The formal demography of kinship III: Kinship dynamics with time-varying demographic rates

Background Kinship models generally assume time-invariant demographic rates, and compute the kinship structures implied by those rates. It is important to compute the consequences of time variation in demographic rates for kinship stuctures. Objectives Our goal is to develop a matrix model for the dynamics of kinship networks subject to arbitrary temporal variation in survival, fertility, and population structure. Methods We develop a system of equations for the dynamics of the age structure of each type of kin of a Focal individual. The matrices describing survival and fertility vary with time. The initial conditions in the time-invariant model are replaced with a set of boundary conditions for initial time and initial age. Results The time-varying model maintains the network structure of the time-invariant model. In addition to the results of the time-invariant model, it provides kinship structures by period, cohort, and age. It applies equally to historical sequences of past rates and to projections of future rates. As an illustration, we analyze the kinship structure of Sweden from 1891 to 2120. Contribution The time-varying kinship model makes it possible to analyze the consequences of changing demographic rates, in the past or the future. It is easily computable, requires no simulations, and is readily extended to include additional, more distant relatives in the kinship network. The method can also be used to show the growth of families, lineages, and dynasties in populations across time and place and between social groups.

1 Introduction 21 We live in a world that changes. It changes in the past as an observed fact of history and 22 we predict its changes in the future as scenarios. Even so, much of demography analyzes 23 change using models that exclude change. For example, calculations of life expectancy or 24 population growth rate based on the mortality and fertility in a consequence for a cohort 25 living out its life under the unchanging conditions of that year. The results can then be (and 26 are often) used to characterize change by comparing them over time or across populations. 27 Time-invariant models are a powerful way to explore a set of conditions by asking what 28 would happen if those conditions were to be maintained (Keyfitz, 1972;Cohen, 1979;Caswell, 29 2001). The early kinship models (Lotka, 1931;Coale, 1965;Goodman, Keyfitz, andPullum, 30 1974, 1975) and the more recent alternative by Caswell (2019Caswell ( , 2020 are all time-invariant. 31 Every individual in the kinship network experiences a fixed set of demographic rates. The 32 kinship structure of a focal individual describes the hypothetical situation in which she has 33 been subject to that fixed set of rates throughout her life. 34 In this paper, we relax the restriction to time-invariant rates, and allow any of the rates to 35 vary over time in any arbitrary fashion. This generalization makes two major contributions.  Goodman, Keyfitz, andPullum (1974, 1975) analyzed kinship structures using a system 44 of integral equations to calculate the expected number of kin, of each type, associated with 45 an individual of a given age. This classical approach is challenging to implement, gives only 46 the numbers, not age distributions of kin, and is difficult to extend beyond the simplest case.
A new approach was introduced by Caswell (2019, 2020), using matrix operations to project 48 the population of each type of kin from one age to the next and account for the production 49 of new kin of one type (e.g., nieces) by the reproduction of a different type (e.g., sisters). 50 The first of these papers provided the general theory for the dynamics of the age structure 51 of any kind of kin; the second extended the theory to multistate models that project the 52 age×stage structure of all types of kin and applied the theory to analyze the age×parity 53 structure of kin. The matrix approach simplifies model notation, facilitates computation, 54 and uses matrix algebra to link kinship relations with population dynamics. 55 We will begin by reviewing briefly the time-invariant kinship model, and use that frame-56 work to develop the time-varying model. We will then use the model to analyze the past and 57 future of kinship structure in Sweden by coupling a long sequence of historical data (1891- Notation. In what follows, matrices are denoted by boldface upper case letters (e.g., U) 64 and vectors by boldface lower case letters (e.g., a). Vectors are column vectors; the vector x T 65 is the transpose of x. The symbol x denotes the 1-norm of x; i.e. the sum of the absolute 66 values of the entries of x. The vector e i is the ith unit vector; i.e., a vector of zeros with a 1 67 in the ith entry. The dimension of e i will be specified if it is not clear from the context. We 68 will sometimes use Matlab notation in which F(i, :) and F(:, j) denote the ith row and the 69 jth column of F, respectively. The operator • denotes the Hadamard, or element-by-element 70 product.

