We consider a model for the evolution of the surnames distribution under a
gender-equality measurement presently discussed in the Spanish parliament (the
children take the surname of the father or the mother according to alphabetical
order). We quantify how this would bias the alphabetical distribution of
surnames, and analyze its effect on the present distribution of the surnames in
Spain

B Sykes

DH Zanette

F Galton

J Panaretos

JBS Haldane

Luis F. Lafuerza

M Gonzalez

PAP Moran

PC Consul

R Fisher

Raul Toral

SC Manrubia

TE Harris

Yamir Moreno

PLoS ONE

English

arXiv

Lafuerza Luis F.

Toral Raul

Public Library of Science (PLOS)

Evolution of Surname Distribution under Gender-Equality Measures

We consider a model for the evolution of surname distribution under a gender-equality measure currently being discussed by the Spanish Parliament (whereby children would adopt their mother's and father's surnames in alphabetical order). We quantify how this would bias the alphabetical distribution of surnames, and analyze its effect on the present distribution of surnames in Spain.The authors acknowledge financial support by the MICINN (Spain) and FEDER (EU) through project FIS2007-60327. L.F.L. is supported by the JAEPredoc program of CSIC. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscriptPeer reviewe

Lafuerza, Luis F.

Toral, Raúl

Digital.CSIC

Evolution of Surname Distribution under Gender-Equality MeasuresLuis F. Lafuerza, Raul Toral*IFISC (Instituto de Fsica Interdisciplinar y Sistemas Complejos), CSIC-UIB, Campus UIB, Palma de Mallorca, SpainAbstractWe consider a model for the evolution of surname distribution under a gender-equality measure currently being discussedby the Spanish Parliament (whereby children would adopt their mother’s and father’s surnames in alphabetical order). Wequantify how this would bias the alphabetical distribution of surnames, and analyze its effect on the present distribution ofsurnames in Spain.Citation: Lafuerza LF, Toral R (2011) Evolution of Surname Distribution under Gender-Equality Measures. PLoS ONE 6(4): e18105. doi:10.1371/journal.pone.0018105Editor: Yamir Moreno, University of Zaragoza, SpainReceived December 22, 2010; Accepted February 21, 2011; Published April 14, 2011Copyright:  2011 Lafuerza, Toral. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permitsunrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.Funding: The authors acknowledge financial support by the MICINN (Spain) and FEDER (EU) through project FIS2007-60327. L.F.L. is supported by the JAEPredocprogram of CSIC. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.Competing Interests: The authors have declared that no competing interests exist.* E-mail: raul@ifisc.uib-csic.esIntroductionIn Spain, as in many other countries, children usually inherittheir father’s surname. As a consequence, the mother’s surname islost in the child’s generation (in Spain, however, the mother’ssurname is kept as a second surname, it is consequently totally lostin the grand-children’s generation). Nowadays, in Spain, parentscan agree upon whether it is the mother’s or father’s surname thatis given to their children, but if parents do not reach an agreement,it will automatically be the father’s surname that is inherited by thechildren. Due to gender-equality issues, a new law is under reviewwhereby, if either, parents do not reach an agreement, or if nowish is expressed, the surname inherited by the children will beselected according to the alphabetical order of the parents’ twosurnames.People have immediately realized that this implies a bias on thesurnames favoring those beginning with the first letters in thealphabet (A,B,…) and could mean that surnames beginning withthe last letters (…,Y,Z) disappear completely. In this short paper,we quantify the effect of this bias on the present distribution ofsurnames in Spain.Evolution of surnames distribution (in the absence of anyalphabetical preference) has been studied previously. Thepioneering work of Galton and Watson used a branching processto study the probability of extinction of surnames in Englisharistocracy[1]. This problem turns out to be mathematicallyequivalent to that of the evolution of non-recombining neutralalleles [2] and several authors[3–5] have used similar ideas in thecontext of biological evolution. Later developments by mathema-ticians and physicists centered on the distribution of family sizes[6–9]. The novelty of our work is that we analyze how thedistribution of surnames evolves when a preference depending ontheir alphabetical position is present. We derive an equation forthe evolution of surnames distribution on these premises. Thesolution of the equations allows us to determine the timescale atwhich the surnames disappear.AnalysisAs a first order model aimed at capturing the essence of theprocess of surname inheritance we propose the following:(i) Initially, a population of N individuals (N=2 male and N=2female) is considered. Each individual has a surname chosenaccording to some prescribed distribution.