Table Of Content

Published as: Physica A 348, 611-629 (2005) A renormalization group theory of cultural evolution Gábor Fáth1,2 and Miklos Sarvary2 1Research Institute for Solid State Physics and Optics, P.O. Box 49, H-1525 Budapest, Hungary 2INSEAD, Boulevard de Constance, 77305, Fontainebleau, France Wepresentatheoryofculturalevolutionbaseduponarenormalizationgroupscheme. Weconsider rational but cognitively limited agents who optimize their decision making process by iteratively 5 updating and refining the mental representation of their natural and social environment. These 0 representations are built around the most important degrees of freedom of their world. Cultural 0 coherence among agents is defined as the overlap of mental representations and is characterized 2 using an adequate order parameter. As the importance of social interactions increases or agents become more intelligent, we observe and quantify a series of dynamic phase transitions by which n cultural coherence advances in the society. A similar phase transition may explain the so-called a “cultural explosion” in human evolution some 50,000 years ago. J 0 PACSnumbers: 87.23.Ge, 89.65.-s,05.45.-a,05.10.Cc 2 ] O I. INTRODUCTION gates an initially random population of biologically and economicallyuniformagents. Socialinteractionislimited A to an imitation process in which agents adapt cultural Culture is the sum of knowledge, beliefs, values and . n behavioral patterns built up by a group of human be- traits stochastically from each other with a bias toward i already similar agents. The interesting question is un- l ings and transmitted from one generation to the next. n derwhatcircumstancesglobalculturaldiversitygetspre- Culturalevolutionfor Homo sapiens startedearlyin the [ serveddespitethehomogenizinginteraction[8,9,10,11]. prehistoric age, and as archeological evidence suggests However, the question can also be posed from the oppo- 2 it has gone through a number of short episodes during v whichitsprevalenceandoverallinfluenceoneverydaylife site direction: in a population with non-negligible het- 0 erogeneity and under social interactions which are not increased abruptly. The most important of these events 7 necessarily imitational but rather reflect selfish individ- occurredapproximately50,000yearsago,giving birthto 0 ual interests how can a coherent culture emerge? art, music, religion and warfare. This episode is usually 2 1 termed the “cultural explosion” [1, 2]. In orderto study this question, ourmodel differs from 3 Theevolutionofhumancultureisinvestigatedinmany thatofAxelrodintwoways. First,weexplicitlyconsider 0 disciplines using different paradigms. Economics uses individualdecisionmakingmechanisms andidentify “di- n/ game theory to understand the developmentof rules,so- mensionreduction”asafundamentalheuristicforcogni- i cialnormsandotherculturalinstitutionsassumingratio- tively limited but otherwise rational agents. This is one nl nal behavior of the constituting individuals (agents) [3]. ofthepossibleimplementationsoftheconceptofbounded : Cognitive and behavioral sciences study how culture is rationality well-known in economics and social sciences v representedinthemind[4],andhowthese“mentalrepre- [3, 12]. Specifically, agents are constrained to describe i X sentations” change under social interactions [5]. Physics andunderstandtheirworldalongafinitenumberof“con- r views culture and its evolution as a complex, dynamic cepts”(Concepts). ThetotalityoftheseConceptsconsti- a system, and aims to identify its basic universal proper- tutestheagent’sculturalprofile. TheConceptsestablish ties using simple minimal models [6, 7, 8, 9, 10, 11]. an interface to the objective world: agents use them to The emergence of culture seems to involve a chicken- build an adequate mental representation of natural and and-eggproblem: culture becomeswhattheconstituting social reality for evaluating decision alternatives. Obvi- agentsmakeittobe,andagents,basedontheircapacity ously, decision making is successful if evaluation is pre- of learning, constantly adapt to the culture they happen cise, thus rational agents face the problem of optimizing tolivein. Inthissensecultureisthedynamicattractorof theirConceptsgiventheactualstateoftheworld. Innate the complex social dynamics, determined by the agents’ oracquiredheterogeneityinindividualpreferences,how- actual constraints and social interactions. The aim of ever,cangiverisetosubstantialdeviationsintheseopti- thispaperistoinvestigatethepropertiesofsuchcultural mal Concepts across members of the population. People attractors in an adequately formulated social dynamics with such differences in cultural profile