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To review briefly here, kinship is defined relative to a focal individual, who we name 72 Focal. The kin, of any type, of Focal are a population, whose structure is governed by births 73 of new members and deaths and aging of current members. Each type of kin in the network 74 of Figure 1 is identified with a letter; e.g., a(x) is the age structure vector of the daughters 75 of Focal at Focal's age x. We reserve the symbol k as a generic kin age structure vector. 76 We write k(x) as the age structure vector for kin of type k at age x of Focal. The kin 77 k(x + 1) at age x + 1 consists of survivors of the kin at age x and new kin: where U is the survival matrix and β(x) is the age structure vector of new kin arriving 81 between x and x + 1. For some types of kin, β(x) = 0, because no new kin are possible. For 82 example, Focal cannot gain any new older sisters. For other types of kin, where F is the fertility matrix and k * is some other type of kin, whose offspring are kin of 85 type k. For example, new granddaughters (b) are produced by the reproduction of daughters 86 (a). The dynamics of granddaughters are then Because the model is a dynamic system, it requires an initial condition k(0) = k 0 , which 89 gives the age structure of the kin at the birth of Focal. For some types of kin,

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For some types of kin, the initial condition is zero (e.g., we know that Focal has no daughters 92 at the time of her birth). For other types of kin, the initial condition is calculated as a mixture 93 over the age distribution π of mothers at the birth of Focal (e.g., the older sisters of Focal 94 at her birth are the daughters of Focal's mother at her age at the birth of Focal). Having carefully specified the model, we are in a position to replace the time-invariant rates 100 by allowing survival and fertility to vary over a sequence of T years: The kin population now depends on both the age of Focal and time, so we write k(x, t) = kin of type k at age x of Focal at time t (8) 105 The dynamics are given by

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Our goal is to calculate these vectors for all the types of kin in the kinship network, over 108 some specified time span for which we have demographic rates.

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As before, the subsidy vector β(x, t) takes one of two forms. if there are no new kin of this type (e.g., older sisters of Focal). Or, 113 which applies the fertility at time t to the age structure vector of the kin that provides the 114 subsidy. For example, the dynamics of granddaughters, given in the time-invariant case by

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(4), now become 117 The distribution π of the ages of mothers at the birth of their children plays an important 118 role in the calculations. In the time-invariant case, this was calculated from the stable The two-dimensional dependence on age and time means that the initial condition k(0) in 131 the time-invariant model is replaced by a set of boundary conditions. The dynamics in (9) 132 requires us to specify the complete age vector at time t = 0, 134 and the initial age vector at each time,

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In the domain shown in Figure 2, with time on the abscissa and age on the ordinate, these 137 boundaries correspond to the bottom and the left margins.

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As always, the boundary conditions require extra information (or assumptions) from 139 outside the system. We use the following procedures 140 Figure 2: Boundary conditions. The figure contains ages from 0 to ω and times from 0 to T . The boundary conditions correspond to k(x, 0) for all x from 0 to ω and k(0, t) for all t from 0 to T .
147 If Focal has possible kin of this type at birth, then the age boundary is calculated as where k * is an appropriate other kind of kin and π i (t) is the proportion of mothers 150 who reproduce at age i at time t. These age boundaries will be presented for each type 151 of kin in Section 2.3 and summarized in Table 1 Including time variation in the demographic rates invites the calculation of period-specific, 154 cohort-specific, and age-specific results, as shown in Figure 3.

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Period results. The kinship structure vector, observed at a specified periodt, as a function 156 of the age of Focal: 158 Cohort results. The kinship structure of Focal as a member of a cohort starting at a 159 specified time t 0 and age x 0 :

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For birth cohorts, x 0 = 0. The cohort can be followed until either x 0 + i = ω or 162 t 0 + i = T , whichever comes first.
163 Figure 3: The period, cohort, and age dimensions of kinship development, within the age×time domain shown in Figure 2 Age results. The kinship structure vector of Focal at a specified agex, observed as a 164 function of time:  Table 1. Focal is assumed to be alive at age x, the subsidy vector is β( where e x is 175 the unit vector for age x. Because we may be sure that Focal has no daughters when 176 she is born, the initial condition is a(0, t) = 0. Thus Focal. At age x of Focal, these daughters have age distribution a(x, t), so β(x, t) = 181 F t a(x, t). Because Focal has no granddaughters at birth, the initial condition is 0; c(x, t) = great-granddaughters of Focal. Similarly, great-granddaughters are the result of reproduction by the granddaughters of Focal, with an initial condition of 0.
The extension to arbitrary levels of direct descendants is obvious. Let k n , in this case, 189 be the age distribution of descendants of level n, where n = 1 denotes children. Then after Focal's birth, so the subsidy term is β(x, t) = 0.

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At the time of Focal's birth, she has exactly one mother, but we do not know her age.

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Hence the initial age distribution d(0, t) of mothers is a mixture of unit vectors e i ; the 204 mixing distribution is the distribution π(t) of ages of mothers given by equation (14).

