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

doi:10.1126/sciadv.1501158

. 2016 Apr 1;2(4):e1501158. doi: 10.1126/sciadv.1501158

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

Javier Borge-Holthoefer ^1,^*,^†, Nicola Perra ^2,^*, Bruno Gonçalves ³, Sandra González-Bailón ⁴, Alex Arenas ^5,^*, Yamir Moreno ^6,^7,^8,^*, Alessandro Vespignani ^2,^8,^9,^*

PMCID: PMC4820379 PMID: 27051875

This work defines the framework to explore the spatiotemporal signature of emergent collective phenomena on social media.

Keywords: collective phenomena, transfer entropy, dynamical transitions

Abstract

Data from social media provide unprecedented opportunities to investigate the processes that govern the dynamics of collective social phenomena. We consider an information theoretical approach to define and measure the temporal and structural signatures typical of collective social events as they arise and gain prominence. We use the symbolic transfer entropy analysis of microblogging time series to extract directed networks of influence among geolocalized subunits in social systems. This methodology captures the emergence of system-level dynamics close to the onset of socially relevant collective phenomena. The framework is validated against a detailed empirical analysis of five case studies. In particular, we identify a change in the characteristic time scale of the information transfer that flags the onset of information-driven collective phenomena. Furthermore, our approach identifies an order-disorder transition in the directed network of influence between social subunits. In the absence of clear exogenous driving, social collective phenomena can be represented as endogenously driven structural transitions of the information transfer network. This study provides results that can help define models and predictive algorithms for the analysis of societal events based on open source data.

INTRODUCTION

A vivid scientific and popular media debate has recently centered on the role that social networking tools play in coordinating collective phenomena; examples include street protests, civil unrests, consensus formation, and the emergence of electoral preferences. A flurry of studies have analyzed the correlation of search engine queries, microblogging posts, and other open data sources with the incidence of infectious disease (1–4), box office returns (5), stock market behavior (6, 7), election outcomes (8, 9), popular votes results (10), crowd sizes (11), and social unrest (12, 13). However, many other studies have also pointed out the challenges that big data present and the likely methodological pitfalls that might result from their analysis (14–19). Previous work suggests that more research is needed to develop methods for exploiting the value of social media data while overcoming their limitations.

Here, we use microblogging data to extract networks of causal influence among different geographical subunits before, during, and after collective social phenomena. To ground our work on empirical data, we analyze five data sets that track Twitter communication around five well-known social events: the release of a Hollywood blockbuster movie, two massive political protests, the discovery of the Higgs boson, and the acquisition of Motorola by Google. We selected these case studies because they represent different points in a theoretical continuum that separates two types of collective phenomena: those that can be represented as an endogenously driven exchange of information and those that respond more clearly to factors that are exogenous to the system. In our context, these phenomena refer to the dynamics of information exchange through social media. In some cases, discussions evolve organically, building up momentum up to the point where the exchange of information is generalized; however, in some other cases, the discussions emerge suddenly as a reaction to some unexpected external event (20). The Motorola-Google case study corresponds to the exogenous type, providing a counterexample and intuitive baseline test for the other cases.

For each case study, we adopt the transfer entropy approach to define an effective social connectivity at the macroscale and study the coordinated activation of localized populations. We address two foundational problems: first, the identification of the characteristic time scale of social events as they develop, gather force, and burst into the public eye. In effect, the determination of the pertinent time scale is one of the fundamental limitations of the analysis of data from social media, namely, to be considered for the posterior processing of information. Symbolic transfer entropy (STE) captures the intrinsic time scale of the information flow and allows a proper diagnosis of the granularity needed to grasp the evolution of social events. Second, we look into the characterization of the structural signature typical of the communication dynamics that underlie social phenomena. We find that the onset of collective phenomena is characterized by the drop of the characteristic time scale; we also show that the emergence of coherent patterns of information flow can be mapped into order-disorder transitions in the underlying connectivity patterns of the transfer entropy network. The methodology we present here can therefore be used to gain new insights on the structural and functional relations occurring in large-scale structured populations, eventually leading to the identification of metrics that might be used for the definition of precursors of large-scale social events.

RESULTS

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.

The spatiotemporal annotation of each tweet in the time series allows the construction of spatially localized activity maps that help identify, as time unfolds, the role that different geographical subunits played in the global exchange of information. For each data set, the definition of the corresponding spatial unit is performed according to administrative and geographical boundaries as specified in Materials and Methods (see Fig. 1 as well). Note that the map only shows how the signal increases in all regions and does not provide evidence of any unexpected transition, pointing out that volume alone is not a good indicator of the evolution of the events.

