Phase diagram of ranking stability in the – plane (: fitness, : noise). For a given ranking system, is a vector of constants () representing the “fitness” of all items of the ranked list, is a parameter measuring the ranking’s noise. Every real ranking system is represented by a line corresponding to the experimentally determined value of . In analogy to the classical phases of statistical mechanics, three phases are identified based on the stability of the top-ranked items: rank stable (solid), score stable (liquid), and unstable or volatile (gas). is the control parameter of the phase transition. The lower panel shows the rank evolution for the top-ranked items of a stable system (diseases diagnosis in Medicare) and a volatile one (page views in Wikipedia).