Hierarchy of families of theories and their rank characteristics

Studying families of elementary theories produces an information on behavior and interactions of theories inside families, possibilities of generations and their complexity. The complexity is expressed by rank characteristics both for families and their elements inside families.
We introduce and describe a hierarchy of families of theories and their rank characteristics including dynamics of ranks. We consider regular families which based on a family of urelements — theories in a given language, and on a step-by-step process producing the required hierarchy. An ordinal-valued set-theoretic rank is used to reflect steps of this process. We introduce the rank RS and related ranks for regular families, with respect to sentence-definable subfamilies and generalizing the known RS-rank for families of urelements, as well as their degrees. Links and dynamics for these ranks and degrees are described on a base of separability of sets of urelements. Graphs and families of neighbourhoods witnessing ranks are introduced and characterized. It is shown that decompositions of families of neighbourhoods and their rank links, for discrete partitions, produce the additivity and the possibility to reduce complexity measures for families into simpler subfamilies.