Modelling the evolution and diversity of cumulative culture

Mälardalen University

This is an accepted version of a paper published in Philosophical Transactions of the

Royal Society of London. Biological Sciences. This paper has been peer-reviewed but

does not include the final publisher proof-corrections or journal pagination.

Citation for the published paper:

Enquist, M., Ghirlanda, S., Eriksson, K. (2011)

"Modelling the evolution and diversity of cumulative culture"

Philosophical Transactions of the Royal Society of London. Biological Sciences,

366(1563): 412-423

URL:

http://dx.doi.org/10.1098/rstb.2010.0132

Access to the published version may require subscription.

Permanent link to this version:

http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-13329

umulative ulture

∗

Abstra t

Previous work on mathemati al models of ultural evolution has

mainly fo used on the diusion of simple ultural elements.

How-ever, a hara teristi featureof human ulturalevolution isthe

seem-inglylimitlessappearan eofnewandin reasingly omplex ultural

el-ements. Herewedevelopa generalmodelingframeworktostudy su h

umulative pro esses, in whi h we assume that the appearan e and

disappearan e of ultural elements are sto hasti events that depend

on the urrent state of ulture. Fives enarios areexplored: evolution

of independent ultural elements, stepwise modi ation of elements,

dierentiation or ombination ofelements, and systemsof ultural

el-ements. Asone appli ation our framework,we study the evolution of

ultural diversity(intime aswell asbetween groups).

Keywords: ultural evolution, umulative ulture, mathemati al model,

ultural diversity, ultural systems

1 Introdu tion

In so ial s ien e and the humanities there is a long tradition of des ribing

omplex ulture(Levi-Strauss,1963;Piaget,1970;Harris,2001;Searle,1995),

andofexploringtheevolutionof ultureand ulturalsystems(Renfrew,1972;

∗

1996; Carneiro, 2003). An important observation about human ulture is

that it evolves ina seeminglyopen-ended manner that,among otherthings,

in ludes the potential for the appearan e of ultural elements of in reasing

omplexity and renement, and to form systems of ulture (Basalla, 1988;

Carneiro, 2003). Our aim in this theoreti al paper isto develop a theory of

ulturala umulationandexplore howthe umulativeaspe t of ulturemay

ontribute to ultural diversity. To a hieve this aim we will on eptualize

umulative ultureina way that lends itselfto mathemati alformalization.

The basi units of our theory willbe ultural elements and dependen ies

between su h elements. By a ultural element we here mean anything that

may or may not be present in a given human so iety at a given time, like

a tool or artifa t, a method, an idea, a pie e of knowledge. Dependen ies

refertorelationshipsbetweenelements,su hthatthepresen eofone ultural

element ae ts the likelihoodthat another element appears ordisappears.

Dependen iesbetween ulturalelementsseemtousto onstitutethe ore

of what umulative ultureis about. Other on eptualizationsof umulative

ulture in the literature seem to be spe ial ases that are too limited. For

instan e, ithasbeen proposed that umulative ultureis ulturethat annot

be reated within a single generation (Tomasello, 1994; Boyd & Ri herson,

1996), whi hex ludes ulturalprogress thatpro eeds inseveral stepswithin

a single generation, and also begs the question why some ulture annot be

reated at su h a high speed. Another example is the metaphor of

umu-lative ulture as a rat het (Tomasello, 1999) or as an irreversible pro ess

(White, 1959;Carneiro, 2003),whi hex ludes the possibility thata ultural

element an be lost and reates a one-dimensional image of progress.

De-penden ies between ultural elements, on the other hand, may des ribe not

onlyrenementorprogress,butalsodierentiation, ombinationofelements,

substitutability(dierentsolutionstothesameproblem),loss,andsoon.

In-deed,weproposethatsu hmulti-dimensionalaspe tsof umulative ulture

are atthe rootof ulturaldiversity between so ieties. Spe i ally,we laim

thata umulationof ultureleadstodiversityonlyifthereareri hbran hing

Wedis uss these aspe ts indetail below.

Isaa Newton's famous statement that If I have seen further it is only

by standing on the shoulders of giants (Southern, 1952) ould serve as a

motto for our theory of umulative ulturalevolution. The general

impli a-tion of this motto is that in order for human ulture (of whi h s ien e and

mathemati s are primeexamples)tobe omein reasinglysophisti ated,itis

not ne essary that in reasingly intelligent and reative individuals are born

(Simonton, 2004). As knowledge,methods, ideas and tools umulate, a new

innovator who has no more talent than his prede essors an still ome up

with aninnovationthat issuperiortowhat anyprede essors haveprodu ed,

pre isely be ause he already has a ess to the ontributions these

prede es-sors made. In our theory we on eptualize this in probabilisti terms; a

ertain ultural element

x

may be very unlikely to appear, or even impossi-ble, withoutthe presen e of some other ultural element

y

;when

y

appears (forwhateverreason), thesubsequentappearan eof

x

maysuddenlybe ome highly likely.

The individuals of so iety are not expli itly represented in our models,

despitetheobviousfa tthathumanagen yis entralindevelopingand

trans-mitting ulture. The virtue ofanagent-less model isof ourseits simpli ity,

and the justi ation for making this simpli ation lies in Newton's

state-ment. Although Newton was an ex eptional s ientist and mathemati ian,

there is no reason to believe that the development of, say, dierential

al- ulus would not have happened withouthim; indeed, buildingonDes artes'

workonanalyti geometry,Newton's ontemporaryGottfriedWilhelm

Leib-niz independently ame up with an equally good solution to the problem

of developing what is known as dierential and integral al ulus. To put

it in general terms, the reason an agent-less model makes sense is that the

innovations madein aso iety atagiven time seem toberelatedmu hmore

strongly tothe so iety's urrent ulturethan to itsspe i individuals. The

framework we are proposing ould be extended toin lude individual agents

and therebya ountforthe degreetowhi ha ulturalelementisestablished

or absent inso iety.

Withtheseambitionsand limitations,our models are very dierentfrom

previous work on modeling ultural evolution. Su h work has typi ally

fo- used onhow apredened set of ulturalelements may ompete and spread

inapopulation(Lave &Mar h,1975; Cavalli-Sforza&Feldman,1981;Boyd

&Ri herson,1985; Rogers,2003;Strimlingetal.,2009a). Asmallernumber

of models allowa umulationof new ulturalvariants,but only alonga

sin-gledimension(Boyd&Ri herson,1985;Henri h,2004;Eriksson etal.,2007;

van derPost & Hogeweg, 2008; Kandler & Steele, 2009). These models are

all individual-based, and the obje t is often to explore oevolution of genes

and ulture(Laland&Brown,2002;Boyd&Ri herson,2005;Mesoudietal.,

2006). Whereas oevolutionary models onsider how ulture both inuen es

and is inuen ed by geneti evolution, we are here interested in the impa t

of ulture onits own evolution.

