Fig. 4: Simple vs. complex contagion in online social media.

a We consider avalanches with size S ≥ S_min = 10 and fit them against the branching process (BP) and the Random Field Ising Model (RFIM). For each time series, we establish whether the fits against the individual models are statistically significant or not; if both fits cannot be rejected, we then select the best model by means of the log-likelihood ratio. We report the fraction of time series that are classified in the RFIM class. This fact may happen because the RFIM fit cannot be rejected whereas the BP is rejected, or both fits cannot be rejected but the RFIM is favored over the BP in terms of log-likelihood ratio. The fraction of time series that are classified as BP is defined in an analogous manner. The fraction of time series that is classified as neither BP nor RFIM is represented by the bar labeled as “None.” Finally, some time series pass both statistical tests. Their fraction is denoted by the label “Both” in the figure. In this case, the log-likelihood ratio test is required for model selection, see Fig. 3c. b We restrict our attention to Twitter hashtags containing characters from the English alphabet only, and display the 30 most popular hashtags classified either in the RFIM (blue) or the BP (red) classes. The font size is proportional to the rank of the hashtag in each class. Hashtags of both classes are selected among those that are sufficiently critical, i.e., \(| \hat{R}-{R}_{c}| /{R}_{c}\le 0.05\) for a time series in the RFIM class or \(| \hat{n}-{n}_{c}| /{n}_{c}\le 0.05\) for a time series in the BP class.