GENUS, LXVIII (No. 1), 1-9
ROBERT SCHOEN*
The kinship web in a simple stationary population
1. INTRODUCTION
The study of family and household demography has grown into a major
demographic area (Bongaarts, Burch, and Wachter, 1987), with the current
state of the field recently reviewed by Willekens (2010). While sophisticated
modeling and simulation procedures are now available, determining the average number of kin mathematically is typically complex even for close relationships such as aunt or grandparent (Keyfitz, 1985; Goodman, Keyfitz and Pullum, 1974). For the most part, the demographic literature also lacks analyses
of the structure of kin ties among members of a population and how they
change over the life cycle.
This research note sets forth population models whose simplicity enables
the kinship web to be described in full detail. We first consider a cohort's kin
structure in the light of population size and exogamy. We then examine each
person's kin web and how it changes over the life course.
2. THE BASIC STATIONARY POPULATION MODEL
To make the analysis tractable, our basic stationary population model
assumes that each person follows a uniform life course. Everyone born survives to exact age 90, and then dies. At exact age 30, each person becomes the
parent of twins, one boy and one girl. There are no other births. Childbearing
relationships are with persons of the same age, making the demographic life
courses of all men and women identical. The kinship web has a reference person (Ego) who can be of either gender. Ego's spouse will have an identical kinship web, but Ego's spouse and in-laws, are not considered as kin here.
For reasons explained below, we assume that each birth cohort has
(4n+2)/3 persons (both male and female). Scale parameter n can be chosen to
reflect the population size of interest. For example, if n=6, cohort size is 1,366
persons. The total number of persons in the basic stationary population is
(4n+2), as there are just 3 cohorts alive at any given time. At time 0, there are
(4n+2)/3 persons at each of ages 0, 30, and 60, while the cohort attaining age
90 has just died.
* Department of Sociology and Affiliate, Population Research Institute, Pennsylvania State
University, U.S.A.; e-mail: [email protected].
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ROBERT SCHOEN
3.
DIAGRAMMING THE KINSHIP ARRAY
To precisely specify kin relationships, consider the consanguinity diagram
shown in Table 1. The superscripted number shown by each relationship is the
"degree" of that relationship, a measure of how closely the person is related to
Ego. Relatives of the first degree, one step away from Ego, are Ego's two parents and two children. Ego's four grandparents, four grandchildren, and one sibling are relatives of the second degree. In the diagram, each can be reached in
two steps from Ego. For example, Ego's sibling is reached by taking one step up
to Ego's parents and a second step down to the other child of Ego's parents. Table
1 can readily be extended to more distant kin, but most contemporary interest,
and the focus here, will be on kin up to and including relatives of the fourth
degree (e.g. first cousins).
Table 1 – Consanguinity Diagram
Note: Superscripted numbers by each relationship indicate degree of kinship to Ego. Numbers in parentheses indicate the genetic relatedness to Ego of relatives of degree four or less.
Source: Structure and nomenclature adapted from Dukeminier et al. (2009: 93); Genetic relatedness from
Brembs (2001).
Mathematical expressions for the number of kin of different kinds date
back at least to Lotka (1931). The most extensive treatment is by Goodman,
Keyfitz and Pullum (1974), which assumed constant mortality and focused on
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THE KINSHIP WEB IN A SIMPLE STATIONARY POPULATION
one-sex (female) populations. Goodman, Keyfitz and Pullum (1974) examined a wide range of kin relationships, including the number of parents, children, sisters, aunts, nieces, and cousins a woman would have. They showed
that the eventual number of kin in a stable (i.e. constant fertility and mortality) population could be expressed in terms of the expected number of sisters
and the population's Net Reproduction Rate (i.e. number of daughters per
woman). However, in the general case, those expressions are rather complex
(e.g. the number of first cousins is given as a quadruple integral). The kin
relationships given here are consistent with those in Goodman, Keyfitz and
Pullum (1974), but the simplicity of the present model allows the number of
male and female kin of different kinds to be specified explicitly and precisely.