Inthefollowingwewillgiveapre iseformulationofourtheoreti al

frame-work, and then present a series of models exploring how dierent types of

dependen ies lead to umulative ultural evolution that results in very

dif-ferent levelsof ulturaldiversity between so ieties.

2 A framework for des ribing how the past

inuen es the future in umulative ulture

It seemsto usthatalmost every partof modernhuman ulture(words,

on-stru tions and artifa ts, behaviours, so ial organization, et .) would have

been alien to prehistori humans. This illustrates that most ultural

ele-ments are not part of any ready-made human repertoire but instead ome

into existen e in a parti ular population at some parti ular point in time.

Here we will use the term appearan e for all su h events, by whi h we

mean that a ulturalelement is established in the populationto the extent

that it an inuen e further ultural hange. Thus we take a ma ros opi

els by tra king the state of ea h individual rather than the populationas a

whole, f. Strimlingetal. 2009b).

The framework we develop here has two main features: it des ribes

ul-tural evolution as (1) a sto hasti pro ess that (2) at any point in time is

partially dependent of the urrent ultural state. To say that a pro ess is

sto hasti meansthatevents o urwithsomeprobability,thus apturingthe

notion that ultural evolution is not ompletely predi table. Nor is ultural

evolution ompletely unpredi table; the more we know about the urrent

state of the world, the more a urately we an predi t oming events. In

other words, future ulturalevolution is not independent of the past.

There are many ways in whi h a urrent ultural element, say

y

, an inuen e the evolution of another ultural element, say

x

. For instan e, if

x

an be obtained as a modi ation of

y

(possibly in ombination with other ultural elements) then learly

x

is mu h more likely to appear if the pre ursor

y

ispresent; the pre ursor ould then eitherbe repla edor remain and ontributeto diversity. There are also more indire t types of inuen e,

su h as an element of general knowledge or an attitude that promotes or

inhibits the appearan e of

x

. An innovation may depend on more than just a single pre ursor and be subje t tomany indire t inuen es. This leads us

tothe notionthat ulturalelementsoftenintera t inlarger ultural systems,

whi hwewill briey explore inone of our models.

Asarst steptoward gainingageneralunderstanding ofpro essesof

u-mulationof omplex ulture, wewillignorethepre isenatureofthe

relation-ships between dierent ulturalelements and only onsider, in the abstra t,

how the probabilities of appearan es and disappearan es hange when new

elements appear or urrent elements disappear. From this perspe tive there

are only threeways inwhi h

b

an inuen e the appearan e ofan element

x

that is urrently not present:

b

may either fa ilitate or inhibit the appear-an e of

x

, or have no ee t at all, i.e., be neutral or independent. These basi inuen es are des ribed and exemplied in Table 1. Disappearan es

of ultural elements follow the same logi : if

x

already is present then its disappearan e may beeither promoted or inhibitedby

b

, or beindependent

of

b

.

The probablity of

x

appearing may of ourse depend on more than one otherelement. Indeed,itmaydepend onalargenumberof ulturalelements

as well as the urrent natural environment. To make possible a general

des riptionofsu hmultipledependen ies,wewillusethe on eptofa urrent

state of the world. We will usually use the symbol

S

to denote the urrent state of the world.

Conditional probability fun tions of appearan e and

disappear-an e. As a means to apture the totality of dependen ies of future ulture

on present ulture, we introdu e the followingpair of mathemati alobje ts:

Pr

(+x|S)

and Pr

(−x|S).

These are mathemati al fun tions that for any ultural element

x

and any state of the world

S

return the onditional probability that

x

will appear resp. disappear (within some given short time period) given that the world

is instate

S

. Wewillpresently dis uss howtomathemati allyrepresent the state of the world.

We want to emphasize that these probability fun tions an a ount not

onlyfor anydependen iesof the urrent stateofthe world butalsoforother

aspe ts thatae t appearan e anddisappearan e,su has ultural

transmis-sion and fun tionality. Forinstan e, eteris paribus,a ulturalelement that

is di ult for individuals to a quire would have a lower appearan e

prob-ability and a higher disappearan e probability; similarly, a useful ultural

element would typi ally have a higher appearan e probability and a lower

disappearan e probability than asimilar but useless element.

Starting onditions and ultural seeds. A spe ial ase to onsideris

the situationwherethereexists no ultureyet. Itseemstousthat almostall

ultural elements are su h that they an arise only in the presen e of other

ulturalelements. Inotherwords,onlyaverylimitedsetof ulturalelements

are su h that they an possibly appear from a situation where there is no

previous ulture. We will all su h elements ultural seeds. In our models

set of ultural seeds and the probabilities for their respe tive appearan es.

These onditions will obviously ae t the start of the umulative pro ess;

for instan e, higher probabilities of appearan e of ulturalseeds will lead to

faster initial a umulation. However, initial onditions may also inuen e

long term evolution, for instan e, by favouring one ulturalsystem over

an-other through path dependen e. (We willdis uss this further in the se tion

on ulturalsystems.)

Representing the state of the world. In prin iple, the variable

S

in the onditional probability fun tion should spe ify every single detail of

the world that an potentially ae t the probability of appearan e or

dis-appearan e of a ultural element

x

. This might in lude ultural fa tors as well as geneti and environmental fa tors. In pra ti e, though, we need to

spe ify modelswhere

S

apturesonlythemost relevantaspe tsofthe world. In the present paper we will only deal with ulturalstates, but it would be

just as easy to in orporate geneti or environmental inu en es (e.g., if one

wanted to model that iron-based artifa ts are more likely to appear in an

environment where iron ore is easilya essible).

To des ribe ultural states we shall use the formalism of set theory. A

state is dened as a set of elements (listed within urly bra kets), so that

for instan e

S = {y, z}

means that

S

onsists of elements

y

and

z

. States hange when new elements are added or oldelements are removed. We will

denote addition of a single element

x

to a set

S

by

S + x

. Thus, for the example above we would have

S + x = {x, y, z}

. Similarly, we will denote removal of an element by the minus sign, so that in our example we would

have

S − y = {z}

. The number of elements in a set

S

is denoted by

|S|

, so in our example we have

|S| = 2

.