end up having model, which integrates the relevant concepts from the unequal views of the world, they evaluate alternatives social sciences with those from the physics perspective. differently, and make different choices accordingly [28]. One of the popular minimal models of cultural evolu- Second, instead of a simple imitation process our tion is that of Axelrod [6, 7]. Axelrod’s model investi- agents are assumed to interact in a competitive game 2 [13]. We think of the Concepts as flexible mental con- structs which are continuously updated and refined ac- … p (x)(a…) context1 contextx contextX Payoffs cording to a “best response dynamics” in order to max- imize the agents’ predictive power in the actual natural v(x) m and social environment. As such, our model treats the evolutionof culturalprofiles (the set ofConcepts) as the result of rational behavior governed by economic bene- 1 2 m K Concepts g ggg fit. The Concepts are continuous variables and agents m are assumed heterogeneous in their fixed preferences by w www (x) default. Conceptually, our theory will be a renormalization a = (a, a, …, a ) Physical Attributes 1 2 D group theory as it is based on the following two key el- ements: (i) it combines the original state variables (attributes of decision alternatives) into important and less important degrees of freedom; agents keep the former FIG. 1: Decision making heuristic using Concepts as renormalized degrees of freedom. (defined to be the Concepts), and discard the latter; (ii) this is done in an iterative way taking into account the otheragents’profilesandaimingtoidentifyafixedpoint built around the world’s K “most important degrees of (associated with Culture). The criterion for truncating freedom”,constitutingthe agent’sConcepts. We assume the degrees offreedom is local: eachagenttries to maxi- again that the approximate payoff that the agent “com- mize its evaluation accuracy, i.e., to minimize the repre- putes” directly is linear (“linear representation assump- sentation/evaluation error within its cognitive limits. tion”) π˜(x)(a)=ω˜(x) a (2) II. A COGNITIVE MODEL i i · with In order to define a simple, tractable model, we will K introduce two simplifying assumptions: we will assume ω˜(ix) = vi(µx)γiµ, (3) that the objective world in which the agents live is lin- µ=1 X ear, and that the subjective heuristics that they use to theagent’sapproximate preferencevector(thattheagent predict their environment is also linear, in a sense to be knows explicitly). ω˜(x) is decomposed using mental made precise below. Clearly,these areimportantsimpli- i ficationsasboththerealworldandtherealmentalmod- weights v(x) in a reduced subspace of dimension K. The iµ els are likely to exhibit substantial non-linearities. For a weight v(x) reflects the significance of Concept µ in con- iµ first analysis, however,this approximation is sufficient. text x. Equation (2) implies that agent i possesses a We consider I agents, each restricted to use a num- number K of concept vectors, γ K , assumed nor- ber K of Concepts only. Agents divide the world into malized γ = 1, which the ag{enitµ}uµs=es1to evaluate al- a number X of contexts in which they evaluate decision | iµ| ternatives. We emphasize that the Concepts are defined alternatives according to their personal preferences. We in a context-independent way, i.e, they have the same assume that alternatives are characterized by their ob- “meaning” in all occurring contexts. jective (physical) attributes a = a ,...,a . Agent i’s 1 D The structure of our model is depicted in Fig. 1. { } theoretical payoff from choosing alternative a in context We have defined three layers: the layer of “Physical At- xispositedtobealinearfunction(“linearworldassump- tributes” of the decision alternatives, the layer of Con- tion”) cepts (defined by concept vectors) which constitute the renormalizeddegreesoffreedom,andthelayerofcontext- π(x)(a)=ω(x) a, (1) and agent-specific “Payoffs”. At the lowest level agents i i · are homogeneous and have identical information about where ω(x) is the agent’s preference vector, characteriz- alternatives. In the highest layer agents are heteroge- i ingtheagent’spersonal(biologicalorotherwiseacquired) neousandhaveindividualpayofffunctionsbasedontheir fixed preference in context x. For each agent there are individual preferences. The middle layer shows partial X preference vectors each of dimension D, which are as- coherence, whose measure, as we will see, is determined sumed fixed in the model. bythestrengthofsocialinteractions. Thisisthelayerwe An agent does not know his preference vectors explic- intend to monitor across society for observing the emer- itly as