Fig. 1 — 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.

The time-stamped series of tweets that originated from each spatial subunit X (supra-urban aggregates) defines the activity time series X_t of the corresponding subunit in the social system. Time stamps are modified for each data set to account for different time zones (see the Supplementary Materials for details). Activity time series encode the role of each geographical subunit, a sort of “who steers whom,” and several techniques can be used to detect directed exchange of information across the social system. Here, we characterize the dominating direction of information flow between spatial subunits using STE (21, 22). This well-established technique has been used to infer directional influence between dynamical systems (23–25) and to analyze patterns of brain connectivity (26).

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, $\hat{X}$ and Ŷ, as

T_{Y, X} = \sum p ({\hat{x}}_{i + δ}, {\hat{x}}_{i}, {\hat{y}}_{i}) {log}_{2} (\frac{p ({\hat{x}}_{i + δ} | {\hat{x}}_{i}, {\hat{y}}_{i})}{p ({\hat{x}}_{i + δ} | {\hat{x}}_{i})})

(1)

where the sum runs over each symbol in the sequence and δ = 1. The transfer entropy refers to the deviations of the cross-Markovian property of the series (independence between them), measured as the Kullback-Leibler divergence (27) (see the Supplementary Materials for all technical details). An important feature of symbolic approaches is that they discount the relative magnitude of each time series; this is important in our case because different geographical units differ largely in population density or Internet penetration rates. Flattening and discretizing the original signal are a key feature to enhancing the sensitivity of our proposal to any changes (even minor ones) in the interaction dynamics between subsystems (see section F in the Supplementary Materials).

Within this framework, we first analyze the temporal patterns characterizing the flow of information. Admittedly, microblogging data can be sampled at different time scales, Δt. To select the optimal sampling rate, we consider all possible pairs (X,Y) of geographical units and measure the total STE in the system T = ∑_XYT_X,Y as a function of Δt. We consider the system-wide characteristic sampling time scale τ as that which maximizes the total information flow T. This quantity provides an indication of the time scale at which the information is being exchanged in the system, which is not necessarily correlated with volume; see the early stages of (A) to (D) in Fig. 2 where the time scale drops by ~50% in a very scarce volume scenario (see section F in the Supplementary Materials as well). The characteristic time scale τ changes as the phenomena under analysis unfold; that is, it decreases as the system approaches the exponential increase in overall activity that signals the onset of the collective event. As shown in the top panels of Fig. 2, τ is a proxy for the internally generated coordination in the system that culminates simultaneously with the time of the occurrence of the social event: the street protest day in the case of political unrest, the movie release date in the case of the Hollywood blockbuster, and the announcement to the press of the Higgs boson discovery. The only clear exception to this behavior is offered by the company acquisition data set: the Google-Motorola announcement is a clear example of collective phenomena driven mostly by an exogenous factor, that is, a media announcement. In this case, the dynamical time scale is constant until the announcement is made public. In the Supplementary Materials, we present the same analysis for the randomized signals, showing that time scale variations are then washed out from the signal, as expected.

Fig. 2 — (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.

The maximized information exchange can be analyzed at the level of geographical subunits by constructing the effective directed network (28) of information flow on a daily basis. This network is encoded in the matrix {T_XY}, which contains pairwise information about how each component in the system controls (or is controlled by) the others. The matrix {T_XY} is asymmetric. The directionality is crucial and denotes that the geographic area x can exert some driving on area y and, at the same time, y might exert some driving on x. For this reason, it is convenient to define the directionality index $T_{X, Y}^{S} = T_{Y, X} - T_{X, Y}$ , which measures the balance of information flow in both directions. This index quantifies the dominant direction of information flow and is expected to have positive values for unidirectional couplings with x as the driver and negative values if y is driving x. For symmetric bidirectional couplings, we expect $T_{X, Y}^{S}$ to be null.

We show in Fig. 3 the temporal evolution of the maximized $\sum_{Y} T_{X, Y}^{S}$ that provides the information flow balance of each specific geographical area. The results show that in the 15M grassroots protests, a limited number of urban areas initially drive the onset of the social phenomenon. These units correspond mostly to major cities; however, the analysis also uncovers hidden drivers, such as Orotava, a less known urban area. Only after the first demonstration day on May 15 does the driving role become much more homogeneously distributed. In the Brazilian case, a set of clear drivers is present only during the onset phase preceding a demonstration on June 6 and becomes fuzzier up until the major demonstration (June 17) and totally blurred afterward. We find a similar behavior in the Higgs boson cases (with rumors around the discovery on July 2 and the final announcement on July 4) (29). The blockbuster case is driven by a steady excitement of the public before the movie release. Again, as expected, we observe completely different patterns in the case of the Google data set.