An example of omplex umulative ultural evolution. As an

example of how our framework might apply in pra ti e, onsider the proof

of the Four-Colour Conje ture (4CC), a long-standing onje ture in

math-emati s stating that the regions of any map an be oloured using at most

four olours so that no two regions sharing a border have the same olour

onje ture was published by Alfred Kempe in 1879, but eleven years later

it was shown by Per y Heawood to ontain a ru ial error. The onje ture

withstoodthe ontinuedatta ksofmanymathemati iansforanother entury

untilitwasproved byKennethAppelandWolfgangHaken in1977. By

om-biningnewideasoftheirownwithanideadevelopedinthe1960sbyHeinri h

Hees h (whi hitselfbuilt onKempe's work),Appeland Haken su eeded in

redu ingthe onje turetoalargebutnitenumberof asesthatwereveried

ase-by- ase bya omputer. Thuswhen Appeland Haken nallyproved the

Four-Colour Conje ture they benetted from several a hievements by other

people. In order to des ribe some ore aspe ts of this instan e of ultural

evolutionwithin our frameworkwedene four states of the world:

S

0

=

the status of mathemati sbefore1842

S

1

= S

0

+

4CC

S

2

= S

1

+

Hees h's idea

S

3

= S

2

+

omputers

We laim that the probability of the appearan e of a orre t proof of the

4CC ought toin rease as the state of the world hanges from

S

0

to

S

3

: Pr

(

+proof

|S

0

) <

Pr

(+

proof

|S

1

) <

Pr

(

+proof

|S

2

) <

Pr

(+

proof

|S

3

)

Theinterpretationoftheseinequalitiesisasfollows. First,aninnovation(the

proof of 4CC) is more likely to happen if there is anexpli it idea that su h

an innovation may bepossible (the 4CC itself). Se ond, an innovation that

is a ombination of several parts is more likely if some part already exists

(Hees h's idea). Third, a labour-intensive innovation is more likely if there

exists adequatelabour-savingtools ( omputers).

Combinations, omponents and fa ilitators. The above example

illustrates two main ways in whi h a pre-existing ultural element

y

an inuen e the likelihood of the arrival of a new ultural element

x

. One possibility is that the new ultural element

x

is a ombination where

y

is

express that

x

is a ombination of

y

and

z

wemay write

x = y ◦ z,

where

◦

denotes the operationby whi hthe parts havebeen ombined. The above example also shows how anelement may fa ilitate the appearan e of

x

withoutitself being part of

x

. The omputers were ne essary to arry out the proof but are not a omponentof the proof itself.

Generally,we willsay thatanelement

y

fa ilitatesthe appearan e of an-otherelement

x

if,forallrelevantstatesoftheworld,itholdsthatPr

(+x|S +

y) ≥

Pr

(+x|S)

; similarly,

y

inhibits the appearan e of

x

if Pr

(+x|S + y) <

P r(+x|S)

.

Representingdependen iesbetween ulturalelementsasagraph.

Relationshipsbetween elementsare oftenee tivelyillustratedusingagraph

withverti esandedgesrespresentingelementsandrelationships,respe tively.

We will draw edges with ontinuous lines for fa ilitating relationships and

dashed lines forinhibitingrelationships;noedge meansnorelationship (i.e.,

independen e). An arrow signies a dire ted edge, whi h means that the

relationship is one-way.

ThegraphsinFigure1representsomespe i modelsthat wewill

inves-tigate. We have hosen these examples toillustrate a diversity of ases. To

explore the extent to whi h ultures tend to diverge or remain similar, we

dene the similarity of two ultural states as the proportion of all elements

present in either state that are shared by both states. See Appendix for

details.

3 Independent ultural elements

Anumberofre entmodelshavestudiedtheevolutionofindependent ultural

elements (Figure 1a; Hahn & Bentley, 2003; Bentley et al.,2004; Bentley &

Shennan, 2005;Enquistetal.,2008;Strimlingetal.,2009b). Althoughthese

elements is not fa ilitated (nor inhibited) by the existen e of other ultural

elements. In our framework this means that the probabilities of appearan e

and disappearan e of a ulturalelement are onstants:

Pr(+x|x /

∈ S) = q

app

(1)

Pr(−x|x ∈ S) = q

dis

(2)

From the point of view of our framework, this is the baseline ase where

there are noother possibleelements than the ultural seedsthemselves(i.e.,

all elementsare a essible from a ulture-less state). Throughout this paper

wedenotethe numberof ulturalseeds by

m

. Hen e, inthis model thereare only

m

elements that are at allpossible.

Let

n

t

denote the expe ted number of elements present at time

t

. The expe ted numberof elementspresent at time

t + 1

an be omputed as

n

t+1

= (1 − q

dis

)n

t

+ q

app

(m − n

t

),

(3)

where the rst term a ounts for the disappearan e of a fra tion

q

dis

of

ele-ments that exist at time

t

, and the se ond term is the expe ted number of elementsthatappear,outofthe

m−n

t

thatdonotexistattime

t

. Assuming that there are no elements at time 0, we have

n

0

= 0

and equation (3) has the unique solution

n

t

=

mq

app

q

app

+ q

dis

1 − (1 − q

app

− q

dis

)

t

.

(4)

As

t

grows, the number of elements approa hes the equilibriumvalue

n

∞

=

mq

app

q

app

+ q

dis

.

(5)

Figure 2 illustrates how the expe ted number of elements varies over

time. Although the expe ted number develops smoothlyfrom0 tothe

equi-libriumvalue,simulatedevolutionarytraje toriespresent han eu tuations

than the probability of disappearan es, then the equilibriumnumberof

ele-ments will be lose to

m

(saturation) and hen e any two ultures will tend to be omequite similar(Figure3). Indeed, assumingthat both ultures are

hara terized by the same twoparameters

q

app

≫ q

dis

, we an obtain an

ex-pli it solution. In this ase the formula for the expe ted ultural similarity

at time

t

, (14) inthe Appendix, simpliesto

Expsim

(X

t

, Y

t

) =

(n

t

/m)

2

2(n

t

/m) − (n

t

/m)

2

=

n

t

2m − n

t

,

whi hat equilibriumtakes the value

Expsim

(X

∞

, Y

∞

) =

q

app

q

app

+ 2q

dis

.

(6)

Here wesee learlythattheexpe ted ulturalsimilarityis loseto1if

q

app

≫

q

dis

. Thesamesaturationphenomenonalsoimpliesthat ulturewillberather

stati ,astypi allyalmostall

m

possibleelementswillbepresentatanygiven time on e equilibriumis rea hed.