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Thus, term β(x, t) = 0. The age distribution of grandmothers at the birth of Focal is the 211 age distribution of the mothers of Focal's mother, at the age of Focal's mother when 212 Focal is born. The age of Focal's mother at Focal's birth is unknown, so the initial age 213 distribution of grandmothers is a mixture of the age distributions d(x, t) of mothers, 214 with mixing distribution π: h(x, t) = great-grandmothers of Focal. Again, the subsidy term is β(x, t) = 0. The 218 initial condition is a mixture of the age distributions of the grandmothers of Focal's 219 mother, with mixing distribution π:

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The extension to arbitrary levels of direct ancestry is clear. Let k n be, in this case, the age distribution of ancestors of level n, where n = 1 denotes mothers. Then the 224 dynamics and initial conditions are

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Note that, because Focal has at most one mother, grandmother, etc., the expected number 228 of mothers, grandmothers, etc. is also the probability of having a living mother, grandmother, 229 etc. the children a(i, t) of the mother of Focal at the age i of Focal's mother. This age 239 is unknown, so the initial condition m(0, t) is a mixture of the age distributions of 240 children with the mixing distribution π(t).
243 n(x, t) = younger sisters of Focal. Focal has no younger sisters when she is born, so the 244 initial condition is n(0, t) = 0. Younger sisters are produced by reproduction of Focal's 245 mother, so the subsidy term is the reproduction of the mothers at age x of Focal.   q(x + 1, t + 1) = U t q(x, t) + F t n(x, t) q(0, t + 1) = 0.

Aunts and cousins
261 Aunts and cousins form another level of side branching on the kinship network; their dy-262 namics follow the same principles as those for sisters and nieces.

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r(x, t) = aunts older than mother of Focal. These aunts are the older sisters of the 265 mother of Focal. Once Focal is born, her mother accumulates no new older sisters, 266 so the subsidy term is β(x, t) = 0. The initial age distribution of these aunts, at the 267 birth of Focal, is a mixture of the age distributions m(x, t) of older sisters, with mixing 278 t(x, t) = cousins from aunts older than mother of Focal. These are the children of 279 the older sisters of the mother of Focal, and thus the nieces of the mother of Focal 280 through her older sisters. The subsidy term comes from reproduction by the older 281 sisters of the mother of Focal.The initial condition is a mixture of the age distributions 282 of nieces through older sisters, with mixing distribution π(t),

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Let there be no confusion between the vector t giving the age structure of cousins and 286 the scalar t indexing time.
287 v(x, t) = cousins from aunts younger than mother of Focal. These are the nieces of 288 the mother of Focal through her younger sisters. The subsidy term comes from repro-289 duction by the younger sisters of the mother of Focal. The initial condition is a mixture 290 of the age distributions of nieces through younger sisters, with mixing distribution π(t).  We obtained mortality and fertility schedules for Swedish females from 1891 to 2018 (the  The future projected by Statistics Sweden is much calmer than the past, especially for 313 fertility. It is interesting to note that the 1918 influenza pandemic created a noticeable dip 314 in life expectancy, the eventual effects of the COVID-19 pandemic were not (and could not 315 have been) projected.

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As discussed in previous demographic work, we expect that the increased survival and 317 reduced fertility over time will have opposing effects on the kinship network: reduced fertility 318 means that fewer kin will be created, but increased survival means that fewer will be lost to 319 2 The mortality rates were obtained from: https://www.statistikdatabasen.scb.se/pxweb/en/ssd/START__BE__BE0401__BE0401F/ BefProgDodstal18/.
As of this writing, the most recent description of the methods used in the projections is Statistics Sweden (2018), in Swedish only. An earlier version in English is available as Statistics Sweden (2012). mortality (Bengtson, 2001;Mare, 2011;Seltzer, 2019;Uhlenberg, 1996Uhlenberg, , 2009. Figure 4-E: Period, cohort, and age results for the numbers of aunts and cousins of Focal; data from Sweden. Aunts older and younger than mother, and cousins through older and younger aunts as in Figure 1, have been combined.

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The analysis here retains the Focal-centric approach of Goodman, Keyfitz, and Pullum (1974) 349 and follows Caswell (2019Caswell ( , 2020   sophisticated branching process models may also be applicable (Pullum, 1982). We look 437 forward to developments in these areas. great-grandmothers i π i g(i) 0 m older sisters i π i a(i) 0 n younger sisters 0 Fd(x) p nieces via older sisters i π i b(i) Fm(x) q nieces via younger sisters 0 Fn(x) r aunts older than mother i π i m(i) 0 s aunts younger than mother i π i n(i) Fg(x) t cousins from aunts older than mother i π i p(i) Fr(x) v cousins from aunts younger than mother i π i q(i) Fs(x)