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Crowd behaviour challenges our fundamental understanding of social phenomena. Involving complex interactions between multiple temporal and spatial scales of activity, its governing mechanisms defy conventional analysis. Using 1.5 million Twitter messages from the 15M movement in Spain as an example of multitudinous self-organization, we describe the coordination dynamics of the system measuring phase-locking statistics at different frequencies using wavelet transforms, identifying 8 frequency bands of entrained oscillations between 15 geographical nodes. Then we apply maximum entropy inference methods to describe Ising models capturing transient synchrony in our data at each frequency band. The models show that

Coordinated activity is a powerful force in creating and maintaining social ties [

The rise of new digital communication tools and network technologies is accelerating fast bidirectional communication, generating new forms of collective communication and action. Digital communications tools increase the autonomy and influence of the social groups making use of them. They do this by promoting forms of mass self-communication [

Propitiously, the rise of social media and digital data-mining creates the opportunity for a novel analysis of human social systems [

We use a data set of 1.5 million Twitter messages to explore transient phase-locking synchronization as a general mechanism explaining interactions within and between temporal scales. In particular, we use a well-known social event of large-scale social and political self-organization: the massive political protest of the 15M movement in Spain, emergent in the aftermath of the 2011 Arab Spring and widely thought to be facilitated through digital social platforms [

Using this data set, we propose phase-locking statistics between geographical nodes at different frequencies as a generic description of coordination in a nationwide social system. This description allows us to use maximum entropy techniques to extract Ising models mapping the statistical mechanics of the system at each frequency band and thus obtain a deeper understanding of the spatiotemporal patterns of coordination within and between frequency bands. Inspecting the properties of the models at each frequency band, we observe that all bands are operating near a critical point but that different frequencies play different roles in the system. While fast bands alternate states of (almost) full synchronization and full desynchronization, bands with slower frequencies display a wide range of possible configurations of metastable states with clusters of partial synchronization. Furthermore, applying transfer entropy in the energy landscape at each frequency described by the Ising models, we characterize cross-scale interactions showing an asymmetry between upward and downward influences, where high-frequency synchronization influences nearby slower frequencies, while slow frequency bands are able to modulate distant faster bands. We argue that our results offer a promising step towards the description of general mechanisms operating at different scales, suggesting the existence of general rules for scaling up and down the dynamics of multitudinous collective systems.

We use a data set of 1,444,051 time-stamped tweets from 181,146 users, collected through the Twitter streaming API between 13 May 2011 and 31 May 2011 [

One of the most prominent features of the

Statistical significance of phase-locking values is determined by comparing them to phase-locking values of surrogate time series obtained using the amplitude adjusted Fourier transform [

As we document in [

For illustrative purposes, in Figure

Phase-locking statistics. Sum of the number of phase-locking links between all cities

In order to inspect how these phase-locked coalitions are operating at each frequency band, we derive from our data statistical mechanics models of the system. With these models we can infer macroscopic properties from microscopic descriptions of the system. Specifically, we use Ising models, which consist of discrete variables that traditionally are assumed to represent the magnetic moments of atomic spins that can be in one of two states (+1 or −1). In our case, positive spins will represent the presence of synchronizing activity of a node at a particular frequency. Spins are connected to other spins in the networks, allowing pairwise interaction between nodes. This is the least-structured (i.e., maximum entropy) model that is consistent with the mean activation rate and correlations of the nodes in the network. Pairwise maximum entropy models have been successfully used to map the activity of networks of neurons [

Using Ising models, we infer the probability distribution of possible states

The maximum entropy distribution consistent with a known average energy is the Boltzmann distribution

From the frequency bands

The accuracy of the inferred models can be evaluated by testing how much of the correlation structure of the data is captured. One measure to evaluate this is the ratio of multi-information between model and real data [

Once we have extracted a battery of models

Signatures of criticality. (a) Ranked probability distribution function of the inferred Ising models for the different frequency bands (solid lines) versus a distribution following Zipf’s law (i.e.,

The Ising model allows us to find further evidence of the critical behaviour of the model by exploring divergences of some variables in its parameter space. By introducing a fictitious temperature parameter

Specifically, a sufficient condition for describing a critical point in the parameter space of an Ising model is the divergence of its heat capacity, which is defined as

The fact that all frequency bands are operating near critical points does not mean that they are displaying the same behaviour. We can extract more information about the behaviour of the system at each frequency by analysing the presence of locally stable or metastable states in the system. Metastable states are defined as states whose energy is lower than any of its adjacent states, where adjacency is defined by single spin flips. This means that in a deterministic state (i.e., a Hopfield network with

