Abstract: We analyzed the interactivity time, defined as the duration between two consecutive tasks such as sending emails, collecting friends and followers and writing comments in online social networks (OSNs).The distributions of these times are heavy tailed and often described by a power-law distribution. However, power-law distributions usually only fit the heavy tail of empirical data and ignore the information in the smaller value range. We argue that the durations between writing emails or comments, adding friends and receiving followers are likely to follow a lognormal distribution, discussing the similarities between power-law and lognormal distributions, and show that binning of data can deform a lognormal to a power-law distribution. An explanation for the appearance of lognormal interactivity times will be discussed and the influence of non-Markovian infection spread on the Susceptible-Infected-References:- C. Doerr, N. Blenn, and P. Van Mieghem. Lognormal Infection Timesof Online Information Spread. PLoS ONE, 8(5):e64349, 05 2013.- E. Cator, R. van de Bovenkamp, and P. Van Mieghem.Susceptible-infected-infection and cure times. Physical Review E, 87:062816, Jun 2013.- P. Van Mieghem and R. van de Bovenkamp. Non-Markovian InfectionSpread Dramatically Alters the Susceptible-Infected-Threshold in Networks. Physical Review Letters, 110:108701, Mar 2013.

Abstract: A number of human activities exhibit a bursty pattern, namely periods of very high activity that are followed by rest periods. Records of these processes generate time series of events whose inter- event times follow a probability distribution that displays a fat tail. The grounds for such phenomenon are not yet clearly understood. In this talk I will present some results concerning the use the freely available Wikipedia?s editing records to unravel some features of this phenomenon. We show that even though the probability to start editing is conditioned by the circadian 24 hour cycle, the conditional probability for the time interval between successive edits at a given time of the day is independent from the latter. We confirm our findings with the activity of posting on the social network Twitter. Our result suggests there is an intrinsic humankind scheduling pattern: after overcoming the encumbrance to start an activity, there is a robust distribution of new related actions, which does not depend on the time of day. Some simulations were performed in order to understand the effect of circadian patterns in the activity, in particular in the stationarity of the power-law inter-event distributions.

Abstract: Empirical studies revealed a number of features of bursty time series [1,2]. We implemented a simple task-queuing model, which produces bursty time series due to the non-trivial dynamics of the task list. We give exact results on the asymptotic behaviour of the model and we show that the inter-event time distribution (IETD) is power-law decaying for any kind of input distributions that remain normalizable in the infinite list limit, with exponents tuneable between 1 and 2. The model satisfies a scaling law between the exponents of inter-event time distribution (?) and autocorrelation function (?):???+???=?2. This law is general for renewal processes with power-law decaying inter-event time distribution and a departure from it indicates long-range dependence between the inter-event times [3]. We investigated another model based on self-exciting point processes with variable memory range. We found that in an intermediate range of memory effect the generated correlated bursts are comparable to empirical findings [4]. Empirical studies show that burstiness has major impact on the spreading processes on networks [5]. We solve the SI model on the infinite complete graph and show that fat tailed IETD causes always acceleration [6]. In order to understand the role of these influencing factors we studied 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 being 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 Barab?si?Albert networks behave very differently from real tree graphs if the IETD is strongly fat-tailed. Furthermore, non-stationarity of the dynamics has a significant effect on the spreading speed for strongly fat-tailed power-law IETDs [7].References:[1] M?rton Karsai, Kimmo Kaski, Albert-L?szl? Barab?si, J?nos Kert?sz: Universal features of correlated bursty behaviour, Scientific Reports, 2 397 (2012)[2] M?rton Karsai, Kimmo Kaski, J?nos Kert?sz: Correlated dynamics in egocentric communication networks, PloS ONE, 7, e40612 (2012)[3] Szabolcs Vajna, B?lint T?th, J?nos Kert?sz: Modelling bursty time series, New Journal of Physics, 15, 103023 (2013)[4] Hang-Hyun Jo, Juan I Perotti, Kimmo Kaski, J?nos Kert?sz: Correlated bursts and the role of memory range, Physical Review E, 92, 022814 (2015)[5] M?rton Karsai, Mikko Kivel?, Raj Kumar Pan, Kimmo Kaski, J?nos Kert?sz, A-L Barab?si, Jari Saram?ki: Small but slow world: How network topology and burstiness slow down spreading, Phys. Rev. E, 83, 025102 (2011)[6] Hang-Hyun Jo, Juan I Perotti, Kimmo Kaski, J?nos Kert?sz: Analytically solvable model of spreading dynamics with non-Poissonian processes, Phys. Rev. X, 011041 (2014)[7] D?vid X. Horv?th, J?nos Kert?sz: Spreading dynamics on networks: the role of burstiness, topology and non-stationarity, New Journal of Physics, 16, 073037 (2014)

Abstract: A diverse variety of processes?including recurrent disease episodes, neuron firing, and communication patterns among humans?can be described using inter-event time (IET) distributions. Many such processes are ongoing, although event sequences are only available during a finite observation window. Because the observation time window is more likely to begin or end during long IETs than during short ones, the analysis of such data is susceptible to a bias induced by the finite observation period. In this talk, I illustrate how this length bias is born, how it can be corrected, and formulate simple heuristic for determining the severity of the bias. This can be donewithout assuming any particular shape for the IET distribution, but one needs to assume that the event sequences are produced by (stationary) renewal processes. I illustrate the method for several well-known empirical communication networks from the literature. It turns out that in these data sets the resulting bias can lead to systematic underestimates of the variance in the IET distributions and that correcting for the bias can lead to qualitatively different results for the tails of the IET distributions.

Abstract: Given a stationary point process, an intensity burst is defined as a short time period during which the number of counts is larger than the typical count rate. It might signal a local non-stationarity or the presence of an external perturbation to the system. In this paper we propose a novel procedure for the detection of intensity bursts within the Hawkes process framework. By using a model selection scheme we show that our procedure can be used to detect intensity bursts when both their occurrence time and their total number is unknown. Moreover the initial time of the burst can be determined with a precision given by the typical inter-event time. We apply our methodology to the mid-price change in FX markets showing that these bursts are frequent and that only a relatively small fraction is associated to news arrival. We show lead-lag relations in intensity burst occurrence across different FX rates and we discuss their relation with price jumps.

Abstract: When modelling dynamical systems on networks, it is often assumed that the process is Markovian, that is future states depend only upon the present state and not on the sequence of events that preceded it. Examples include diffusion of ideas or diseases on social networks, or synchronisation of interacting dynamical units. In each case, the dynamics is governed by coupled differential equation, where the coupling is defined by the adjacency matrix of the underlying network. The main purpose of this talk is to challenge this Markovian picture. We will argue that non-Markovian models can provide a more realistic picture in the case of temporal networks where edges change in time, or in situations when pathways can be measured empirically. We will focus on the importance of non-Poisson temporal statistics, and show analytically the impact of burstiness on diffusive dynamics, before turning to applications and incorporating memory kernels in predictive models of retweet dynamics.