The dynamics of information-driven coordination phenomena: A transfer entropy analysis
Javier Borge-Holthoefer, Nicola Perra, Bruno Gonçalves, Sandra González-Bailón, Alex Arenas,Yamir Moreno, Alessandro Vespignani. Science Advances, 2016, 2(4): e1501158.
Our data sets consist of time-stamped and geolocalized time series of tweets associated to the following events: the Spanish 15M social unrest in 2011, the Outono Brasileiro (“Brazilian Autumn”) in 2013, the discovery of the Higgs boson in 2012, the release of a Hollywood blockbuster in 2012, and the acquisition of Motorola by Google in 2011.
STE quantifies the directional flow of information between two time series, X and Y, by first categorizing the signals in a small set of symbols or alphabet (see section B.3 in the Supplementary Materials), and then computing the joint and conditional probabilities of the sequence indices from the relative frequency of symbols in each sequence, and Ŷ, as
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 (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), , 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.