Moreover, if we count the number of nodes that are phase-locked (i.e., the sum of all nodes with

Distribution of metastable states by frequency and synchronized nodes. Distribution of the number of active (i.e., synchronized) states of the inferred Ising models for each frequency band. The horizontal axis represents the frequency band, the vertical axis represents the number of active states of the metastable states, and the colour represents the count of metastable states with the same number of active states. The dashed line is the result of a least squares second-order polynomial fit (higher orders show a similar trend) over the number of active nodes of metastable states (excluding states with zero active nodes) with respect to the frequency index, showing a decrease of the number of synchronized nodes in the metastable states in favour of local clusters of activity.

These results suggest that fast and slow synchronization frequencies in the network operate in complementary regimes, all operating near critical points, the former rapidly propagating information to all the network and the latter sustaining a variety of configurations responding to specific situations. Systems in critical points present a wide range of dynamic scales of activity and maximal sensitivity to external fluctuations. These features may be crucial for large systems that are self-organized in a distributed fashion. The presence of these complementary modes of critical behaviour at different frequency bands suggests that the system might be operating in a state of self-organized criticality, in which frequency bands adaptively regulate each other in order to maintain a global critical behaviour.

Modelling phase-locking statistics provides a characterization of the interactions within frequency bands of synchronization. Furthermore, differences in the metastable states at each frequency band suggest what kind of interactions take place between distinct temporal scales. Because our definition of phase-locking statistics is restricted to interactions within the same frequency, we cannot use the computed phase-locking statistics to directly model interscale phase-locking between different frequencies (e.g., 2 : 1 phase-locking). However, we can use the thermodynamic descriptions of the system provided by maximum entropy models to simplify the analysis of interscale relations in real data.

Analysis of multiscale causal relations is typically a difficult task, and in our case we have to deal with a system of a high number of dimensions (

We characterize the information flow between frequency bands using transfer entropy [

In order to compute transfer entropy over energy values between timescales, we discretize the values of energy

To simplify the interpretation of the data, we compute the average value of transfer entropy (across the logarithmic range of

Average transfer entropy. (a) Average transfer entropy

These results show an interesting picture of cross-scale interactions. While in upward interactions energy at each frequency band only influences neighbouring slower bands, in downward interactions slow frequency bands modulate distant faster bands. We also observe this in the schematic in Figure

Cross-scale interactions in social coordination. Schematic displaying the results presented in previous figures. (a) Interactions in terms of average transfer entropy between energy levels at different frequencies described by

It is appealing to think that general coordinative mechanisms may be suited to explain the behaviour of social systems at different scales. Here, using a large-scale social media data set, we have shown how the application of maximum entropy inference methods over phase-locking statistics at different frequencies offers the prospect of understanding collective phenomena at a deeper level. The presented results provide interesting insights about the self-organization of digitally connected multitudes. Our contribution shows that phase-locking mechanisms at different frequencies operate in a state of criticality for rapidly integrating the activity of the network at fast frequencies while building up an increasing diversity of distinct configurations at slower frequencies. Moreover, the asymmetry between upward and downward flows of information suggests how social systems operating through distributed transient synchronization may create a hierarchical structure of temporal timescales, in which hierarchy is not reflected in a centralized control but in the asymmetry of information flows between the coordinative structures at different frequencies of activity. This offers a tentative explanation of how a unified collective agency, such as the 15M movement, might emerge in a distributed manner from mechanisms of transient large-scale synchronization. Of particular interest would be to test the extent our findings about the structural and functional relations of social coordination apply to other self-organizing social systems, or their relation with mechanisms of cross-scale interactions known from large-scale systems neuroscience. A new generation of experimental findings based on statistical mechanics models may provide the opportunity to discover the mechanisms behind multitudinous social self-organization.

The data employed in this study was kindly provided by the authors of [

Ising models are inferred using an adapted version of the coordinate descent algorithm described in [

The author declares no competing financial interests.

This research was supported in part by the Spanish National Programme for Fostering Excellence in Scientific and Technical Research Project PSI2014-62092-EXP, Projects TIN2016-80347-R and FFI2014-52173-P funded by the Spanish Ministry of Economy and Competitiveness, and the UPV/EHU postdoctoral training program ESPDOC17/17.

Table S1: frequencies of salient synchronization. Table representing the frequency values corresponding to the peaks represented in Figure S1(B). Figure S1: peaks of salient synchronization. (A) Sum of total phase-locking links