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Zha Y, Zhou T, Zhou C. Unfolding large-scale online collaborative human dynamics. Proc Natl Acad Sci U S A 2016; 113:14627-14632. [PMID: 27911766 PMCID: PMC5187734 DOI: 10.1073/pnas.1601670113] [Citation(s) in RCA: 30] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
Large-scale interacting human activities underlie all social and economic phenomena, but quantitative understanding of regular patterns and mechanism is very challenging and still rare. Self-organized online collaborative activities with a precise record of event timing provide unprecedented opportunity. Our empirical analysis of the history of millions of updates in Wikipedia shows a universal double-power-law distribution of time intervals between consecutive updates of an article. We then propose a generic model to unfold collaborative human activities into three modules: (i) individual behavior characterized by Poissonian initiation of an action, (ii) human interaction captured by a cascading response to previous actions with a power-law waiting time, and (iii) population growth due to the increasing number of interacting individuals. This unfolding allows us to obtain an analytical formula that is fully supported by the universal patterns in empirical data. Our modeling approaches reveal "simplicity" beyond complex interacting human activities.
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Affiliation(s)
- Yilong Zha
- CompleX Lab, Web Sciences Center, University of Electronic Science and Technology of China, Chengdu 611731, People's Republic of China
- Department of Physics, Hong Kong Baptist University, Kowloon Tong, Hong Kong
- Centre for Nonlinear Studies, Institute of Computational and Theoretical Studies, Hong Kong Baptist University, Kowloon Tong, Hong Kong
- The Beijing-Hong Kong-Singapore Joint Centre for Nonlinear and Complex Systems (Hong Kong), Institute of Computational and Theoretical Studies, Hong Kong Baptist University, Kowloon Tong, Hong Kong
| | - Tao Zhou
- CompleX Lab, Web Sciences Center, University of Electronic Science and Technology of China, Chengdu 611731, People's Republic of China;
- Beijing Computational Science Research Center, Beijing 100084, People's Republic of China
- Big Data Research Center, University of Electronic Science and Technology of China, Chengdu 611731, People's Republic of China
| | - Changsong Zhou
- Department of Physics, Hong Kong Baptist University, Kowloon Tong, Hong Kong;
- Centre for Nonlinear Studies, Institute of Computational and Theoretical Studies, Hong Kong Baptist University, Kowloon Tong, Hong Kong
- The Beijing-Hong Kong-Singapore Joint Centre for Nonlinear and Complex Systems (Hong Kong), Institute of Computational and Theoretical Studies, Hong Kong Baptist University, Kowloon Tong, Hong Kong
- Beijing Computational Science Research Center, Beijing 100084, People's Republic of China
- Research Centre, Hong Kong Baptist University Institute of Research and Continuing Education, Shenzhen 518000, China
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Jamali T, Jafari GR, Vasheghani Farahani S. Patterns for the waiting time in the context of discrete-time stochastic processes. Phys Rev E 2016; 94:032110. [PMID: 27739745 DOI: 10.1103/physreve.94.032110] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/16/2015] [Indexed: 11/07/2022]
Abstract
The aim of this study is to extend the scope and applicability of the level-crossing method to discrete-time stochastic processes and generalize it to enable us to study multiple discrete-time stochastic processes. In previous versions of the level-crossing method, problems with it correspond to the fact that this method had been developed for analyzing a continuous-time process or at most a multiple continuous-time process in an individual manner. However, since all empirical processes are discrete in time, the already-established level-crossing method may not prove adequate for studying empirical processes. Beyond this, due to the fact that most empirical processes are coupled; their individual study could lead to vague results. To achieve the objectives of this study, we first find an analytical expression for the average frequency of crossing a level in a discrete-time process, giving the measure of the time experienced for two consecutive crossings named as the "waiting time." We then introduce the generalized level-crossing method by which the consideration of coupling between the components of a multiple process becomes possible. Finally, we provide an analytic solution when the components of a multiple stochastic process are independent Gaussian white noises. The comparison of the results obtained for coupled and uncoupled processes measures the strength and efficiency of the coupling, justifying our model and analysis. The advantage of the proposed method is its sensitivity to the slightest coupling and shortest correlation length.
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Affiliation(s)
- Tayeb Jamali
- Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran 19839, Iran
| | - G R Jafari
- Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran 19839, Iran and Center for Network Science, Central European University, H-1051, Budapest, Hungary
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Colman ER, Vukadinović Greetham D. Memory and burstiness in dynamic networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:012817. [PMID: 26274235 DOI: 10.1103/physreve.92.012817] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/22/2015] [Indexed: 06/04/2023]
Abstract
A discrete-time random process is described, which can generate bursty sequences of events. A Bernoulli process, where the probability of an event occurring at time t is given by a fixed probability x, is modified to include a memory effect where the event probability is increased proportionally to the number of events that occurred within a given amount of time preceding t. For small values of x the interevent time distribution follows a power law with exponent -2-x. We consider a dynamic network where each node forms, and breaks connections according to this process. The value of x for each node depends on the fitness distribution, ρ(x), from which it is drawn; we find exact solutions for the expectation of the degree distribution for a variety of possible fitness distributions, and for both cases where the memory effect either is, or is not present. This work can potentially lead to methods to uncover hidden fitness distributions from fast changing, temporal network data, such as online social communications and fMRI scans.