(ii) Males and females reproduce in random pairs in such a waythat, on average, the total population remains constant.(iii) With probability a it is assumed that parents reach anagreement, so that the surnames of their children are chosen atrandom between those of the parents (the proportion of whetherthe father’s surname or the mother’s is preferred is irrelevant onthe results). With probability 1{a, parents do not reach or do notexpress an agreement, and children adopt their surname by thealphabetical order rule.We measure time t in average reproductions per person, orgenerations. In a generation, parents are replaced by their childrenin the population.The population evolves according to a bisexual Galton andWatson branching process[10]. The statistics of the number ofpeople as a function of time [11] and the distribution of thefrequency of surnames in a model similar to this one when thesurnames are chosen at random have been studied previously[8].The model introduced above is a minimal model and does notconsider some realistic issues such as geographical distribution ofsurnames, etc. but those are expected to be second order effectswith little impact in the overall trend. The effect of immigrationand population growth is analyzed later in the text.Let us define p(n,t) as the proportion of individuals (both males andfemales) with surname in the alphabetical position n~1, . . . ,M, beingM the total number of surnames. It evolves according to:Lp(n,t)Lt~(1{a)p(n,t)XMk~nz1p(k,t){Xn{1k~1p(k,t)" #ð1ÞPLoS ONE | www.plosone.org 1 April 2011 | Volume 6 | Issue 4 | e18105~(1{a)p(n,t) 1{P(n,t){P(n{1,t)½ , ð2Þwhere P(n,t)~Pnk~1 p(k,t) is the cumulative distribution. The firstterm in the square brackets of Eq.(1) represents the increase inprobability of surname n due to the pairing with surnamesk[½nz1,M which are further forwards the end in the alphabeticalorder, while the second term represents the loss in probability due topairings with surnames k[½1,n{1 earlier in the alphabetical order. Itfollows that:LP(n,t)Lt~(1{a)P(n,t) 1{P(n,t)½ , ð3Þwhose solution is:P(n,t)~P(n,0)e(1{a)t1zP(n,0)(e(1{a)t{1): ð4ÞThe distribution of surnames at time t is thenp(n,t)~P(n,t){P(n{1,t) for n§1 with the conventionP(0,t)~0. Approximating the difference by a derivativep(n,t)^ LP(n,t)Ln , we obtain:p(n,t)~p(n,0)e(1{a)t1zP(n,0)(e(1{a)t{1)½ 2: ð5ÞEq. (4) shows that the distribution of surnames approaches aKronecker-delta at n~1 (P(n,t)~1,Vn) exponentially quickly witha characteristic time 1=(1{a). Assuming, for instance, thatcouples reach, and express, an agreement in 50% of the cases(a~1=2), we find from Eq.(5) that the frequency of a surnametowards the end of the alphabetical table would be decreased by afactor 10 in around 4:6 generations(*115 years). If, on the otherhand, couples do not reach an agreement in 5% of the cases(a~0:95), then the decrease by a factor 10 occurs in 46generations.Evolution of current distributionWe have applied the above results to the current distribution ofSpanish surnames. Besides the analytical result of Eq.(5), we haveperformed a numerical simulation of the model by which N~107couples have probabilities (0:05,0:2,0:5,0:2,0:05) of having(0,1,2,3,4) children (average value is 2). The probability of parentsreaching an agreement is set at a~0:5. Whether an agreement hasbeen reached or not, the rule applied to the first-born child is usedfor all subsequent children. We have used as the initial conditionp(n,0) the distribution of the M~100 most common surnames inSpain, after ordering them in alphabetical order. The data appearin the INE webpage www.ine.es (INE stands for ‘‘InstitutoNacional de Estadı´stica’’). Similar data is available for othercountries. Our simulation results only consider those 100 surnamesfor which data are publicly available. In figure 1 we plot (symbols)the probability distribution p(n,t) resulting from this numericalsimulation after n~4 (top panel) and n~10 (bottom panel)generations. In the same figure we also plot the theoreticalprediction, Eq. (5) using the same initial condition and for thesame number of generations. As it can be seen in the figure (notethe logarithmic scale for better viewing of data in the case t~10)the concurrence between the simulation and the analytical result isexcellent. It can be noted that the relative importance of surnamesmoves towards the surnames which are earlier in the alphabet astime increases.The evolution in the frequency of a surname does not have to bemonotonous, as it can first increase and then decrease in time. Let ustake, for instance, the most common surname in Spain: ‘‘Garca’’.According to the INE data, there are 1,481,923 people bearing thissurname and the cumulative distribution is P(n,0)&0:291. Hence,if there is a 50% agreement, the frequency or this surname wouldfirst increase up to 1:8|106 in two generations, to then decrease to1:1|105 in 10 generations. In the case of the surname ‘‘Toral’’,there are nowadays 3,190 people bearing this surname and thecumulative distribution is P(n,0)&0:97. According to the previousanalysis, and considering again 50% agreement, it would decreaseto 1,980 in one generation and to 23 in 10 generations. The samestudy for ‘‘Lafuerza’’ (very rare surname, only 122 people bear it inSpain currently and the cumulative distribution is P(n,0)&0:465),shows that it would stay practically constant in the first generation tothen decrease to 4 (practical extinction) in 10 generations. Finally, ifwe take a surname high in the alphabetical order such as ‘‘Aguilar’’,of which there are 58,771 people at the moment, it would increasepractically exponentially, as the cumulative distribution is very smalland can be neglected in the denominator of Eq.