this would require a detailed understanding of gence and evolution of a common Culture. the effect of all attributes on his payoffs. However, by An example may be enlightening at this point. Think collecting experience on choices he has made previously, about the world of chess where decision alternatives are he learns to approximate the payoffs using an appropri- the possible moves a player can choose in a given situ- ate mental representation. The mental representation is ation. These moves are characterized by the position of 3 thepiecesontheboard(physicalattributes). Calculating Theagents’goalistominimizetheerrorE byoptimally i thepayoffofamove(theprobabilityofwinningthegame choosing their concept vectors γ and mental weights iν withthatmove)is basicallyatwostepprocedurefor hu- v(x). Recall that ω(x) is assumed fixed in the model. iν i man players and chess programs alike. First the move is evaluated along some general Concepts like “material advantage”,“positionaladvantage”,“pinned piece”,etc. III. OPTIMAL CONCEPTS AS A PCA Indeed, research in psychology has demonstrated that a PROBLEM human grandmaster possessestens ofthousands of such chess-related concepts [14, 15]. Second, these concept Theoreticallythe minimizing parameterscanbe easily scores are weighted (mental weights) in an appropriate determined. Differentiatingw.r.t.v(x) andassumingthat mental model. The weighting scheme depends on the iµ the K K metric tensor Γ = γ γ is invertible player’s personal skills/style (agent heterogeneity) and × iνµ iν · iµ (note that the Concepts are not necessarily orthogonal), on the actual adversary (context dependence). There the optimal mental weights read is a theoretically best move (preference vector) for that player with that adversary in that situation, but that v(x) =ω(x) γµ, (7) is not known explicitly to the player. Mastering chess iµ i · i amounts to finding an optimal mental representation, where we have introduced the “dual” concept vectors i.e.,optimalConceptsandoptimalweights(approximate γµ = K [Γ−1] γ . Writing this back to Eq. (6) i ν=1 i µν iν preference vectors) which allow a decision close to the we can write the error as theoretical best. In chess there are different “schools” P (cultures) where the applied Concepts have somewhat X X K E = ω(x) 2 (ω(x) γµ)(ω(x) γ ). (8) different meaning. Of course, decision in chess is very i | i | − i · i i · iµ non-linear,andoursimplifiedlinearmodelcanonlygrasp x=1 x=1µ=1 X XX the big picture. In this way the representationerror is written as a func- The hierarchical structure depicted in Fig. 1 formally tion of the γ concept vectors only. Conceptually this is resembles a linear two-layer (concept vectors, mental equivalent to the reasonable assumption that the men- weights) neural network. Note, however, that it is not talweightsaccommodatemuchfasterthanthe Concepts a microscopic neural network model of a cognitive func- themselves. tion, but a phenomenological (macroscopic) model of a Neglectingthefirstterm,whichisanunimportantcon- generic decision making strategy. stant, we define the agent’s utility as the negative of his Under this cognitive architecture the number of vari- overallrepresentation error ables defining an agent is (X + D)K, much less than the total number of parameters describing the world, X K XD. However, there is a price that should be paid for U = (ω(x) γµ)(ω(x) γ ), (9) i i · i i · iµ bounded rationality: due to the reduction of dimension- x=1µ=1 XX ality, K <D, the approximate payoff π˜(x) deviates from thetheoreticalpayoffπ(x). Theagents’gioalistofindthe which should now be maximized for {γiν}Kν=1. The util- i ity can be written in a more compact form best possible set of Concepts and mental weights which minimize the error of the mental representation under K the constraint that only K Concepts can be used. The U = γµ W γ . (10) natural measure of agent i’s representation error is the i i · i iµ µ=1 X variance by introducing a positive semi-definite, D D dimen- X × E = (π(x) π˜(x))2 , (4) sional matrix, to be called the world matrix, i i − i x Xx=1D E X where h.ix is the averageover alternatives in context x. Wi = ω(ix)◦ω(ix), (11) We restrict our attention to the case, where the at- x=1 X tributes are delta correlated which encompasses all information about agent i’s (ob- had(x)ix =0, ha(dx)a(dx′)ix =δdd′ (5) jective) personal relationship to the world. The maximization problem in Eq. (10) is the well- and nontrivial structure is assumed in the agent-specific known Principal Component Analysis (PCA) problem. preference vectors only. This simplifies the forthcoming According to this, a particular solution for the optimal analysiswithout