In general, the evolving effective networks reveal a transition from hierarchical causal relationships to symmetric (if rather fluctuating) networks where information is flowing symmetrically among all subunits. If information flows mainly in one direction (that is, if the subsystems are arranged in a highly hierarchical structure), a subunit dominates another, with no or little information flowing in the opposite direction. In this situation, a convenient manipulation of the matrix (T → T^†) based on the ranking and reordering of the elements according to their directionality index yields an upper triangular matrix (see Materials and Methods). The transition from such hierarchical or centralized driving to a symmetric scenario can be clearly identified by monitoring the ratio $θ = T_{l}^{†} / T_{u}^{†}$ between the sum of all elements of T^† in the lower triangle and the same quantity evaluated in the upper triangle. As schematically illustrated in Fig. 4, in a regime of perfectly directed driving, all the elements below the diagonal are zeros; that is, θ ≈ 0. In the opposite situation (that is, in the perfectly symmetric regime), the values below and above the diagonal are comparable; that is, θ ≈ 1. Hence, the quantity θ can be considered a suitable order parameter to characterize this order-disorder transition and thus helps in the identification and differentiation of communication patterns across the subunits of a system.

Fig. 4 — If, for any given pair (x, y), $T_{x, y}^{S} ~ T_{x, y}$ , 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.

We show in Fig. 5 the behavior of the parameter θ as a function of time in our five data sets. In all of the cases, we initially observe a highly asymmetric effective network, where a few subunits have a dominant directional coupling to the rest of the system and θ ≪ 1. As the systems approach the onset date of the collective event, the quantity $T_{l}^{†} / T_{u}^{†}$ undergoes a quick transition to θ ≈ 1, identifying a regime in which the couplings indicate the existence of collective phenomena where all subunits mutually affect each other. We see that in four of the five data sets, the system has a clear order-disorder transition occurring in the proximity of the collective event. In the case of the Brazilian protests, the measure significantly increases before the main event (June 17). Such behavior probably results from the effects of small precursor protests taking place from June 6 onward. The same behavior is observed in the Higgs boson data set, given that rumors started circulating after July 2. Once more, the Google data set behaves in a completely different way, never showing a clear signature of a collective regime for the couplings network. In the Supplementary Materials, we report the same analysis using the randomized signal for both the 15M and Brazil events, and we observe no order-disorder transition. Similarly, no transition exists for the Twitter unfiltered stream case study (also in the Supplementary Materials).

All data sets cover a time span preceding and following the event, and details on data collection, spatial aggregation (including keyword selection and the geolocalization of messages), and sensitivity analysis of the methodology can be found in Materials and Methods and in the Supplementary Materials.

DISCUSSION

The mapping of influence networks using an information theoretical approach offers a new lens to analyze the emergence of collective phenomena. Through this lens, we have revealed the existence of a double transition—in the time scales (slow to fast) and directional couplings (hierarchical to distributed)—in systems that gather around some sort of collective action. Regarding the first, we bring to light that time series analysis should pay attention to the time scales of the underlying dynamical processes if it is to provide a reliable account of those dynamics—a fact that resonates beyond societal analysis. We also uncover the effective network of information flow between spatially defined subunits of the social system and study the structural changes of the network connectivity pattern as the system goes through different collective states. In addition, the effective network lends itself to further analysis that can lead to the identification of structural hubs, coordinated communities, influence pathways of geographical or cultural characteristics, and geographical subunits that may have recurrent roles in the onset of social phenomena. The methodology we present here can therefore be used to gain new insights on the structural and functional relations occurring in large-scale structured populations, eventually leading to the identification of metrics that might be used for the definition of precursors of large-scale social events.

Additionally, the methodology presented here opens interesting paths to advance in the analysis of social phenomena and the identification of generative mechanisms; however, this advance should not be conflated with the possibility of forecasting the emergence of social events, but as a natural complement of techniques along this line, such as network change-point detection (30). The evidence we discuss is agnostic with regard to the predictive potential of online networks and microblogging platforms. A real predictive approach cannot be disentangled from an automatic selection of the relevant discussion topics. Our analyses use data sets that were already zooming into the right conversation domain and monitoring specific keywords/hashtags in the Twitter stream. However, we believe that the general methodological framework we put forward is a first step toward a better understanding of the temporal and spatial signatures of large-scale social events. This advancement might eventually inform the development of tools that can help us anticipate the emergence of macroscopic phenomena. In the meantime, our method offers a valuable resource to analyze how information-driven transitions unfold in socially relevant contexts.