To relate the consanguinity diagram to members of the stationary population, consider the kin structure of each birth cohort. If the stationary population has been in existence for some time, no less than n generations, then
everyone in each cohort is related, and the most distant relationship is (n-1)st
cousin.
To demonstrate that, we can trace all of the cohort kin ties. Ego has 2
parents and 1 sibling. Ego's 4 grandparents (2 couples), have 4 children.
Two of them are Ego's parents, and the other 2 are an aunt (Ego's father's sister) and an uncle (Ego's mother's brother). It is not rare for a mother's brother to marry a father's sister, but such unions are atypical in contemporary
Western societies. To simplify matters the model assumes that such pairings
do not occur. Hence Ego's aunt's 2 children and Ego's uncle's 2 children give
Ego a total of 4 first cousins. Next, tracing the descendants of Ego's 8 great
grandparents, we find that Ego has 4 great aunts/uncles who, marrying other
people, produce 8 first cousins once removed. In turn, those 8 relatives of
degree 5 provide Ego with 16 second cousins (relatives of degree 6).
If parents are the first ascendant generation, grandparents the second,
and so on, continuing the above analyses shows that the nth ascendant generation gives Ego 4n-1 (n-1)st cousins. Our birth cohort thus consists of Ego,
Ego's sibling, 4 first cousins, 16 second cousins, ... , and 4n-1 (n-1)st cousins.
Using the equation for the sum of a geometric progression we find that
[1 + (1+4+16+...+4n-1)] gives a total of (4n+2)/3 persons, the assumed cohort
size. For example, if we only go back three generations to Ego's great
grandparents, then cohort size is 22 and is comprised of Ego, Ego's sibling,
Ego's 4 first cousins, and Ego's 16 second cousins. In this population of 66
persons, Ego finds a partner from among Ego's 8 other-sex second cousins.
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ROBERT SCHOEN
4.
EGO'S KINSHIP WEB OVER THE LIFE COURSE
We can use the above stationary population relationships to determine what
kin Ego has at every age. Table 2, Panel A, summarizes that kinship web for each
kin relationship through degree 4, i.e. for "close" kin. At birth, Ego has 17 close
kin. Of them, 7 are of degree 1 or 2, i.e. "very close" kin. The number of close
kin is fairly stable over Ego's life course, varying only from 13 to 20, but the type
of kin shifts as Ego ages. At age 30, when Ego becomes a parent, Ego has 4 older
kin, 5 kin the same age, and 4 younger kin. At age 60, when Ego becomes a
grandparent, most of Ego's close kin are younger and the older kin have died.
Table 2 – Living kin of Ego, by relationship, in the stationary population
...Cont’d...
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THE KINSHIP WEB IN A SIMPLE STATIONARY POPULATION
Table 2 – Cont’d
Note: The symbol "--" indicates "just died". Ego's spouse and in-laws are not included.
Source: See text.
Table 2, Panel B, summarizes Ego's kin web if life expectancy is 60 years
(2 generations) instead of 90 years (3 generations), with all other features
remaining the same. The shorter life expectancy shrinks Ego number of close
kin to only 9 at ages 0 and 30 and 12 at age 60. Other things equal, the more
generations in Ego's life expectancy, the more close kin Ego has.
Table 3 summarizes Ego's kinship web in a similar but growing population, one where every person has four children (2 boys and 2 girls) at exact
age 30. That population has a Net Reproduction Rate (NRR) of 2, as every
woman has 2 daughters, and has a generation length (T) of 30 years. From
Lotka's relationship NRR=exp(rT), we find that the population's average
annual growth rate (r) is a moderate 2.3% (cf. Schoen, 2006).
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ROBERT SCHOEN
In Panel A, life expectancy is 90 years (3 generations). In any cohort, tracing as described above shows that Ego has 3 siblings, 24 first cousins, 192 second cousins, and (cohort size permitting) 3•8n-1 (n-1)st cousins. Since the NRR=2,
cohort size (and population size) doubles every generation. Here, when fertility
doubles, the number of kin triples or more. Again, more kin are younger as Ego
gets older. At birth and at age 30, Ego has 51 close kin, though only 9 are of the
first or second degree. However, at age 60, Ego has 23 very close kin and a total
of 107 close kin.