4 Stepwise modi ation

There are fewformal studiesthat onsider ultureas more than a olle tion

of independent elements. To our knowledge, the only relatively wellstudied

ase is that of an ordered su ession of elements, representing su essive

modi ation ofanan estral element(Figure1b). Forexample,toolssu has

hammersmaybearrangedinasu essionofin reasingfun tionality(Basalla,

1988). Element 0of thesu ession would des ribe la k ofhammers, element

1 very rude hammers su h as unmodied stones, and so on. Alternatively,

a su ession an represent the stepwise development of a one-dimensional

quantitativetrait (e.g.,the lengthof aspear)whi hismost e ientatsome

parti ular value (Boyd& Ri herson,1985;Henri h,2004). Wehere onsider

the former ase for illustration. Formally, we write

x

i

for the element at position

i

, and we assume that it an appear if the pre eding element

x

i−1

Pr(+x

i

|x

i

∈ S) =

/











q

app if

x

i−1

∈ S

0

otherwise (7)

As above, ea h element has an independent probability

q

dis

of disappearing.

Inthis model,progress alongthedimensionislinearintime,provided

q

app

>

q

dis

(Figure4a). Thisistheonlypossiblepathof ulturalevolution,hen ethis

model too produ es onsiderable similarity between independently evolved

ultures (Figure3).

It is easy to modify the model so that at any given time the best of all

presentvariantshavealowerdisappearan eprobabilitythaninferiorvariants,

whi h might be more realisti . However, in simulations we have found that

su h a modi ationdoes not lead to any qualitative hange in results.

5 Dierentiation

Inmodelsof ultureasanorderedsu ession,ea helement anbeelaborated

upon along a single dimension only. Typi ally, however, ultural elements

an be modiedinmany ways. Hammers,for instan e, an bespe ializedto

servedierentpurposes, withanin rease inthediversity ofhammertypesas

wellasanin reaseinthee ien yofea htype. Su hanin reaseindiversity

an bereferred to as ultural dierentiation.

A ulturethat evolves by dierentiationof existing elements an be

rep-resented as a bran hing tree that originates from one of the ultural seeds

(Figure1 ). Atany time duringthe dierentiation pro ess,any element an

potentiallydierentiateintoone ormorenewversions. Here weassumethat

all su h dierentiation events o ur independently of ea h other with

prob-ability

q

app

. Thus, any element

x

that is not a ultural seed has a unique (dire t)prede essor,

π(x)

. Onlyelements thatare urrentlypresent an dif-ferentiate, so similar to the previous model of su essive modi ations we

Pr(+x|x /

∈ S) =











q

app if

π(x) ∈ S

0

otherwise (8)

The dieren e to the previous model is that an element may be ome the

dire t prede essor of more than one new element. In our simulations we

havesetthe numberofpotentialsu essors totwo. Underourusual

assump-tion that ea h element has an independent probability

q

dis

of disappearing,

the expe ted path of ulturalevolution isexponentialgrowth inthe number

of elements, provided that

q

app

is su iently larger than

q

dis

(see Enquist

et al. 2008 for a related model and empiri al examples of exponential

ul-tural growth)). An interesting observation is that individual runs of the

model show onsiderablevariationin the time of onset of a umulation but

thereafter grow in a quite regular manner (Figure 4b). Exa tly whi h

el-ements appear among the many possible hoi es, however, is a matter of

han e. This results in little similarity between independently evolved

ul-tures (Figure3).

6 Combinations

We now onsider a model inwhi h new ultural elements an be formed by

ombinationof existingelements. Forinstan e, afood

y

andaspi e

z

anbe physi ally ombinedtoprodu eadish

x = y ◦z

. However, by ombinationwe alsorefermorebroadly toany ulturalelementthat an ariseonlyif two (or

more) omponent elements both must be present (su h as the ombination

ofmathemati alideaswith omputerte hnologyinthe theproofofthe

four- olor onje ture, asdis ussed earlier).

For simpli ity, we here assume that only pairwise ombinations an be

formed, and that any two elements an be ombined in only one way (

y◦ =

z ◦ y

). However, we will assume that the order of su essive ombinations is ru ial for the result. For example, ooking foodstu

y

with te hnique

z

and then adding foodstu

w

results in dish

(y ◦ z) ◦ w

, whi h is in general dierentfromdish

(y ◦ w) ◦ z

obtained by rst ombiningthe rawfoodstus

y

and

w

and then ookingwith te hnique

z

.

As usual we will assume that there are

m

ultural seeds that an be invented dire tly from a ulture-less state. Any other element an only be

formed as a ombinationoftwoother elements, and an appear only ifboth

omponents are present:

Pr(+y ◦ z|y ◦ z 6= S) =











q

app if

y ∈ S

and

z ∈ S

0

otherwise (9)

As in our previous models, we assume a onstant disappearan e probability

of

q

dis

.

Culturalgrowthinthismodelisveryfast,evenfasterthantheexponential

growth we saw in the model of ultural dierentiation. The reason is that

the number of elements that an be invented by pairwise ombinationsof

n

elements is on the order of

n

2

, while the number of elements that an be

invented by dierentiation is proportional to

n

.

Figure 4 shows the expe ted number of elements, omputed as an

av-erage over 100 simulations. Similarly to the ase of dierentiation above,

in individual runs of the model growth isinitiallyerrati , but be omes very

regular after a few elements have appeared. At this stage, losses be ome

negligible ompared tothe very high numberof innovationsthat an appear

by ombiningelements.

Figure 3 illustratesthe average similarity of two independently evolving

ultures. During the initial stages of growth, expe ted similarity in reases

due to the relatively high probability of the ultures inventing the same

ultural seeds and some of the simplest ombinations. After growth pi ks

up, however, the likelihood that the two ultures invent the same omplex

ombinations is very small, hen e similarity between ultures tends to drop

qui kly toward zero.

Lastly,Figure5billustratesthegrowthin omplexity,denedasthe

num-ber of evolutionary events ( reations of ultural seeds and ombinations of

omplex-ity of a ultural seed is 1,the omplexity of

y ◦ z

is 3, et . As shown in the gure, the average omplexity in reases rapidly.

7 Cultural systems

Inallpreviousmodels, theappearan e anddisappearan eof agiven ultural

element have depended on at most one other element. In this se tion we

onsider the ulturalevolutionof systems of ulture, inthe senseof sets of

interdependent ulturalelements.

Werst onsideramodelinwhi hthe probabilitythata ulturalelement

appears depends on all elements in the ulture. Here we assume that there

is a set of

N

potential ultural elements, ea h of whi h may stand in either a fa ilitating orinhibiting relationshipwith any otherelement(as dis ussed

in se tion 2). For instan e, a te hnology for melting iron ore may fa ilitate

the appearan e of iron tools; the pra ti e of keeping an animal spe ies for

ompanionship may inhibit onsumptionof its meat.