losingessentialfeatures. With this pro- Concept vectors is provided by the K most significant viso, the representation error turns out to be (largesteigenvalue)eigenvectorsof W . Thus to achieve i 2 the best possible mentalrepresentationthe agentshould X K E = ω(x) v(x)γ . (6) choose his concept vectors according to the eigenvectors i i − iν iν! ofhis worldmatrixinthe orderoftheir significance,and x=1 ν=1 X X 4 then adapt his mental weights according to Eq. (7). No- pair interactions will be considered. In social contexts tice, however, that this solution is not unique: any non- the understandingofanotheragent’svaluationanddeci- singular basis transformation in the subspace of the K sionmakingmechanismisanasset. Agentihasstrategic most significant eigenvectors yields a different solution advantage from being able to estimate j’s payoff for her withidenticalutility. Wecanfreelydefinelineartransfor- alternatives. Again, chess is an illustrative example. Al- mationsinthe subspaceofConcepts; the mentalweights waysplayingagainstthesameunchangingcomputersoft- adapt, and the overall representation error remains un- ware is an individual context, whereas playing against a changed. ItisobviousthatifW isanon-singular,rank- humanadversarycapableoflearningfromher errorsis a i D matrix, then for K < D the error is strictly positive social context. even with an optimally chosen set of Concepts. In a given context, agent j bases her evaluations on her mental representation which is characterized by her approximate preference vector ω˜(x). This is necessarily j Relationship with the DMRG within her actualConceptsubspace. Agenti tries to approximate this within his Concept subspace. The accu- The identification, sorting and truncation of the de- racyofthisapproximationdepends onthe overlapofthe greesoffreedominourmodeliscloselyanalogoustowhat two subspaces. Thus social contexts introduce a “force” occursinWhite’sDensityMatrixRenormalizationGroup which tries to deform i’s Concept subspace towards that method (DMRG) [16] – a numerical technique widely of j in order to improve i’s predictive power in j-related used in the simulation of strongly correlated quantum social contexts. The interplay between the two compet- systems. In the DMRG the optimally renormalized de- ing goals – optimizing the mental representation for so- grees of freedom turn out to be the K most significant cial contexts vs. individual ones – is the ultimate factor eigenvectors of the reduced density matrix of the quan- which determines the society’s cultural coherence. tumsubsystemembeddedintheenvironmentwithwhich In terms of the above classification, agent i’s overall it interacts. world matrix is a sum of individual and social contribu- AformalmappingbetweentheDMRGandourmethod tions. As seen above, individual contexts contribute can be established by identifying the DMRG’s density X matrix for block i with the world matrix W for agent i i W0 = ω(x) ω(x); x=individual, (12) (bothmeasuringthesubsystem’srelationshiptotheenvi- i i ◦ i x=1 ronment),andtherenormalized(kept)quantummechan- X icaldegreesoffreedomwiththe Concepts. TheDMRG’s andthecontributionofallj-relatedsocialcontextsisnow “truncation error”is analogous to our representationerror. There, the error measures how precisely the en- Wj = ω˜(x) ω˜(x) x=j-related social. (13) i j ◦ j tangled quantum mechanical wave function (written in x X a matrix form) is reconstructed after the truncation of The interpretation of Wj is that agent j’s approximate the degrees of freedom; here, the role of the wave func- i preference vectors (which determine her decisions) are tion is played by the D X dimensional matrix formed by the preference vecto×rs ω(x). The two problems are fully presented to agent i (they build into his world ma- i trix), but – in accordance with the central thesis of our conceptually similar in that both seek an optimal linear model – agent i can only pick up from this what his in- reduction of dimensionality for the subsystem, which is ternal Concepts span. mathematically equivalent to the PCA problem. We realize that Wj is a rank-K operator, which can i be expressed explicitly in terms of j’s concept vectors γ K and mental weights v(x) using Eq. (3). This IV. SOCIAL INTERACTIONS { jµ}µ=1 jµ explicit representation has the structure As shown above, the representation error is minimal Wj =BS , (14) i j if the agent learns to approximate his world matrix in the K dimensionalsubspace spanned by the most signif- i.e., the product of icant eigenvectors of his world matrix. If agents i and j K havedifferentpreferences,andthusdifferentworldmatri- S = γ γµ , (15) ces,they wouldnecessarilyendupwithdifferentoptimal