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bioRxiv · 10.64898/2026.03.31.715515

The probable numbers of kin in a multi-state population: a branching process approach

Butterick, J.
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Abstract

Recent progress in mathematical kinship modelling has allowed one to predict the probable numbers of kin for a typical population member. In the models, kin may be structured by age and sex, both in static or time-variant demographies. Knowing the probable numbers of kin in different stages - such as parity, health status, or geographic location - however, remains an open challenge in Kinship Demography. Knowing how population structure delimits kin to distinct stages is an advance - for instance, the probability of having one sister at home and one sister away has different social implications from the probability of having two sisters. We present a novel analytical framework, grounded in branching process theory, that provides kin-number distributions jointly structured by age and stage. Using recursive compositions of probability generating functions (PGFs), we derive the joint age, stage, and age x stage kin-number distributions. All marginal distributions over either dimension naturally emerge. Simple extensions of the PGF approach additionally yield: the joint distribution of an individuals own stage and their kins stage; the probable numbers of kin deaths, both in total and by generation number; and the probabilities of being kinless and/or orphaned. We demonstrate the framework through novel results in an application using UK parity-specific fertility and mortality data. HighlightsO_LIA new method calculates probability generating functions for the number of kin structured by age and stage C_LIO_LIThe model allows predicting the probable numbers of kin organised by age and stage C_LIO_LIRecursive nesting of probability generating functions in branching processes is used C_LIO_LIAn application is presented highlighting the novel results C_LI

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Butterick, J.. 2026-04-02. The probable numbers of kin in a multi-state population: a branching process approach. https://doi.org/10.64898/2026.03.31.715515

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