(5). Of course, allthese predictions are for the mean values, and significant statisticaldeviations could occur for low-frequency surnames.Effect of immigration and population growthIn the previous analysis we assumed that the population remainsconstant (there is an average of two children per couple) and thatthe only changes in the distribution of surnames correspond to theapplication of the alphabetical order rule. We now consider theeffect that, both, population growth and new surnames brought inby immigration have in the distribution of surnames. This impliesmodifying condition (ii) in the model, allowing the number ofchildren per couple to have any average value r and settingimmigration events with rate lIN(t), proportional to the totalpopulation number. The alphabetical position of the surname ofthe immigrant is chosen at random according to some probabilitydistribution pI (n). The total population increases exponentially asN(t)~N(0)e(lIzlp)t with lp~(r{2)=2.Let N(n,t)~p(n,t)N(t) be the number of people with surnamein alphabetical position n. It evolves according to:N(n,tzDt){N(n,t)~12N(t)DtN(n,t)N(t)N(n,t)N(t)|(r{2)z12N(t)Dt2N(n,t)N(t)N(t){N(n,t)N(t)|ar{22z(1{a)N(t)(1{P(n,t))N(t){N(n,t)r{1ð ÞzN(t)P(n{1,t)N(t){N(n,t)({1)  zlIN(t)pI (n)DtzO(Dt2):ð6ÞThe first term corresponds to mating of two people (one male, onefemale) both with surname in alphabetical position n, in this case theaverage increase in N(n,t) equals the average increase due to themating, which is r{2. The second term corresponds to the mating of aperson with surname n with a person with a different surname: if, withprobability a, they agree on the surname to be assigned to the children,the average increase in N(n,t) equals (r{2)=2; otherwise, the averageincrease in N(n,t) is either r{1 (if the other surname is later in thealphabet) or {1 (if it is earlier). This derivation neglects the possiblefluctuations that can appear in the distributions of surnames in malesEvolution of Surname DistributionPLoS ONE | www.plosone.org 2 April 2011 | Volume 6 | Issue 4 | e18105and females. The last term corresponds to the addition of newindividuals brought in by immigration.From this equation, and after some algebra, one can obtain theevolution of the number of people N(sƒn,t)~Pns~1 N(s,t) withsurname in alphabetical position smaller or equal to n:N(sƒn,tzDt){N(sƒn,t)~ r 1{ a2 {1h iDtN(sƒn,t){r2(1{a)DtN(sƒn,t)2N(t)zlIN(t)PI (n)DtzO(Dt2),ð7Þbeing PI (n) the cumulative distribution of the immigrantssurnames. Dividing by N(tzDt)~N(t)(1z(lpzlI )Dt)zO(Dt2)and taking the limit Dt?0, we obtain:LP(n,t)Lt~lI PI (n){P(n,t)ð Þz(1{a) r2P(n,t) 1{P(n,t)½ , ð8Þwhose solution is:P(n,t)~12{lI(1{a)rzC(n)(1{a)rtanhC(n)tzargtanh(1{a)r½P(n,0){1=2zlIC(n)  ,ð9Þwith C(n):ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2(1{a)rlIPI (n)z½(1{a)r=2{lI 2q.We see from Eq.(8) that the intrinsic growth, r, of the populationonly changes the timescale of the dynamics of the surnamedistribution. Immigration, however, might have a greater impact.Let us focus on the asymptotic, t?? distribution, which has the form:P(n)~12zC(n){lI(1{a)r: ð10ÞAn analysis of this expression, shows that a critical valuelc:(1{a)r=2 exists, such that the delta-like singularity thatappeared in the non-immigration case disappears for lIwlc. Thesituation then, is that for lIvlc there is an accumulation atsurnames close to the lower limit n~1, such that a fraction1{2lI(1{a)rof people bear the surname first in the alphabeticalorder. So, the low immigration rate produces results which are thesame, qualitatively, as in the case without immigration. ForlIwlc, however, the fresh distribution of surnames brought in byimmigration is enough to overcome the accumulation at n~1towards which the probability distribution would tend in theasymptotic limit due to the alphabetical order rule. In both cases,the tail of the stationary probability distribution, behaves asp(n)*pI (n)=½PI (n){1=2. If, for instance, the distribution of newsurnames were uniform in the alphabet, PI (n)*n, then the tail ofthe stationary distribution would behave as a power-law of slope{1=2.DiscussionWe have developed a mathematical model for the evolution ofthe surnames distribution when an alphabetical-order rule on theprogenitor’s surnames is applied. The premises of the model leadto a differential equation governing the evolution of the probabilitydistribution. As initial condition we have considered the data forthe present