Fig. 2: Universality and criticality of information propagation in social media.

a Avalanche size distribution. Different colors/symbols indicate data obtained from different social media. Acronyms are defined as in Fig. 1. Dotted lines represent the maximum likelihood estimators of the exponent τ obtained by fitting the Random Field Ising Model (RFIM), in red, and the branching process (BP), in teal. The RFIM was fitted using N = 10⁹, R = 0.8 and considering the same number of avalanches as the Twitter sample. The BP was fitted using n = 1.0 and sampling 10⁶ avalanches. Distributions are displayed via logarithmic binning of the data. To make distributions collapse one on the top of the other, the size is multiplied by the factor 1/〈S〉 and probabilities are multiplied by the factor 〈S〉. Cumulative distributions are reported in SI E. b Distribution of avalanche duration for the same data as in panel a. To make distributions collapse one on the top of the other, duration is multiplied by the factor 1/〈T〉 and probabilities are multiplied by the factor 〈T〉. Dashed lines represent the maximum likelihood estimators of the exponent α obtained by fitting the RFIM (red) and the BP (teal). c Average size of avalanches with given duration. Data are the same as in a and b. To make the curves collapse one on top of the other, the abscissa of each curve is rescaled as T/〈T〉 and the ordinate is rescaled as 〈S〉(T)/〈S〉. Solid lines represent the hyperscaling exponent \((\hat{\alpha }-1)/(\hat{\tau }-1)\) obtained using the maximum likelihood estimators of τ and α for the RFIM (red) and for the BP (teal). d Maximum likelihood estimates of the exponents \(\hat{\tau }\), \(\hat{\alpha }\) and \(\hat{\gamma }\), see SI C for details. We also display the ratio \((\hat{\alpha }-1)/(\hat{\tau }-1)\). Error bars are always smaller than the size of the symbol. Dotted lines correspond to the best fit of the exponent τ to the RFIM (red) and to the BP (teal), as shown in panel a. Analogously for dashed lines, representing the best fit of α shown in b and for solid lines, representing the hyperscaling relation shown in c.