Impact of individual actions on the collective response of social systems

In a social system individual actions have the potential to trigger spontaneous collective reactions. The way and extent to which the activity (number of actions—A) of an individual causes or is connected to the response (number of reactions—R) of the system is still an open question. We measure the relationship between activity and response with the distribution of efficiency, a metric defined as η=R/A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta =R/A$$\end{document}. Generalizing previous results, we show that the efficiency distribution presents a universal structure in three systems of different nature: Twitter, Wikipedia and the scientific citations network. To understand this phenomenon, we develop a theoretical framework composed of three minimal statistical models that contemplate different levels of dependence between A and R. The models not only are able to reproduce the empirical activity-response data but also can serve as baselines or null models for more elaborated and domain-specific approaches.