Research ArticleINFORMATION SCIENCE

The dynamics of information-driven coordination phenomena: A transfer entropy analysis

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Science Advances  01 Apr 2016:
Vol. 2, no. 4, e1501158
DOI: 10.1126/sciadv.1501158
  • Fig. 1 Spatiotemporal activity as observed from the microblogging platform Twitter.

    Spain’s 15M protest growth in time shows that the protest did not transcend the online sphere until May 15 when the political movement emerged on the streets. Broadcasting traditional media started reporting about it soon after; by that time, demonstrations had been held in the most important cities of the country.

  • Fig. 2 Characteristic time scale τ.

    (A to E) The panels report the variation of the characteristic time scale (blue) that maximizes the STE flow as the social event is approached. Red lines correspond to activity volume (number of tweets). Light red vertical lines correspond to the onset of the main social event. Gray vertical lines (B and C) indicate a smaller precursor event. (A) The 15M event shows a progressive decline of the characteristic time scale well before the actual social event; the same is observed for the Outono Brasileiro in (B) (note a data blackout between days 10 and 11). The patterns for the Higgs boson discovery data set in (C) and the Hollywood blockbuster data (D) also reveal a drop in the characteristic time scale, although this is smoother in the movie case. Overall, in (A) to (D) (endogenous activity), the time scale has already dropped to 50% by the time the absolute volume signals a system-wide event. Finally, the Google-Motorola deal triggers a high volume of microblogging activity without actual change in the time scale of the information flow (E). In this case, the decline is observed in the aftermath of the announcement. As discussed in the main text, this event is the only one that is clearly elicited by an exogenous trigger.

  • Fig. 3 Evolution of information flow balance between geographical locations for the analyzed events.

    (A to E) The color goes from dark blue to dark red (white corresponds to null driving), with the former standing for negative values of Embedded Image (that is, driven locations) and the latter corresponding to positive information flow balances (that is, drivers). The size of the circles is log-proportional to the number of messages sent from the location at that time, and the vertical bars mark the day of the main event. The geographical locations are ordered according to population size, except for (C), in which countries are ranked with the amount of Higgs-related tweets produced.

  • Fig. 4 Schematic representation of a transition from a centralized to a decentralized information flow scenario.

    If, for any given pair (x, y), Embedded Image, all existent dynamical driving is net driving; that is, subsystems present a highly hierarchical structure. In this scenario, if a subsystem dominates another one, the former is not dominated by the latter. This is well illustrated in (A) and (B). Note, however, that only a few subsystems play an active (dynamical) role in (A),whereas the situation has reached a perfectly hierarchical structure in (B). Indeed, in this idealized situation, the net transfer entropy reaches its maximum: any further addition in terms of dynamical driving will decrease the amount of net transfer entropy [as in (C)]. Furthermore, (B) and (C) illustrate that there exists a tipping point beyond which the event has necessarily gone global. The extreme case where every subsystem exerts some amount of dynamical driving results in a “null driving” scenario [as in (D)]. In this schematic representation, the color scales go from dark blue to red, that is, zero to maximum transfer entropy, respectively.

  • Fig. 5 Order parameter θ as a function of time for the five events analyzed.

    The figure represents the behavior of the ratio Embedded Image characterizing the order/disorder of the effective connectivity matrix as a function of time (note a point missing in the Brazilian data set because of a data blackout between days 10 and 11). For each data set, two or three matrices T are plotted, considering one or two times before and one after the main event (signaled with a red vertical bar). A clear transition from a hierarchical directed to a distributed symmetrical scenario is observed for events (A), (B), (C), and (D). The Google data set, depicted in (E), behaves differently by not showing the same evidence of transition effects.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/2/4/e1501158/DC1

    Data, context, and chronology of the events analyzed

    Methods used in the analysis

    Sensibility analysis of the parametrization

    Validation of results (I): Time series randomization

    Validation of results (II): Unfiltered Twitter stream

    Validation of results (III): Synthetic time series generation

    Table S1. List of keywords used to find tweets related to the Outono Brasileiro.

    Fig. S1. Schematic representation of the algorithm used to gather geographical coordinates of the Hollywood movie release and the Google-Motorola acquisition data sets.

    Fig. S2. Sample order pattern for m = 3.

    Fig. S3. Schematic view of the sliding window scheme.

    Fig. S4. Evolution of the order parameter θ for thresholded (green) and raw (red) T matrices.