To onstru t a simple model, let us say that an element

x

is inhibited in ultural state

S

if

S

ontains more inhibitors than fa ilitators of

x

, and assume that

x

an appear in state

S

only if itis not inhibitedin this state:

Pr(+x|S) =

(

q

app if

x

isnot inhibited in

S

0

if

x

isinhibitedin

S

(10)

We also assume, as usual, that an element has a xed probability

q

dis

of

disappearing at ea h time step. The out ome of this model depends on the

probabilitythatelementsfa ilitateorinhibitea hother. Whenfewinhibiting

dependen ies exist,most elements anappearinmost ulturalstates,

result-ing in ultures with most of the

m

possible elements are present. In turn, this results in a high level of similarity between independently evolved

ul-tures (Figure8). Whenthe probabilityof inhibitingdependen ies in reases,

ultures evolve to ontain a smaller number of elements, and onsequently

are more dierent fromea h other.

disap-fa ilitated instate

S

annotdisappear fromthat state:

Pr(−x|S) =

(

q

dis if

x

is not fa ilitatedin

S

0

if

x

is fa ilitatedin

S

(11)

An inuen e of ultural state ondisappearan e of elements may giverise to

new phenomena, among whi h rivaling systems and ombinations of

inde-pendent systems.

Rivaling systems. Figure6shows the dependen ies between 8 ultural

elementsrepresented asagraphinwhi hedgesrepresentfa ilitating

relation-ships, and absen e ofedges represents inhibitingrelationships. Inthis graph

we an identify two ultural systems:

A = {1, 2, 3, 4}

and

B = {5, 6, 7, 8}

. By this we mean that elements within

A

typi ally fa ilitate ea h other and inhibit elements outside

A

, and the same goes for

B

. Neither system, how-ever,isperfe tlyfreefrom oni t. Forinstan e,withinsystem

A

elements3 and 4inhibitea hotherand fa ilitateoutsiders 5and6,respe tively.

Never-theless, we expe t ultural evolutionto establish either system

A

or system

B

. Whi h of these two systems be omes established is a matter of han e events inthe beginningof the evolutionary pro ess.

In1000simulationsofthis example,usingtheappearan eand

disappear-an e rules inequations (10) and (11), the ultureended up in system

A

406 times andinsystem

B

594times. Thus, owingtothestronginuen e of ul-tural state onthe appearan e and disappearan e of elements, the similarity

between two independently simulated ultures was always either0 or 1.

Combinationsof independent systems. When therearealsoneutral

relationshipsbetweenelements, olle tionsofseveralsmaller ulturalsystems

an emerge and oexist independently of ea h other. An example isgiven in

Figure 7. Here, there are four identiable systems:

C = {1, 3}

,

D = {2, 4}

,

E = {5, 7}

and

F = {6, 8}

. Systems

C

and

D

are mutually ex lusive, as are systems

E

and

F

. However, the rst two systems are independent of the se ondtwo. Inthissituationwethereforeexpe t anyof thefourpossible

Insummary, omplexwebsofpositiveandnegativedependen iesbetween

potential ultural elements will give rise to emergent ultural systems or

olle tions ofsystems. Chara teristi s ofsu h systemsare that: (a) they are

highly path-dependent, so that dierent ultural groups may develop very

dierent ultural systems despite the same initialpotentialfor ulture, and

(b)thesystems onsistofelementsthatareonthewholemutuallysupporting

butwheresome oni tbetween elementsmaybeunavoidable(e.g. elements

3 and 4 insystem A in Figure6).

8 Dis ussion

Inthis paperwehavedeveloped atheoreti alframeworkforexploring

umu-lative ulturalevolution. Itisbasedonthe simpleideathatexisting ultural

elements an fa ilitate or inhibit the appearan e of new elements as well as

the disappearan e of present elements (see Table 1). With many ultural

elements, the set of interdependen ies an be ome arbitrarily omplex (e.g.,

norms about what an be eaten an inuen e farming, breeding, household

pra ti es,et .,and anbeinuen edbyreligionandothertraditions),sothat

evolution an onlybe understood if the whole system is studiedtogether.

Bydes ribing how dierent ultural states inuen e the appearan e and

disappearan eof ulturalelementswe anexplore long-term umulative

ul-tural evolution as asu ession of appearan e and disappearan e events. By

studying a series of dierent s enarios, we have shown in this paper that

the nature of dependen ies between ultural elements has dramati ee ts

on the pattern of ultural evolution. For example, when any given element

fa ilitatesthe appearan e of several similarelements weobserve apro ess of

ulturaldierentiationinwhi h thenumberofelementsgrows exponentially

in time. In ontrast, if elements an vary only along a single dimension we

observe linear growth. We stress that both the des ription of dependen ies

between elements and the pro ess of ultural evolution that emerges from

su h dependen ies lend themselves naturally to mathemati al formulation.

Belowwe rst dis uss ourresults onthe evolutionof ulturaldiversity, then

we dis uss some open issues.

8.1 The evolution of ultural diversity

The expression  ultural diversity an referto several phenomena. Thus in

developinga theoryof ulturaldiversity we an askmanydistin tquestions,

su h as:

1. Why dodierent ultures exist, andwhat determineshowmany

dier-ent ultures there are?

2. Given that distin t ultures exist, what governs their similarities and

dieren es?

3. What determines the numberof ultural elements withina ulture?

4. What determinesthe diversityof ulturalelementswithina ulture, in

terms of similaritiesand dieren es between the elements?

5. What determines the extent to whi h individuals within a ultural

group arry the same ordierent ultural elements?

Here we have mainly onsidered questions 2 and 3. Our analysis has

high-lighted a numberof fa tors, summarized in Table 2, that inuen e both the

numberof elementsin a ulture and the extent to whi h two independently

evolved ultures share ommon elements. The rst fa tor is the number of

bran hing possibilities, that is, the possibilities to reate new ultural

ele-ments from existing ones (e.g, by dierentiation or ombination). If there

areplentysu hpossibilities,wehaveseenthat ulturestendtobe omelarger

and less similar to ea h other (Figures 3 and 5). Figure 4 also shows that,

whenmanyinnovationsarepossible, ulturalelementsa umulatelonger

his-tories, i.e., they arise from many evolutionary steps. Table 2 alsopoints to

the omplementaryinuen esof fa ilitationand inhibitionon ultural

elements eventuallyappear indierent ultures, even if the ultures initially

ontain dierent elements. Mutual inhibition in reases ultural dieren es

be ausedierent ultures mayestablishdierentsubsetsof mutually

in om-patible elements (Figure8).

