Sensitivity of collective outcomes identifies pivotal components*

The relation between collective outcomes and individual behavior is a central question in social science, biology, and statistical physics. Using the information geometry of minimal models from statistical physics, we develop a general approach for identifying key "pivotal" components on which aggregate statistics depend most sensitively. For political voting, pivotal blocs are like swing voters on whom the distribution of majority-minority divisions depends most sensitively. Analogously, collective market movement may be characterized by a few important stock indices, or the identity of a community on Twitter may hinge on a few individuals. In neural networks, pivotal components may be important for determining collective states. Our approach may help evaluate how political bodies change with membership or analyze the robustness of social institutions and biological networks to targeted perturbation.