j jµ◦ j µ=1 Concepts. Cultural evolution boils down to the interac- X tion of these different Concepts in a social network. In projectingontoagentj’sConceptsubspace,andanother order to introduce social interactions we cast contexts operator B acting within this subspace. This latter de- into two basic categories: those where preference vectors (x) pends on the actual v weights, but for simplicity in jµ only depend on a single agent (“individual contexts”), what follows we assume that it is proportional to the andthosewherethepreferencevectorsforagentidepend identity operator on the (simultaneous) decision/preference of at least another agent j (“social contexts”). For simplicity only B =h 1. (16) ij 5 This assumption simply states that all directions in j’s average of Concept subspace projector operators Conceptsubspacehaveequalimportance fori,andisrea- K sonable if i has to predict j’s behavior in many different O = S = γ γµ . (20) j-related contexts which average out the weight depen- h jiI * jµ◦ j+ dence. µX=1 I Eventually, agent i’s overall world matrix to be used OurCOPisatensororderparameterwhichmeasuresthe in the utility Eq. (10) takes the form average overlap of the individual Concept subspaces. In factitisahigh-dimensionalgeneralizationofdeGennes’s W =W0+ h S , (17) tensororderparameterintroducedinthetheoryofliquid i i ij j crystals [17]. The eigenvalue structure of O is useful to jX∈Ni characterize the level of coherence in the society and to the sum being over the agent’s social network . The distinguishdifferentphases. Obviously,ifthere isperfect i N parameters h measure the relative strength (impor- order andthe individual Conceptsubspaces are all lined ij tance) of social interactions with agent j. The equal up, the COP has eigenvalues o = o = ... = o = 1, 1 2 K importance assumptionassures that our W remains in- and o = o = ... = o = 0. In the opposite i K+1 K+2 D variant w.r.t. nonsingular local linear transformations of extremeofcomplete disorder alldirectionsareequivalent the individual concept vectors. This means that in our andthe eigenvaluesareo =o =...=o =K/D. The 1 2 D model the actual choice of Concepts do not count, only trace of the COP is always K. the subspace they span do. UsingEq.(17)inEq.(10)allowsustowritetheagent’s utility function in an explicit form, which only depends V. CULTURAL ORDERING AS A PHASE on the concept vectors, TRANSITION K K The equilibrium and dynamic properties of the social Ui = γµi ·W0i γiµ+ hij (γνi ·γjµ)(γiν·γµj). system depend crucially on the parameters D,X,K, the µX=1 jX∈Ni µX,ν=1 statistical properties of W0i, and the social interaction (18) matrix h (connectivity structure). In the following we ij analyze the case when the agents’ individual preference vectorsω(x)areGaussianrandomvectors withzeromean Adjustment dynamics i andunit variance,i.e., withoutinteractionagentschoose random Concept subspaces (complete disorder). This is The introduction of social interactions in our model a benchmark case which can demonstrate in its purest of mental representations results in an economic game form how culture can emerge spontaneously in an inter- where the agents’ optimization problems are potentially acting social system. With this assumption W0, which i in conflict. We can view the society as a dynamical sys- isquadraticinthepreferencevectors,hasWishartdistri- tem in which agents continuously act as best-response bution[19,20]. We restrictour attentionto a mean-field optimizers [18], in each step maximizing their utility network (all agents are connected) and set h = h/I. ij given the actual state of nature and other agents (best- With this the world matrix simplifies to response dynamics). Alternatively,and this is more con- sistentwithboundedrationality,wecanpostulateaslow, Wi =W0i +hO. (21) continuous adjustment dynamics whichdrives agents to- As discussed above there are various possibilities for wards ever better responses. In this spirit the time evo- reasonable evolutionary dynamics. We have analyzed lution of the concept vectors is defined to be the social behavior under two: the best response dynamics and the continuous gradient adjustment dynam- δγ ∂U iµ =const i , (19) icsdefinedinEq.