Athirdsour eof ulturaldiversityis han e. Be ausetheappearan eand

disappearan e ofelements have sto hasti omponents,wegenerallyobserve

random variationboth inthe time of appearan e of spe i elementsand in

whatelementsa tuallyappear. Someee tsof han e anbeappre iatedby

ontrastingaveragepathsof ulturalevolutionwithsinglesimulationruns in

Figure4. Whenmanypossibilitiesforinnovationexist,andinthepresen eof

inhibitory dependen ies between elements, han e is parti ularly important

in hoosing whi h of the many possible paths a parti ular ulture a tually

takes. Thismeansthatanytwo ulturesareunlikelytotakeexa tlythesame

path (multilinearevolutionCarneiro,2003). Even inthepresen e of random

fa tors, however, there an be surprising regularities. One example is the

regular growth inamountof ulture inthe model of ulturaldierentiation,

reminis ent of a a steady rate of geneti hange in geneti evolution, the

mole ular lo k metaphor (Futuyma,1998).

It is possible to extend our framework to address questions 4, 5, and 1

above. Tota klequestion4weneedtodeneameasureofsimilaritybetween

ultural elements, e.g., similarity in fun tion, appearan e, or history.

Simi-larity measures based ondierent riteria may sometimes agree,though not

always ( f.analogy and homologyingeneti evolution,Futuyma,1998). For

instan e, two hammers that are derived fromthe same, preexisting hammer

will often be similar in fun tion, appearan e, and, of ourse, history. On e

a measure of similarity between ultural elements is dened, it is possible

to use it within our framework to study the similarity of evolved ultural

elements. Whether this an be a fruitful line of resear h is a question for

future work.

Question 5 on erns how ulture is distributed among individuals in a

group. Although we have not onsidered how individuals arry ulture, it

fa t that ea h individual has limited memory). Thus onditions that favor

a large ulture are also expe ted to foster within-group ultural diversity,

i.e., spe ialization. We also point out that spe ialization itself may favor

ulturalgrowth,be auseifindividualsdonotneedto arryallofthegroup's

ulture they may have more resour es to reate new ulture within their

spe ialization.

Question 1 is akin to asking how dierent biologi alspe ies evolve, and

what determines their number and abundan e. Cultural diversity in this

sense, therefore,may bequantied usingmeasures ofbiodiversity(Purvis &

He tor, 2000). A true understanding of howsu h ultural diversity evolves,

however, requires signi anttheoreti al developments, whi h lie beyond our

present s ope. Note that Questions 1 and 5, while seemingly at opposite

ends of a spe trum rangingindividuals to ultures, may a tually be stri tly

interrelated. The reason is that a omplete understanding of how distin t

ultures emergerequires understanding how ulturaldieren es develop

be-tween individuals. Addressing questions1and 5requiresa renementof our

frameworkinwhi h theappearan e anddisappearan eof traitsistra ked at

the levelof individuals(Strimlingetal.,2009b; A erbi etal., 2009).

8.2 Causes of the appearan e and disappearan e of

ul-tural elements

We have assumed in our models that ultural elements appear and

disap-pear solely based ontheir dependen ies onother elements. In reality, many

other fa tors ontribute. For instan e, the appearan e and disappearan e of

a ultural element are inuen ed by its fun tionality. For example, many

ombinations of ultural elements seem unlikely to appear simply be ause

they an serve no fun tion (think of the possible ombinations of pasta,

tomato sau e, hammer, and  omputer). Introdu ing su h ultural

se-le tion based on fun tion may or may not hange the general patterns of

growth analyzed above. Consider for instan e the model of ultural

determinant of element appearan e and disappearan e, we expe t evolved

ultures to onsist mostly of the few e ient elements; we nolonger expe t

the numberofelementstogrowlinearlyintime on ethefun tionalelements

have appeared(Boyd &Ri herson, 1985). Some of our results, however,

ap-pearmorerobust. Forexample, onsiderthemodelof ulturaldierentiation

and suppose that only a fra tion of the elements that an be derived from

any given element is fun tional. We still expe t an exponential in rease in

the number of elements, albeit at a slower rate. Thus a system in whi h

ultural elements an dierentiate would still produ e more diversity than

a system that develops along one dimension only, and less diversity than a

system inwhi h ultural elements an ombine.

Wehavealsoleftouttheee tofenvironmentalvariationandgeneti

fa -torson ulturalevolution. Theenvironment anbein orporatedinthestate

of the world so that,for example, a ultural element maybemore likely to

appearinoneenvironmentthaninanother. Similarly,geneti predispositions

may inuen e appearan e and disappearan e probabilities. Forinstan e,

fa- ilitationandinhibitionbetween ulturalelements ouldbeviewedas

ree t-ing the impa tof evolved mental stru ture. An extension of our framework

toindividualswouldenablestudyofthe interplaybetweenindividualgeneti

variationand umulative ultural evolution.

8.3 Cultural omplexity

The greatest hallenge in studying the evolution of ultural diversity lies

perhaps in the omplexity of ultural systems. We have only tou hed upon

this topi in our last model, but our framework an over a wider range of

ases wherepro essesofrenement,dierentiation, ombination,fa ilitation

and inhibition,whi hwehave studiedseparately,o ur simultaneously.

We believe that an advantage of our approa h in the study of omplex

ulture is a stronger fo us on reativity and ultural history, ompared to

most urrent theory whi h emphasize so ial learning as the main for e in

Ri h-maintained in time (in luding why some elements may be more easily

re-tained), but the most spe ta ular feature of human ulturalevolutionis the

open-ended pro ess of reationof novel, often in reasingly omplex ulture.

Although individual reativity has been the subje t of mu h investigation

(Simonton, 2004; Sternberg, 2000), very little is learly understood about

how reativity shapes long-term ulturalevolution. We believea framework

like ours is helpful, possibly even ne essary, for real progress tobe made on

this topi .

There are many steps left to be taken, for whi h our framework an

be a starting point. For instan e, the issue of the onsequen es of human

intentionality ould be explored through studies of the intera tion between

dierent kinds of ultural elements, su h as ideas (about what is possible),

opinions (about what isimportant),and goals (for what toa hieve).

Another obvious route to go is to in orporate more ne-grained aspe ts

aboutthepopulationtomakeitpossibletodealwithissueslikespe ialization

and sub ultures withingroups, and intera tions between ulturalgroups.

Ourmodelsalsopointtotheimportan eofwhatwehave alled ultural

seeds, i.e., ultural elements that an appear in the absen e of preexisting

ulture. It may very well be the ase that there does not exist a very large

set of ultural elements that are all essentially independent of ea h other

and that an evolve from a situation without any ulture. Theoreti al and

empiri alexplorationsof this issueare, toour knowledge,extremely limited,

withthepossibleex eptionofideaswithinstru turalanthropology(Barnard,

2000).

