mathematical framework for time-variant multi-state kinship modelling.

Recent research on kinship modelling in demography has extended age-structured models (i) to include additional characteristics, or "stages" (multi-state kinship), and (ii) to time-variant situations. A wide variety of population structures can affect kinship networks. However, only one prior model has comprehensively considered such effects, and under specific assumptions relating to the nature of individuals' stages. As such, the leading multi-state framework for kin is theoretically limited in scope, and moreover, has yet to be implemented under time-variant demographic rates. Generalising kinship models to encompass arbitrary population characteristics and extending them to time-dependent processes remain open challenges in demography. This research proposes a methodology to extend multi-state kinship. We present a model which theoretically accounts for any stage, both in time-variant and time-invariant environments. Drawing from Markov processes, a concise mathematical alternative to existing theory is developed. The benefits of our model are illustrated by an application where we define stages as spatial locations, exemplified by clusters of local authority districts (LADs) in England and Wales. Our results elucidate how spatial distribution - a demographic characteristic ubiquitous across (and between) societies - can affect an individual's network of relatives.