Universality and correlations in individuals wandering through an online extremist space

Z. Cao, M. Zheng, Y. Vorobyeva, C. Song, and N. F. Johnson

Phys. Rev. E 97, 032315 – Published 26 March 2018

Abstract

The “out of the blue” nature of recent terror attacks and the diversity of apparent motives highlight the importance of understanding the online trajectories that individuals follow prior to developing high levels of extremist support. Here we show that the physics of stochastic walks, with and without temporal correlation, provides a unifying description of these online trajectories. Our unique data set comprising all users of a global social media site reveals universal characteristics in individuals' online lifetimes. Our accompanying theory generates analytical and numerical solutions that describe the characteristics shown by individuals that go on to develop high levels of extremist support and those that do not. Going forward, it is conceivable that a deeper understanding of these temporal and many-body correlations may eventually contribute to the important task of better countering the spread of radical propaganda online.

Received 20 June 2017
Revised 8 August 2017

DOI:https://doi.org/10.1103/PhysRevE.97.032315

Physics Subject Headings (PhySH)

Econophysics Random walks

Social networks

Fokker–Planck equation

Interdisciplinary Physics

Authors & Affiliations

Z. Cao¹, M. Zheng¹, Y. Vorobyeva², C. Song¹, and N. F. Johnson¹

¹Physics Department, University of Miami, Coral Gables, Florida 33146, USA
²Department of International Studies, University of Miami, Coral Gables, Florida 33146, USA

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Vol. 97, Iss. 3 — March 2018

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Images

Figure 1
Schematic of possible individual trajectories in $d - ℓ - b$ space bounded by absorbing barriers at $d_{abs}, ℓ_{abs}, b_{abs}$ such that $d (t) < d_{abs}$ , $ℓ (t) < ℓ_{abs}$ , and $b (t) < b_{abs}$ . These illustrate the three possible outcomes of interest.
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Figure 2
Panels (a), (c), and (d) show the distribution of clock-time lifetimes for individuals with the three types of future outcomes shown in Fig. 1, together with our theoretical predictions. The legend in (a) applies to all. Panel (b) enlarges (a) for short lifetimes. One-standard-deviation error bands are shown. The results with the TC (i.e. temporal correlation) effect for (c) and (d) are similar to the respective no-TC case and hence are not shown.
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Figure 3
(a) Exact diagrammatic expansion [26] of the probability that a randomly chosen individual ends up as banned (denoted as “b”), self-deleting (“d”), or latent (“l”) after joining $L = 0, 1, 2, \dots$ online groups of any type. $L = 0, 1, 2$ terms are shown explicitly. (b) Terms from (a) evaluated empirically by counting the fraction of individuals who end up as “b,” “d,” or “l” after joining exactly $L$ online groups (i.e., event-time lifetime is $L$ ). [(c)–(e)] Expansion terms from (a) for a specific event-time lifetime (different subscripts for $L$ are used to distinguish the user types), where here $n$ only counts the number of future-banned groups (B) joined (see text) and hence the maximum $n$ appears bounded from above by the event-time lifetime. Points are empirical results. Solid lines are our temporal correlation (TC, i.e., memory) model results determined using maximum likelihood estimation (MLE). Dashed lines are null model (i.e., binomial distribution). (f) MLE $q$ values vs. event-time lifetime in our model. MLE $p$ value $\approx 0.73$ in all cases.
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Figure 4
Evolution of the many-body correlation measure $〈 IQR 〉$ (i.e., average IQR) between different subpopulations of individuals. As in Fig. 3, “b” means future banned, “l” means future latent, “d” means future self-deleting, and “any” means irrespective of individual type. For example, “b—d” means between future banned and future latent individuals. One standard deviation error bands are shown around the null model result. In principle, each empirical curve has its own null model, obtained by randomizing the time stamps $t$ of the time series $s_{i} (t)$ [and $s_{j} (t)$ ]. Since these null model results are all very similar, we only show the result for the “any—any” case.
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