8.4 Con lusion

We sought to apture in a lear formal framework what we believe is the

essen e of ulturala umulation: the unlimitedpotentialforinnovation and

the omplex dependen ies between ultural elements. Our approa h oers,

to our knowledge for the rst time, a way to model at least some of the

a -however, ourapproa hmustbe onne tedprodu tivelytoempiri al

observa-tions of ultural dependen ies, ultural evolution, and ultural history. We

believe this is possible through investigations of a tual traje tories of

ul-tural evolution and studies of relationships between ultural elements. As

an example of empiri aldata that are relevant here, it has been shown that

the number of ultural elements in some domains has grown exponentially

(Enquistetal.,2008),suggestingthatdierentiationhasbeenamajor

under-lyingpro ess in these ases. One example of empiri alstudieswe would like

to see done isanalyses of absen e of parti ular elements in ultural systems

in terms of presen e of inhibitingelements.

There has so far been little ommon ground between mathemati al

the-oryof ultural hangeandmainstreamworkon ultural hangein

anthropol-ogy and other so ials ien es (Carneiro, 2003). Our framework may help to

strengthen the onne tion, as the evolving ultural systems presented here

ould be used to model many existing notions within the human s ien es

(e.g., withinthe eldsofethni ity,sexand gender,so ialnorms,world views

and subsisten esystems)abouthowvariousideasand pra ti esmaysupport

or be in oni t with ea h other.

A knowledgments

We thank Kevin Laland for many insightful omments. Work supported by

European Commission grant FP6-2004-NEST-043434(CULTAPTATION).

A Appendix

To quantify ultural diversity we dene the similarity of two ultural states

as the proportion of all elements present in either state that are shared by

both states. Formally, if

X

and

Y

are the sets of elements representing the twostates,andassumingthat atleast oneisnot empty,then theirsimilarity

s(X, Y ) =

|X ∩ Y |

|X ∪ Y |

= Pr(x ∈ X ∩ Y |x ∈ X ∪ Y ),

(12) where

x

is a random element drawn uniformly from the set of all possible elements (assumed to be nite). If the ulturalstates

X

and

Y

arise froma sto hasti pro ess, theyarethemselvesrandomvariables,and wedenetheir

expe ted similarity by Expsim

(X, Y ) =

E

(s(X, Y )|X ∪ Y 6= ∅).

Then we have Expsim

(X, Y ) = Pr(x ∈ X ∩ Y |x ∈ X ∪ Y ) =

Pr(x ∈ X ∩ Y )

Pr(x ∈ X ∪ Y )

.

(13) Assuming that states

X

and

Y

have evolved independently of ea h other, the last expression an berewritten as

Expsim

(X, Y ) =

Pr(x ∈ X) Pr(x ∈ Y )

Pr(x ∈ X) + Pr(x ∈ Y ) − Pr(x ∈ X) Pr(x ∈ Y )

.

(14) Thus we an al ulate the expe ted similarity between ultures if we know

the probabilitythat an element ispart of a ulturalstate.

Referen es

A erbi,A.,Ghirlanda,S.&Enquist, M.2009Culturalevolutionand

individ-ualdevelopmentofopennessand onservatism.Pro eedingsoftheNational

A ademy of S ien es of the USA106, 1893118935.

Barnard, A.2000Historyand theoryin anthropology. Cambridge,UK:

Cam-brdige University Press.

Basalla,G.1988 Evolution OfTe hnology. Cambridge,UK: Cambridge

hange. Pro eedingsRoyal So iety London B 271, 14431450.

Bentley,R.A.&Shennan,S.J.2005Random opyingand ulturalevolution.

S ien e 309, 877879.

Boyd,R. &Ri herson,P.2005 Theorigin and evolution of ultures. Oxford:

OxfordUniversity Press.

Boyd, R. & Ri herson, P. J. 1985 Culture and the evolutionary pro ess.

Chi ago: University of Chi ago Press.

Boyd, R. & Ri herson, P. J. 1996 Why ulture is ommon, but ultural

evolution is rare. Pro eedingsof the British A ademy 88, 7793.

Carneiro, R. L. 2003 Evolutionism in ultural anthropology. Boulder:

West-view.

Cavalli-Sforza, L. & Feldman, M. 1981 Cultural transmission and evolution.

Prin eton, NJ: Prin eton university press.

Enquist,M.,Ghirlanda,S.,Jarri k,A.&Wa htmeister,C.A.2008Whydoes

human ulturein rease exponentially? Theoreti al PopulationBiology 74,

4655. Unpublished manus ript.

Eriksson, K., Enquist, M. & Ghirlanda, S. 2007 Criti al points in urrent

theory of onformist so ial learning. Journal of Evolutionary Psy hology

5,6788.

Futuyma, D. J. 1998 Evolutionary biology. Sunderland, Massa husetts:

Sin-auer Asso iates, In .

Galtung, J. & Inayatullah,S., eds. 1997 Ma rohistory and Ma rohistorians.

London: Praeger.

Goudsblom, J., Jones, E. & Mennell, S., eds. 1996 The Course of Human

example from baby names. Pro eedings Royal So iety London B (Suppl.)

270, 00.

Harris, M. 2001 Cultural materialism: The struggle for a s ien e of ulture.

Walnut Creek: AltamiraPress, 2nd edn.

Henri h,J.2004 Demographyand ulturalevolution: Howadaptive ultural

pro esses anprodu emaladaptivelossestheTasmanian ase. Ameri an

Antiquity 69(2), 197214.

Kandler, A.&Steele,J.2009 Innovationdiusionintime andspa e: Ee ts

of so ialinformationand of in omeinequality. diusion-fundamentals.org

11(3), 1117.

Laland, K. N. & Brown, G. R. 2002 Sense and nonsense. Evolutionary

per-spe tives on human behaviour. Oxford: Oxford University Press.

Lave, C. W. & Mar h, J. G. 1975 An introdu tion to models in the so ial

s ien es. New York: Harper and Row.

Levi-Strauss, C. 1963 Stru tural anthropology. London: An hor Books.

Mesoudi, A., Whiten,A. & Laland,K.N. 2006 Towards a unieds ien e of

ulturalevolution. Behavioral and Brain S ien es 29, 329383.

Mokyr,J.1990Twenty-ve enturiesof ultural hange. London: Routledge.

Piaget, J. 1970 Stru turalism. London: Harper &Row.

Purvis, A. & He tor, A. 2000 Getting the measure of biodiversity. Nature

405, 212219.

Renfrew, C. 1972 The emergen e of ivilisation: The Cy lades and the

Aegean in the third millennium B.C. London: Methuen.

Rogers, E. M. 2003 Diusion of innovations. Tampa, FL: Free Press, 5th

Simonton, D. K. 2004 Creativity in s ien e. Cambridge, UK: Cambridge

University Press.

Southern, R.W.1952Themakingof the MiddleAges. Yale: YaleUniversity

Press.

