is because the activity of a node is the superposition of the interaction processes on the links incident to the node and the superposition of independent bursty point processes is not bursty in general. Here we introduce a temporal network model based on bursty node activation and show that it leads to heavy-tailed inter-event time distributions for both node dynamics and link dynamics. Our analysis indicates that activation processes intrinsic to nodes give rise to dynamical correlations across links. Our framework offers a way to model competition and correlation between links, which is key to understanding dynamical processes in various systems.

A Multiscale Survival Process for Modeling Human Activity Patterns.

Human activity plays a central role in understanding large-scale social dynamics. It is well documented that individual activity pattern follows bursty dynamics characterized by heavy-tailed interevent time distributions. Here we study a large-scale online chatting dataset consisting of 5,549,570 users, finding that individual activity pattern varies with timescales whereas existing models only approximate empirical observations within a limited timescale. We propose a novel approach that models the intensity rate of an individual triggering an activity. We demonstrate that the model precisely captures corresponding human dynamics across multiple timescales over five orders of magnitudes. Our model also allows extracting the population heterogeneity of activity patterns, characterized by a set of individual-specific ingredients. Integrating our approach with social interactions leads to a wide range of implications.

Emergence of long-range correlations and bursty activity patterns in online communication.

Research has suggested that the activity occurring in a variety of social, economic, and technological systems exhibits long-range fluctuations in time. Pronounced levels of rapidly occurring events are typically observed over short periods of time, followed by long periods of inactivity. Relatively few studies, however, have shed light on the degree to which inhomogeneous temporal processes can be detected at, and emerge from, different levels of analysis. Here we investigate patterns of human activity within an online forum in which communication can be assessed at three intertwined levels: the micro level of the individual users; the meso level of discussion groups and continuous sessions; and the macro level of the whole system. To uncover the relation between different levels, we conduct a number of numerical simulations of a zero-crossing model in which users' behavior is constrained by progressively richer and more realistic rules of social interaction. Results indicate that, when users are solipsistic, their bursty behavior is not sufficient for generating heavy-tailed interevent time distributions at a higher level. However, when users are socially interdependent, the power spectra and interevent time distributions of the simulated and real forums are remarkably similar at all levels of analysis. Social interaction is responsible for the aggregation of multiple bursty activities at the micro level into an emergent bursty activity pattern at a higher level. We discuss the implications of the findings for an emergentist account of burstiness in complex systems.

Universal bursty behavior in the air transportation system.

Social activities display bursty behavior characterized by heavy-tailed interevent time distributions. We examine the bursty behavior of airplanes' arrivals in hub airports. The analysis indicates that the air transportation system universally follows a power-law interarrival time distribution with an exponent α=2.5 and an exponential cutoff. Moreover, we investigate the mechanism of this bursty behavior by introducing a simple model to describe it. In addition, we compare the extent of the hub-and-spoke structure and the burstiness of various airline networks in the system. Remarkably, the results suggest that the hub-and-spoke network of the system and the carriers' strategy to facilitate transit are the origins of this universality.

Correlated bursts and the role of memory range.

Inhomogeneous temporal processes in natural and social phenomena have been described by bursts that are rapidly occurring events within short time periods alternating with long periods of low activity. In addition to the analysis of heavy-tailed interevent time distributions, higher-order correlations between interevent times, called correlated bursts, have been studied only recently. As the underlying mechanism behind such correlated bursts is far from being fully understood, we devise a simple model for correlated bursts using a self-exciting point process with a variable range of memory. Whether a new event occurs is stochastically determined by a memory function that is the sum of decaying memories of past events. In order to incorporate the noise and/or limited memory capacity of systems, we apply two memory loss mechanisms: a fixed number or a variable number of memories. By analysis and numerical simulations, we find that too much memory effect may lead to a Poissonian process, implying that there exists an intermediate range of memory effect to generate correlated bursts comparable to empirical findings. Our conclusions provide a deeper understanding of how long-range memory affects correlated bursts.

