Relating the microscopic rules in coalescence-fragmentation models to the cluster-size distribution

Coalescence-fragmentation problems are now of great interest across the physical, biological, and social sciences. They are typically studied from the perspective of rate equations, at the heart of which are the rules used for coalescence and fragmentation. Here we discuss how changes in these microscopic rules affect the macroscopic cluster-size distribution which emerges from the solution to the rate equation. Our analysis elucidates the crucial role that the fragmentation rule can play in such dynamical grouping models. We focus our discussion on two well-known models whose fragmentation rules lie at opposite extremes. In particular, we provide a range of generalizations and new analytic results for the well-known model of social group formation developed by Eguíluz and Zimmermann, [Phys. Rev. Lett. 85, 5659 (2000)]. We develop analytic perturbation treatments of this original model, and extend the analytic analysis to the treatment of growing and declining populations.