Sternberg, R.J.2000 Handbook of intelligen e. Cambridge: Cambridge

Uni-versity Press.

Strimling, P., Eriksson, K. & Enquist, M. 2009a Repeated learning makes

ulturalevolutionunique. Pro eedingsoftheNationalA ademyof S ien es

of the USA0.

Strimling,P.,Sjöstrand,J.,Enquist, M.&Eriksson, K.2009b A umulation

of independent ultural traits. Theoreti al Population Biology 76, 7783.

Tomasello, M. 1994 Cultural transmission in the tool use and

ommuni a-torysignaling of himpanzees? In &quot;Language&quot; and intelligen e

in monkeys and apes (eds. S. Taylor Parker & K. Gibson), pp. 274311.

Cambridge: CambridgeUniversity Pres.

Tomasello, M. 1999 The Cultural Origins of Human Cognition. London:

Harvard University Press.

van derPost, D. J. & Hogeweg, P. 2008 Diet traditionsand umulative

ul-tural pro esses asside-ee ts of grouping. Animal Behavior 75, 133144.

White, L. 1959 The evolution of ulture. New York: M Graw-Hill.

Table 1: Kinds of dependen ies of a ulturalelement,

x

, uponanother ulu-tral element,

y

. Dependen e Histories Examples Fa ilitation

S

0

y

x

x

y

isatool,material,orknowledge ne essaryto reate

x

x

isamodi ationof

y

x

isa ombinationof

y

andanother element(e.g., theharpoon ombines spearand rope)

y

isaso ialinstitution thatpromotes

x

y

isate hnologythatmakes

x

heaper

Neutral

S

0

y

x

x

y

iswhollyunrelatedto

x

Inhibition

S

0

y

x

x

y

isataboothatforbids

x

y

isanalternativeto

x

,e.g.,asolution tothesameproblem

Symbols:

x, y

: ulturalelements;

S

0

: ulturestatewithout

x

and

y

;thi kerlines indi- atehigherprobabilityoftransition.

Fa tor

ultural elements between ultures

Bran hing possibilities

+

+

Fa ilitation

+

−

Inhibition

−

+

Table 2: Ee t of several fa tors onthe number of ultural elements and on the dieren es between ultures.

d

e

c

a

b

Figure 1: Examples of dependen ies between ultural elements: a) indepen-dentelements;b)linearsu essionof elements; )dierentiationofelements; d)pairwise ombinationsofelements;e)systemsof ulturalelements(empty arrows representinhibitoryrelationships). Theopen ir lerepresentsastate in whi h no ulture is present.

0 50 100 150 0 20 40 60 80 100 Independent elements P r o p o r t i o n o f e l e m e n t s ( % ) Generations

Figure 2: The expe ted path of ultural evolution (equation 4) when all possible ulturalelementsappearanddisappear independentlyofea hother. Parameter values:

q

app

= 0.05

,

q

dis

= 0.01

.

0 50 100 150 0.0 0.2 0.4 0.6 0.8 1.0 Independent elements Modification Differentiation Combinations Cultural system S i m i l a r i t y Generations

Figure 3: Average similarity between two independently evolved ultures a ording to models in the text. To allow omparison between dierent models, we hose a number

m = 2

of ultural seeds in all ases, with the ex eption of the model of ultural systems in whi h we onsidered

M =

100

possible elements. Similarity is al ulated analyti ally in the ase of independent elements (see Appendix), and as an average of the similarity observedinpairsofindependentsimulationsintheother ases(modi ations: 500 simulations; dierentiation, ombinations, and ultural systems: 100 simulations). Parameter values:

q

app

= 0.05

,

q

dis

= 0.01

0 50 100 150 1 10 100 0 50 100 150 0 2 4 6 0 50 100 150 1 10 100 1000 N u m b e r o f e l e m e n t s Generations Differentiation Average Exampl es Average Exampl e Modification Combinations Medi an Exampl es

Figure 4: The expe ted path of ultural evolution, together with sample paths from individual simulations, for ea h of three models of umulative ultural evolution. Parameter values for allmodels:

q

app

= 0.05

,

q

dis

= 0.01

. a) Cultural evolution by su essive modi ations of elements. The number of seeds is

m = 1

. The average path is omputed over 500 simulations. b) Cultural evolution by dierentiation of elements. The number of seeds is

m = 2

. Ea h element an dierentiate into

2

elements. The average path is omputed over 100 simulations. ) Cultural evolution by pairwise ombinationsof elements. The numberof seeds is

m = 2

. The median path is al ulated over100 simulations.

0 50 100 150 0 2 4 6 8 10 12 0 50 100 150 1 10 100 1000 Independent Modification Differentiation Com bination

b

H i st o r y Independent Modification Differentiation Combination Generations

a

N u m b e r o f e l e m e n t s

Figure 5: Comparison between the models of umulative ulture dis ussed in the text. a) The number of elements in the ulture (the verti al axis is logarithmi ). b)Average history of elements. The history of anelementis the numberof evolutionaryevents that reated the element, startingfroma ultural seed. A value of 1means that the element is a ultural seed, whi h evolved independently of other elements. Parameter values asin Figure4.

4

₆

8

7

5

3

1

2

B

A

Figure 6: Example of a system of relationships between eight ultural ele-ments. Edges indi atefa ilitation,missingedges inhibition. We an identify two ulturalsystemsA andB, i.e., sets of ulturalelementswhi h,typi ally, fa ilitate ea hother and inhibit elements outsidethe set.

5

C

3

1

2

4

D

8

6

F

7

E

Figure 7: A system of relationships between eight ultural elements. Edges with lled arrows indi ate fa ilitation, edges with empty arrows indi ate inhibition,and missing edges indi ateneutral relationships. The sets C and D are mutually ex lusive (elements in C inhibit elements in D, and vi e-versa), asare systems E andF. Sets C and D are, however, ompatible with sets Eand F, asonly neutralrelationships exists.

0.0 0.2 0.4 0.6 0.8 1.0 0 20 40 60 80 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Proportion of inhibitory dependencies b N u m b e r o f e l e m e n t s Independent elemens t Independent elemens t a S i m i l a r i t y

Figure 8: Simulation of ultural evolution when appearan e of a ultural element depends onthe number of fa ilitatingvs. inhibitingelementsin the urrent ultural state (equation 10 with

q

dis

= 0.05

), and disappearan e of element is randomwith probability

q

dis

= 0.01

). Left: similarityof indepen-dently evolved ulture as a fun tion of the probability that the relationship betweenany twoelementsisinhibitingvs.fa ilitating. Right: sizeof evolved ultures underthe same onditions. A numberof

m = 100

possibleelements was onsidered.