Fig. 2: Institutions densify at different rates but their size and frequency follow universal patterns.

a Internal and (b) external collaborations versus institution size for three arbitrarily chosen institutions with more than 10³ cumulative researchers in each field or simulation. Circle markers correspond to simulation data; solid lines, medium dashed lines, long dashed lines, and short dashed lines correspond to data from the fields of computer science (CS), physics, math, sociology, respectively. Dash-dotted lines report linear scaling, showing that institutions' scaling laws are super-linear. Insets: distribution of exponents across thousands of institutions (cf. Supplementary Note 1). c The distribution of researchers in each institution as of 2017 (Zipf’s law), and (d) the number of unique institutions versus the total number of researchers in each field (Heaps' law). Closed circle markers correspond to simulation data; open circles, squares, diamonds, and triangles correspond to computer science (CS), physics, math, sociology, respectively. In addition, light dash-dotted lines indicate empirical trends while darker dashed lines indicate theoretical scaling law exponents −1 − ν/ρ and ν/ρ for (c) and (d) respectively²⁵. Simulation data in (a) and (b) are collected from four realizations and in (c) and (d), from fifteen realizations (individual realizations show similar trends). Simulation parameters are ρ equals 4, ν equals 2, μ_p equals 0.6, and σ_p equals 0.25.