(19). Inthebestresponsedynamicswe δt ∂γ iµ startedfromacertaininitialconditionofindividualCon- cepts. In each step, we first calculated the actual COP, where U is given by Eq. (18) (gradient adjustment dy- i thenupdated(withacertainprobabilitytomaketheup- namics). Since in the general case U depends on all i date asynchronous) each agent’s Concepts according to agents’ all Concepts, this in fact amounts to I K cou- the K largest principal components of W . This gave × i pled equations. As we will see below the two dynamics agent i’s best response for the actual environment (indi- lead to the same result. vidual plus social). In the gradient adjustment method we discretized the differential equation in Eq. (19). Fixed points of the best response dynamics are Nash The cultural coherence order parameter equilibria,where agents haveno incentive to unilaterally deform their mental representations, because these are In order to monitor cultural coherence we introduce alreadymutuallyoptimal. Fixedpointsattaineddynam- a coherence order parameter (COP) as the population ically by gradient adjustment, in turn, are necessarily 6 1 completely ordered: o = 1 if k K, zero otherwise, k ≤ meaning that agents share all K Concepts, i.e., we can s talk about a coherent culture. In between we observe a e0.8 u al seriesofdynamic phasetransitions,eachassociatedwith v eigen0.6 o1 o2 o3 o4 tEhaechsutdradnensiteiomnercgaennbceeaosfsaonciaatdedditwioitnhalasfihrasrteodrdCeornjcuemptp. er o in the eigenvalues through a subcritical bifurcation. In et 1 m accordancewith the subcriticalnature of the bifurcation a0.4 ar scheme hysteresis is observed. Figure 2 only shows how p der h o1 behavesas h diminishes from its maximum value, but Or0.2 c the behavior is similar for the other eigenvalues too. A D = 10, X = 10, K = 4, I = 2000 o5 ... o10 qualitatively identical picture, with a series of first or- 0 der transitions and hysteresis,was found numerically for 15 20 25 30 35 40 45 Strength of social interactions h other parameter values, as well. Our results suggest that the social dynamics has a FIG. 2: Fixed point spectrum of the order parameter matrix unique equilibrium only for small couplings. Above the O for D = X = 10, K = 4, I = 2000. Solid lines apply for critical value h spontaneous ordering occurs in the sys- c increasing h, dashed line (only shown for o1) for decreasing tem, and the number of equilibria increases. On the one h. The phasetransitions occur as subcritical bifurcations. hand, there is a discrete number of possible fixed point “families” as represented by the eigenvalue structure of theCOP:anadiabaticallyslowchangeinhleadstobifur- stable against small collective perturbations in the con- cations at some critical values, where the system jumps cept vectors, i.e., when severalagents consider slight de- from one equilibrium to another. These transitions are viationssimultaneously,butnotnecessarilyoptimalona irreversible and give rise to hysteresis, which assures the global scale. It is a special feature of our model (which coexistence of at least two different equilibrium families is usually not the case in other models) that these two withinthehysteresisloop. Onthe otherhand,eachfam- dynamics have identical fixed points, i.e., the attained ily of equilibria is infinitely degenerate in itself, since fixed point of the gradientdynamics is also a Nash equi- the associated eigenvectors of the COP can point in es- librium, andvice versa. This feature stemsfromthe fact sentially any direction (respecting orthogonality). Since that the landscape of the PCA’s quadratic error func- we assumed that preferences are random, there are no tion is known to be smooth with a unique minimizing preferred directions on the social level by default. This subspace [21]. This is enough to assure the equivalence symmetry of the D-space is brokenspontaneously above of fixed points. However, in the case of multiple agents hc. As h there is only one possible structure for → ∞ (coupledPCAproblems)thenumberoffixedpointsmay the eigenvalues of the COP (there is only one family), not be unique. In fact our results demonstrate that it is meaningthatthe societyis fully coherent. However,this typical to have many equilibria, the structure of which coherentK-dimensionalsubspace,representingthe most depends strongly on h. important dimensions of the society is not fixed a priori. Incaseofmultiple equilibria,equilibriumselectionbe- The ultimate state of culture gets selected as a result of comes important. In the following we analyze a situa- the idiosyncratic fluctuations present at the time of the tion when the strength of social interactions h is varied symmetry breaking bifurcations. slowly. An adiabatic change in the exogenous parameters assures that the fixed point the system has reached Calculating the critical point is tracked analytically, except for possible bifurcations. We start with zero coupling, h = 0, where the (unique) fixed point arises as the collection of individual PCA so- The critical interaction strength hc, where the com- lutions. pletely disordered phase loses stability for increasing