Identifying exogenous and endogenous activity in social media

Kazuki Fujita, Alexey Medvedev, Shinsuke Koyama, Renaud Lambiotte, and Shigeru Shinomoto

Phys. Rev. E 98, 052304 – Published 13 November 2018

Abstract

The occurrence of new events in a system is typically driven by external causes and by previous events taking place inside the system. This is a general statement, applying to a range of situations including, more recently, to the activity of users in online social networks (OSNs). Here we develop a method for extracting from a series of posting times the relative contributions that are exogenous, e.g., news media, and endogenous, e.g., information cascade. The method is based on the fitting of a generalized linear model (GLM) equipped with a self-excitation mechanism. We test the method with synthetic data generated by a nonlinear Hawkes process, and apply it to a real time series of tweets with a given hashtag. In the empirical dataset, the estimated contributions of exogenous and endogenous volumes are close to the amounts of original tweets and retweets respectively. We conclude by discussing the possible applications of the method, for instance in online marketing.

Received 2 August 2018

DOI:https://doi.org/10.1103/PhysRevE.98.052304

Physics Subject Headings (PhySH)

Stochastic inference

Statistical Physics & Thermodynamics

Authors & Affiliations

Kazuki Fujita^1,*, Alexey Medvedev^2,3,†, Shinsuke Koyama^4,‡, Renaud Lambiotte^5,§, and Shigeru Shinomoto^1,∥

¹Department of Physics, Kyoto University, Kyoto 606-8502, Japan
²NaXys, Universite de Namur, 5000 Namur, Belgium
³ICTEAM, Universite Catholique de Louvain, 1348 Louvain-la-Neuve, Belgium
⁴The Institute of Statistical Mathematics, Tokyo 190-8562, Japan
⁵Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom

^*fujita.kazuki.37n@st.kyoto-u.ac.jp
^†alexey.medvedev@unamur.be
^‡koyama0526@gmail.com
^§renaud.lambiotte@maths.ox.ac.uk
^∥shinomoto@scphys.kyoto-u.ac.jp

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Vol. 98, Iss. 5 — November 2018

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Images

Figure 1
Comparison of intensity self-reinforcement between (a) the linear Hawkes process and (b) the nonlinear Hawkes process. Both processes have background rates equal to 1 and exponential memory kernels. We consider the hypothetical situation where events get realized at the same times in each case and compare the resulting intensities. Linear reinforcement generates a constant number of secondary events, while multiplicative effect is stronger if subsequent events arrived closer in time, e.g., around $t = 4$ .
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Figure 2
Fitting the GLM to synthetic data. (a) Exogenously modulated rate process [Eq. (10), $γ_{0} = ln 0.01$ ; $b_{0} = 0.2$ ; $T = 43200$ ]. (b) The nonlinear Hawkes process with small self-excitation [Eq. (11), $γ_{0} = ln 0.01$ ; $α_{0} = 10$ ; $τ_{0} = 300$ ). The inset magnifies the rate profile. (c) The nonlinear Hawkes process with the larger self-excitation [Eq. (11), $γ_{0} = ln 0.01$ ; $α_{0} = 25$ ; $τ_{0} = 300$ ]. (d) The system receiving external fluctuations and self-excitation [Eq. (12), $γ_{0} = ln 0.01$ ; $b_{0} = 0.2$ ; $T = 43200$ ; $α_{0} = 20$ ; $τ_{0} = 300$ ]. Left panel: Contour plot of the log-likelihood [Eq. (7)]. Right panel: The solid blue line shows the optimal time histogram, the red line below shows the original exogenous activity $exp [γ (t)]$ , the black curve represents the inferred exogenous activity $exp [\hat{γ} (t)]$ , and green is the total rate given by the GLM, $λ_{GLM} (t)$ [Eq. (9)].
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Figure 3
Contour plots of the log-likelihood functions of various GLMs. Panels (a), (b), and (c) correspond to exponential kernels $h (t) = 1 / τ exp (- t / τ)$ of timescales $τ = 30$ , 300, and 600 seconds, respectively. Panel (d) corresponds to a power law kernel $h (t) = c {(t + a)}^{- 3}$ ( $c = 2 a^{2}, a = 300$ s). The original data were derived from the nonlinear Hawkes process of the system receiving external fluctuations and self-excitation of the exponential kernel of timescale $τ_{0} = 300$ seconds as in Fig. 2.
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Figure 4
Tweeting rate in the whole dataset from January to September 2017. The solid blue line shows the tweeting rate of the tweets that contain a hashtag related to “bitcoin.” The dashed red line shows the rate of appearance of original tweets with the same hashtags. Both rates were approximated from daily bins.
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Figure 5
Original tweeting rate estimation for one-week samples of tweets: (a) between January 9 and January 16, 2017, (b) between May 1 and May 8, 2017, and (c) between July 13 and July 20, 2017. The solid blue line shows the tweeting rate of all tweets, the red line below shows the rate of original tweets, the black curve represents the inferred exogenous activity $exp [\hat{γ} (t)],$ and green is the total rate given by the GLM. The total and original tweeting rates were approximated using 20 min bins and the estimated rates are given using 1 min bins.
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Figure 6
Contour plots of the log-likelihood function for timescales $τ = 5$ , 60, and 150 seconds. The optimal value found by GLM method is depicted as $(\hat{α}, \hat{β})$ .
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