MATERIALS AND METHODS

Data

The first data set focuses on the Spanish 15M movement, which emerged in 2011 (31, 32). The data cover a dormant period of low microblogging activity that is followed by an explosive phase in which the movement gained the attention of the general public and was widely covered by traditional media sources (see Fig. 1). The second data set contains more than 2.5 million geolocalized tweets associated to the Outono Brasileiro, a set of political protests that emerged in Brazil in June 2013. Similar to the Spanish case, the Brazilian data include an initial phase of low activity followed by a gradual escalation toward the high volumes of general attention that accompanied the street protests. The third data set tracks communication on the discovery of the Higgs boson before and after it was officially announced to the press in July 2012; this data set has been used before to assess how rumors spread through online social networks (29). The fourth data set contains messages related to the release of a Hollywood blockbuster, announced months before its premiere to stir momentum among the fan base. Finally, we also consider a data set that tracks communication on the acquisition of Motorola by Google, which came as sudden and unexpected news and immediately triggered a high volume of public attention.

Spanish Twitter activity is spatially coarse-grained according to the list of metropolitan areas defined by the European Spatial Planning Observation Network (33). This process yields 56 aggregated time series, each of which corresponds to a different geographical area. In addition, there is an extra signal that accounts for any activity not included in those areas; that is, the system is made up of n = 57 components. The data from Brazil are aggregated in 97 basins, which correspond roughly to metropolitan areas (34, 35). The data that track rumors about the Higgs boson are aggregated at the country level, including only the n = 61 most active basins around this topic. Finally, the Motorola-Google and the blockbuster data are classified in 52 U.S. areas: 50 states plus Washington, DC, and Puerto Rico.

Order-disorder transition

In real data sets, the transition between centralized and decentralized information-transfer scenarios can be visually inspected with a convenient sorting of the rows and columns of the T_x,y matrix. We did so in Fig. 5, ranking each subunit of the system. The rank for a subunit x is assigned according to the number of times x is dominant over the rest of the subunits. Once the ranking is settled, any $T_{x, y} < \frac{1}{2} T_{x, y}^{max}$ is set to 0 to improve the visual understanding of the figure. We then obtained a transformed matrix, that is, $T_{x, y} \to T_{x, y}^{†}$ . Beyond visualization, the sorted matrix allows us to define a monitoring measure $θ = \frac{\sum_{x > y} T_{x, y}^{†}}{\sum_{x < y} T_{x, y}^{†}} = \frac{T_{l}^{†}}{T_{u}^{†}}$ (that is, the ratio between the sums of all the matrix’s elements in the lower and upper triangles), which provides a quantification of the state of the system (as explained in the main text and illustrated in Fig. 4). For completion, we have also plotted the same figures without threshold (see section B.6 in the Supplementary Materials).

Supplementary Material

http://advances.sciencemag.org/cgi/content/full/2/4/e1501158/DC1

supp_2_4_e1501158__index.html^{(3.8KB, html)}

Acknowledgments

We thank D. Allen, R. Compton, and T.-C. Lu at HRL Laboratories LLC for assistance with the Brazilian data set and useful discussions; we also thank A. Lima for sharing the Higgs boson data. Funding: B.G. thanks the Moore and Sloan Foundations for support as part of the Moore-Sloan Data Science Environment at New York University. A.A. acknowledges the support of the European Union MULTIPLEX (317532) and the Spanish Ministry of Science and Innovation (FIS2012-38266-C02-01) and partial financial support from the ICREA Academia and the James S. McDonnell Foundation. Y.M. acknowledges support from MINECO through grant FIS2011-25167, Comunidad de Aragón (Spain) through a grant to the group FENOL, and the European Commission Future and Emerging Technologies Proactive Project MULTIPLEX through grant 317532. For the analysis of data outside of the United States, A.V. and N.P. acknowledge the Intelligence Advanced Research Projects Activity (IARPA) via Department of Interior National Business Center (DoI/NBC) contract number D12PC00285. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of IARPA, DoI/NBE, or the United States Government. The funders had no role in the study design, data collection and analysis, decision to publish, or preparation of the manuscript. Author contributions: All authors contributed equally to this work. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper, the Supplementary Materials, and/or www.jbh.cat/data/twitter_timeseries.tar.gz. Additional data related to this paper may be requested from the authors.

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.

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