    Fig. S5. Dependence of τ with the sliding window size ω, considering the Spanish 15M protest.

    Fig. S6. Normalized directionality index for each geographical unit in the 15M data set for different ω.

    Fig. S7. Fraction of false nearest neighbors as a function of m for the Spanish data set and the Madrid time series.

    Fig. S8. Normalized directionality index for each geographical unit in the 15M data set for different m.

    Fig. S9. Characteristic time scale τ for four data sets at alternative geographical aggregation levels.

    Fig. S10. Normalized directionality index for four data sets at alternative geographical aggregation levels.

    Fig. S11. Behavior of θ as a function of time for four data sets at alternative geographical aggregation levels.

    Fig. S12. Average total amount of STE for some Δt (top panel) and time scale profile τ (bottom panel) for 15M data set amplitude adjusted Fourier transform surrogates (50 randomizations).

    Fig. S13. Behavior of θ as a function of time for 15M and Outono Brasileiro data sets randomized surrogates.

    Fig. S14. Average total amount of STE for some Δt (top panel) and time scale profile τ (bottom panel) for 15M data set constrained surrogates (20 randomizations).

    Fig. S15. Evolution of τ as a function of time.

    Fig. S16. Thresholded T matrices corresponding to different moments in the Twitter unfiltered data set.

    Fig. S17. Raw time series for Twitter unfiltered stream for Δt = 600 s and Δt = 45 s (left and right, respectively).

    Fig. S18. Evolution of two nonlinear systems under four changing scenarios: from dynamic independence (β = 0) to strong asymmetric coupling (β = 20.0).

    Part 1. Minimalist example: Disentangling volume and time scales (Δt).

    Part 2. Nonlinear Lorentz oscillators: Time scales, volume, and dynamical coupling.

    References (3658)

  • Supplementary Materials

    This PDF file includes:

    • Data, context, and chronology of the events analyzed
    • Methods used in the analysis
    • Sensibility analysis of the parametrization
    • Validation of results (I): Time series randomization
    • Validation of results (II): Unfiltered Twitter stream
    • Validation of results (III): Synthetic time series generation
    • Table S1. List of keywords used to find tweets related to the Outono Brasileiro.
    • Fig. S1. Schematic representation of the algorithm used to gather geographical coordinates of the Hollywood movie release and the Google-Motorola acquisition data sets.
    • Fig. S2. Sample order pattern for m = 3.
    • Fig. S3. Schematic view of the sliding window scheme.
    • Fig. S4. Evolution of the order parameter θ for thresholded (green) and raw (red) T† matrices.
    • Fig. S5. Dependence of τ with the sliding window size ω, considering the Spanish 15M protest.
    • Fig. S6. Normalized directionality index for each geographical unit in the 15M data set for different ω.
    • Fig. S7. Fraction of false nearest neighbors as a function of m for the Spanish data set and the Madrid time series.
    • Fig. S8. Normalized directionality index for each geographical unit in the 15M data set for different m.
    • Fig. S9. Characteristic time scale t for four data sets at alternative geographical aggregation levels.
    • Fig. S10. Normalized directionality index for four data sets at alternative geographical aggregation levels.
    • Fig. S11. Behavior of θ as a function of time for four data sets at alternative geographical aggregation levels.
    • Fig. S12. Average total amount of STE for some Δt (top panel) and time scale profile τ (bottom panel) for 15M data set amplitude adjusted Fourier transform surrogates (50 randomizations).
    • Fig. S13. Behavior of θ as a function of time for 15M and Outono Brasileiro data sets randomized surrogates.
    • Fig. S14. Average total amount of STE for some Δt (top panel) and time scale profile τ (bottom panel) for 15M data set constrained surrogates (20 randomizations).
    • Fig. S15. Evolution of τ as a function of time.
    • Fig. S16. Thresholded T† matrices corresponding to different moments in the Twitter unfiltered data set.
    • Fig. S17. Raw time series for Twitter unfiltered stream for &Delat;t = 600 s and Δt = 45 s (left and right, respectively).
    • Fig. S18. Evolution of two nonlinear systems under four changing scenarios: from dynamic independence (β = 0) to strong asymmetric coupling (β = 20.0).
    • Part 1. Minimalist example: Disentangling volume and time scales (Δt).
    • Part 2. Nonlinear Lorentz oscillators: Time scales, volume, and dynamical coupling.
    • References (36–58)

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