distribution of the 100 most common surnames inFigure 1. Evolution of the distribution of surnames after n~4 (top) and n~10 (bottom) generations, taking as initial condition p(n,0)the actual distribution of theM~100 most common surnames in Spain. For n~10 we have used a logarithmic scale for a better viewing ofthe data. The dots are the result of the numerical simulation of the model described in the main text, and solid lines correspond to the analyticalresult (4).doi:10.1371/journal.pone.0018105.g001Evolution of Surname DistributionPLoS ONE | www.plosone.org 3 April 2011 | Volume 6 | Issue 4 | e18105Spain, obtained from the National Statistics Institute of Spain(INE), that is publicly available at its webpage www.ine.es. Similardata is available for other countries. We have also performednumerical simulations of an agent-based model, which agree withthe analytical result.In our minimal model for surname transmission, we prove thatthe adoption of the alphabetical rule leads to an exponentialdecrease in the surnames that begin with letters that are towardsthe end of the alphabet, with a characteristic decay time of1=(1{a) generations, being a the fraction of parents that reach anagreement. This quantifies the decrease in the frequency of thosesurnames. We have also considered the effect of surnames broughtin by immigration and found that, below a critical value of theimmigration rate, the results are the same, qualitatively, as in thecase without immigration. For large immigration rates, the delta-like singularity that appeared at names earlier in the alphabet,disappears.We believe that this study offers an example in which statisticalmethods and mathematical modeling can be used to quantitativelycalculate the consequences of a political measure and, conse-quently, it can serve as a guide to institutions and policy makers.AcknowledgmentsWe thank Susan Meal for useful suggestions related to presentation.Author ContributionsAnalyzed the data: LFL RT. Contributed reagents/materials/analysistools: LFL RT. Wrote the paper: LFL RT.References1. Galton F, Watson HW (1874) On the probability of the extinction of families.J Roy Anthropol Inst 4: 138–144.2. Sykes B, Irven C (2000) Surnames and the y chromosome. Am J Hum Genet 66:1417–1419.3. Fisher R (1922) On the dominance ratio. Proc Roy Soc Edin 42: 321–341.4. Haldane JBS (1927) A mathematical theory of natural and artificial selection,part v: selection and mutation. Proc Camb Philos Soc 26: 838–844.5. Moran PAP (1962) The statistical processes of evolutionary theory. Oxford:Clarendon Press.6. Panaretos J (1989) On the evolution of surnames. Int Stat Rev 57: 161–167.7. Consul PC (1991) Evolution of surnames. Int Stat Rev 59: 271–278.8. Zanette DH, Manrubia SC (2001) Vertical transmission of culture and thedistribution of family names. Physica A 295: 1–8.9. Manrubia SC, Zanette DH (2002) At the boundary between biological andcultural evolution: the origin of surname distributions. J Theor Biol 216:461–477.10. Harris TE (1989) The Theory of Branching Processes. Dover Pub., 2nd edition.11. Gonzalez M, Molina M (1997) Some theoretical results on the progeny of abisexual Galton-Watson branching process. J Serdica Math 23: 15–24.Evolution of Surname DistributionPLoS ONE | www.plosone.org 4 April 2011 | Volume 6 | Issue 4 | e18105

We consider a model for the evolution of surname distribution under a gender-equality measure currently being discussed by the Spanish Parliament (whereby children would adopt their mother’s and father’s surnames in alphabetical order). We quantify how this would bias the alphabetical distribution of surnames, and analyze its effect on the present distribution of surnames in Spain

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Evolution of Surname Distribution under Gender- Equality Measures

We consider a model for the evolution of surname distribution under a gender-equality measure currently being discussed by the Spanish Parliament (whereby children would adopt their mother's and father's surnames in alphabetical order). We quantify how this would bias the alphabetical distribution of surnames, and analyze its effect on the present distribution of surnames in Spain

Luis F Lafuerza

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Evolution of surname distribution under gender-equality measures.

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A mathematical theory of natural and artificial selection, part v: selection and mutation.

At the boundary between biological and cultural evolution: the origin of surname distributions.

Evolution of surnames.

On the dominance ratio.

On the evolution of surnames.

Some theoretical results on the progeny of a bisexual Galton-Watson branching process.

Surnames and the y chromosome.

The statistical processes of evolutionary theory.

The Theory of Branching Processes. Dover Pub., 2nd edition.

Vertical transmission of culture and the distribution of family names.

Watson HW (1874) On the probability of the extinction of families.

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