  Coalescence-fragmentation problems are of great interest across the physical,
biological, and recently social sciences. They are typically studied from the
perspective of the rate equations, at the heart of such models 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. More generally, our work
elucidates the crucial role that the fragmentation rule can play in such
dynamical grouping models. We focus on two well-known models whose
fragmentation rules lie at opposite extremes setting the models within the
broader context of binary coalescence-fragmentation models. Further, we provide
a range of generalizations and new analytic results for a well-known model of
social group formation [V. M. Eguiluz and M. G. Zimmermann, Phys. Rev. Lett.
85, 5659 (2000)]. We develop analytic perturbation treatment of the original
model, and extend the mathematical to the treatment of growing and declining
populations.