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Affiliation(s)
- Ewan R Colman
- Centre for the Mathematics of Human Behaviour, Department of Mathematics and Statistics, University of Reading, RG6 6AX, United Kingdom
| | - Danica Vukadinović Greetham
- Centre for the Mathematics of Human Behaviour, Department of Mathematics and Statistics, University of Reading, RG6 6AX, United Kingdom
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Aguinis H, O'Boyle E, Gonzalez-Mulé E, Joo H. Cumulative Advantage: Conductors and Insulators of Heavy-Tailed Productivity Distributions and Productivity Stars. PERSONNEL PSYCHOLOGY 2015. [DOI: 10.1111/peps.12095] [Citation(s) in RCA: 40] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
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Escobar JV, Sornette D. Dynamical signatures of collective quality grading in a social activity: attendance to motion pictures. PLoS One 2015; 10:e0116811. [PMID: 25612292 PMCID: PMC4303319 DOI: 10.1371/journal.pone.0116811] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/16/2014] [Accepted: 12/15/2014] [Indexed: 11/18/2022] Open
Abstract
We investigate the laws governing people’s decisions and interactions by studying the collective dynamics of a well-documented social activity for which there exist ample records of the perceived quality: the attendance to movie theaters in the US. We picture the flows of attendance as impulses or “shocks” driven by external factors that in turn can create new cascades of attendances through direct recommendations whose effectiveness depends on the perceived quality of the movies. This corresponds to an epidemic branching model comprised of a decaying exponential function determining the time between cause and action, and a cascade of actions triggered by previous ones. We find that the vast majority of the ~3,500 movies studied fit our model remarkably well. From our results, we are able to translate a subjective concept such as movie quality into a probability of the deriving individual activity, and from it we build concrete quantitative predictions. Our analysis opens up the possibility of understanding other collective dynamics for which the perceived quality or appeal of an action is also known.
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Affiliation(s)
- Juan V. Escobar
- Instituto de Física, Universidad Nacional Autónoma de México, P.O. Box 20-364, México City, 04510, México
- Physics Department, Universidad Autónoma Metropolitana-Iztapalapa, México City, 09340, México
- * E-mail:
| | - Didier Sornette
- ETH Zurich, Department of Management, Technology, and Economics, Switzerland and Swiss Finance Institute, Zürich, Switzerland
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Formentin M, Lovison A, Maritan A, Zanzotto G. Hidden scaling patterns and universality in written communication. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:012817. [PMID: 25122352 DOI: 10.1103/physreve.90.012817] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/20/2014] [Indexed: 06/03/2023]
Abstract
The temporal statistics exhibited by written correspondence appear to be media dependent, with features which have so far proven difficult to characterize. We explain the origin of these difficulties by disentangling the role of spontaneous activity from decision-based prioritizing processes in human dynamics, clocking all waiting times through each agent's "proper time" measured by activity. This unveils the same fundamental patterns in written communication across all media (letters, email, sms), with response times displaying truncated power-law behavior and average exponents near -3/2. When standard time is used, the response time probabilities are theoretically predicted to exhibit a bimodal character, which is empirically borne out by our newly collected years-long data on email. These perspectives on the temporal dynamics of human correspondence should aid in the analysis of interaction phenomena in general, including resource management, optimal pricing and routing, information sharing, and emergency handling.
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Affiliation(s)
- M Formentin
- Dipartimento di Fisica "Galileo Galilei", Università degli studi di Padova, Via Marzolo 8, 35131 Padova, Italy
| | - A Lovison
- Dipartimento di Matematica, Università di Padova, via Trieste 63, I-35121 Padova, Italy
| | - A Maritan
- Dipartimento di Fisica "Galileo Galilei", Università degli studi di Padova, Via Marzolo 8, 35131 Padova, Italy
| | - G Zanzotto
- Dipartimento di Psicologia Generale, Università di Padova, via Venezia 12, I-35131 Padova, Italy
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Saichev A, Sornette D. Fertility heterogeneity as a mechanism for power law distributions of recurrence times. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:022815. [PMID: 23496576 DOI: 10.1103/physreve.87.022815] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/26/2012] [Revised: 02/04/2013] [Indexed: 06/01/2023]
Abstract
We study the statistical properties of recurrence times in the self-excited Hawkes conditional Poisson process, the simplest extension of the Poisson process that takes into account how the past events influence the occurrence of future events. Specifically, we analyze the impact of the power law distribution of fertilities with exponent α, where the fertility of an event is the number of triggered events of first generation, on the probability distribution function (PDF) f(τ) of the recurrence times τ between successive events. The other input of the model is an exponential law quantifying the PDF of waiting times between an event and its first generation triggered events, whose characteristic time scale is taken as our time unit. At short-time scales, we discover two intermediate power law asymptotics, f(τ)~τ(-(2-α)) for τ<<τ(c) and f(τ)~τ(-α) for τ(c)<<τ<<1, where τ(c) is associated with the self-excited cascades of triggered events. For 1<<τ<<1/ν, we find a constant plateau f(τ)=/~const, while at long times, 1/ν</~τ, f(τ)=/~e(-ντ) has an exponential tail controlled by the arrival rate ν of exogenous events. These results demonstrate a novel mechanism for the generation of power laws in the distribution of recurrence times, which results from a power law distribution of fertilities in the presence of self-excitation and cascades of triggering.
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Affiliation(s)
- A Saichev
- Department of Management, Technology and Economics, ETH Zurich, Scheuchzerstrasse 7, CH-8092 Zurich, Switzerland.
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