Table 3, Panel B keeps the NRR at 2 but reduces life expectancy to 60 years
(2 generations). The number of close kin drops substantially from the previous
panel, with a total of 35 close kin at age 0, 43 close kin at age 30, and 80 close
kin at age 60. At ages 0 and 30, most close kin are Ego's age, the majority of them
being first cousins.
Table 3 – Living kin of Ego, by relationship, in the NRR=2 stable population
...Cont’d...
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THE KINSHIP WEB IN A SIMPLE STATIONARY POPULATION
Table 3 – Cont’d
Note: The symbol "--" indicates "just died". Ego's spouse and in-laws are not included.
Source: See text.
Another way to measure kinship is by using a person's genetic relatedness
to Ego, i.e. by the probability that a random gene at a specified locus is identical
by descent (cf. Brembs, 2001; Hamilton, 1963). The figure in parentheses in
Table 1 gives the genetic relatedness of relatives up to the fourth degree. Using
those fractions, a summary Index of Aggregate Kinship can be created by adding
up the genetic relatedness of all of Ego's extant kin.
Table 4 shows the Index of Aggregate Kinship, by Ego's age, for the 4 models in Tables 2 and 3. The model where life expectancy is 60 years and the NRR
is 1 has the lowest values of the Index, but at every age Ego still has at least the
equivalent of 5 parents/siblings/children. With the NRR remaining at 1 but with
life expectancy at 90 years, the Index is 4.0 at every age, the equivalent of 8 parents/siblings/children.
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ROBERT SCHOEN
Table 4 – Index of Aggregate Kinship, by Age of Ego, for Net Reproduction
Rates (NRRs) of 1 and 2 and for Life Expectancies of 90 years and 60 years
Note: The Index of Aggregate Kinship is the sum of the genetic relatedness fractions of all of Ego's living kin of degree 4 or less. Those fractions are shown in Table 1.
The Net Reproduction Rate is the number of daughters a woman has in her lifetime under a specified
schedule of fertility and mortality.
Source: See text.
The Index of Aggregate Kinship increases substantially when the NRR is 2.
With life expectancy at 60 years the Index varies from 7.0 to 15.0, and with life
expectancy at 90 years the Index goes from 9.5 to 23.
5.
SUMMARY AND CONCLUSIONS
Kin ties can be described in detail for simple populations where every person has either two children (one boy and one girl) or four children (two boys and
two girls) at age 30. All persons in a birth cohort are related, but in a population
of any size most will be distant cousins. The focus here is on "close kin", that is
relatives who are no more distant than first cousins (i.e. relatives of the fourth
degree).
In a stationary population with a life expectancy of 90 years, Ego has at
least 13 close kin at every age. Very close kin, i.e. from grandparents to grandchildren, number from 5 to 7. If life expectancy is only 60 years (two generations), Ego still has at least 9 close kin at every age. If fertility doubles so that
every person has 4 children at age 30, the number of kin greatly expands.
The models presented here have deliberately been made as simple as possible so that a complete, explicit representation of the kinship web could be provided. That approach necessarily involves a number of limitations and departures from reality. The models here are completely deterministic, with no uncertainty, no random variation, and no overlap of generations. Mortality and fertility are very narrowly restricted, unchanging over time, and identical for men and
women. The sex ratio at birth is always 1. There is no population heterogeneity
and no migration. The overall impact of those restrictions on the nature of the
kinship web is likely to vary from population to population. Simulation
approaches (cf. Willekens, 2010) can address them, but at the cost of simplicity
and precision.
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THE KINSHIP WEB IN A SIMPLE STATIONARY POPULATION
The models presented here reflect how a person's kinship web varies with
the level of fertility and the number of generations in the average lifespan. In a
stationary population of (4n+2) persons, no one is more distant than an (n-1)st
cousin. Since the kin numbers presented here do not include spouses or in-laws,
even a stationary population with a modest life expectancy can provide enough
close kin to allow substantial kin-based social interaction over the life course.
Acknowledgements
Valuable referee comments and helpful observations on kinship by Diane J.
Klein are acknowledged with thanks.
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