Understanding human dynamics in microblog posting activities

Human activity patterns are an important issue in behavior dynamics research. Empirical evidence indicates that human activity patterns can be characterized by a heavy-tailed inter-event time distribution. However, most researchers give an understanding by only modeling the power-law feature of the inter-event time distribution, and those overlooked non-power-law features are likely to be nontrivial. In this work, we propose a behavior dynamics model, called the finite memory model, in which humans adaptively change their activity rates based on a finite memory of recent activities, which is driven by inherent individual interest. Theoretical analysis shows a finite memory model can properly explain various heavy-tailed inter-event time distributions, including a regular power law and some non-power-law deviations. To validate the model, we carry out an empirical study based on microblogging activity from thousands of microbloggers in the Celebrity Hall of the Sina microblog. The results show further that the model is reasonably effective. We conclude that finite memory is an effective dynamics element to describe the heavy-tailed human activity pattern.

Emergence of Bursts and Communities in Evolving Weighted Networks

Understanding the patterns of human dynamics and social interaction and the way they lead to the formation of an organized and functional society are important issues especially for techno-social development. Addressing these issues of social networks has recently become possible through large scale data analysis of mobile phone call records, which has revealed the existence of modular or community structure with many links between nodes of the same community and relatively few links between nodes of different communities. The weights of links, e.g., the number of calls between two users, and the network topology are found correlated such that intra-community links are stronger compared to the weak inter-community links. This feature is known as Granovetter's “The strength of weak ties” hypothesis. In addition to this inhomogeneous community structure, the temporal patterns of human dynamics turn out to be inhomogeneous or bursty, characterized by the heavy tailed distribution of time interval between two consecutive events, i.e., inter-event time. In this paper, we study how the community structure and the bursty dynamics emerge together in a simple evolving weighted network model. The principal mechanisms behind these patterns are social interaction by cyclic closure, i.e., links to friends of friends and the focal closure, links to individuals sharing similar attributes or interests, and human dynamics by task handling process. These three mechanisms have been implemented as a network model with local attachment, global attachment, and priority-based queuing processes. By comprehensive numerical simulations we show that the interplay of these mechanisms leads to the emergence of heavy tailed inter-event time distribution and the evolution of Granovetter-type community structure. Moreover, the numerical results are found to be in qualitative agreement with empirical analysis results from mobile phone call dataset.

Impact of heterogeneous human activities on epidemic spreading

Recent empirical observations suggest a heterogeneous nature of human activities. The heavy-tailed inter-event time distribution at the population level is well accepted, while whether the individual acts in a heterogeneous way is still under debate. Motivated by the impact of temporal heterogeneity of human activities on epidemic spreading, this paper studies the susceptible-infected model on a fully mixed population, where each individual acts in a completely homogeneous way but different individuals have different mean activities. Extensive simulations show that the heterogeneity of activities at the population level remarkably affects the speed of spreading, even though each individual behaves regularly. Furthermore, the spreading speed of this model is more sensitive to the change of system heterogeneity compared with the model consisted of individuals acting with heavy-tailed inter-event time distributions. This work refines our understanding of the impact of heterogeneous human activities on epidemic spreading.

Heavy-Tailed Statistics in Short-Message Communication

Short-message (SM) is one of the most frequently used communication channels in modern society. Based on the SM communication records provided by some volunteers, we investigate the statistics of SM communication pattern, including the inter-event time distributions between two consecutive short messages and two conversations, and the distribution of message number contained in a complete conversation. In the individual level, the empirical result raises strong evidence that the human activity pattern, exhibiting a heavy-tailed inter-event time distribution, is driven by a non-Poisson process.

Empirical analysis on temporal statistics of human correspondence patterns

Recently, extensive empirical evidence shows that the timing of human behaviors obeys non-Possion statistics with heavy-tailed interevent time distribution. In this paper, we empirically study the correspondence pattern of a great Chinese scientist, named Hsue-Shen Tsien. Both the interevent time distribution and response time distributions deviate from the Poisson statistics, showing an approximate power-law decaying. The two power-law exponents are more or less the same (about 2.1), which strongly support the hypothesis in [A. Vázquez, J.G. Oliveira, Z. Dezsö, K.-I. Goh, I. Kondor, A.-L. Barabási, Phys. Rev. E 73 (2006) 036127] that the response time distribution of the tasks could in fact drive the interevent time distribution, and both the two distributions should decay with the same exponent. Our result is against the claim in [A. Vázquez, J.G. Oliveira, Z. Dezsö, K.-I. Goh, I. Kondor, A.-L. Barabási, Phys. Rev. E 73 (2006) 036127], which suggests the human correspondence pattern belongs to a universality class with exponent 1.5.