h can be calculated analytically for I . Starting from The COP spectrum in the fixed point, determined by →∞ the disordered phase we can write the COP in step l as a numerical simulation with adiabatic increase (and de- crease) of h for D =10, X =10, K =4 and I =2000 is O =(K/D)1+εδO , (22) l depicted in Fig. 2. The finite population of agents with where δO is an arbitrary perturbation. The dynamics random world matrices were artificially “symmetrized” l to assure W0 =K/D1 exactly even in a finite sample defines a mapping h ii at h = 0. This symmetrization procedure, however, is δO δO . (23) l l+1 irrelevant if the sample is large enough. → If the disordered state is stable we have δO 0 as According to the figure, for small h there is complete || l|| → l , otherwise δO diverges. We can identify h as disorder, and all eigenvalues are equal, ok = K/D, ∀k. a→fix∞ed point of th|i|s mal|p|ping c In other words there is no systematic overlap between the agents’ Concepts. For h the fixed point is δO =δO , (24) l+1 l → ∞ 7 and determine δO as a function of δO and h using l+1 l firstorderperturbationtheoryinε. Thecalculationnec- essarily involves the spectral properties of the random h W0 ensemble, and leads to (see the Appendix for de- s i n o tails) cti Ordered a D2 nter Coherent Culture hc = 2 Kν=1ξν, (25) of social i hc with P h gt ξ = D 1 , (26) Stren DNois Courldtuerreed ν λ λ * iν im+ mX6=ν − I Agent intelligence K where λ are eigenvalues (in descending order) of the im Wishart matrix W0, and ν is the index for Concepts. i When ν << D,X we find that ξ is ν-independent, FIG. 3: Schematic phase diagram on the h vs. K plain ν (D,X =fixed) for themean-field model with random prefer- and takes the form (see the Appendix) ences. Solid line indicates the principal transition calculated in the text,dashed lines theadditional transitions. 1 (X D 1) ξ = 1 − − . (27) ν 2 − D+X +2√DX (cid:18) (cid:19) In the limit D,X with X/D=r >0 and K <<D evolution goes through a number of phase transitions, →∞ this leads to the critical coupling which we found first order in the mean-field model. The critical value of the most relevant transition was calcu- D2 1+√r h . (28) lated analytically using linear stability arguments and c ≈ K 2 random matrix theory. Keeping r fixed, h increases in D (quadratically) and Transition to a phase with substantially higher cul- c decreases in K. When the World is complex (large D tural coherence can be triggered in two different ways andX)andtheagentsareprimitive(smallK)thecritical (see the arrowsinFig.3): socialcontextscanincreasein couplingishighandthesocietyislikelytostayculturally significance, or alternatively, agents may develop better disordered. A schematic phase diagram on the h vs K cognitive skills. In any case when one of the transition plain is presented in Fig. 3. The h curve cuts the plain linesiscrossed,thesubspacesspannedbytheagents’con- c into two basic domains: one which is disorderedand one cepts rearrangeinto a state with higher averageoverlap, which is spontaneously ordered. This latter is divided and we can speak about a boost in cultural coherence. further into sub-domains with different levels of order, A similar transition may explain the sudden cultural ex- separated by first order transition lines. plosions for humanity approximately 50,000 years ago, andafurther jumpabout10,000yearsagoinconnection with the emergence of agriculture. Since in our model VI. CONCLUSIONS the ordering involves a spontaneous symmetry breaking, the emerging fixed point strongly depends on the initial In summary, in this paper we have proposed a theory perturbation (a phenomenon usually termed “path de- of cultural evolution built upon a renormalization group pendence” in the social sciences), implying that cultural scheme. We assumed that bounded rationality forces evolutionisnotcompletelydeterministic butdependson agents to choose a limited set of concepts in describing a number of idiosyncratic factors. their world. These concepts and the way they are used Our theory differs conceptually from some traditional are continuously updated and refined in order to find an thinking on the possible causes underlying human cul- optimalrepresentation. Individualinterestinmakingthe tural explosion. Most of these traditional explanations best possible decisions in natural (individual) and social presume a sudden and substantial biological change (ge- (collective) contexts act as a driving force for cultural netic mutations [2, 25], the integrationofcognitivemod- evolution and, at least in the particular cases studied, ules in the brain [1], etc.) as the initial trigger. In con- drive the system towards a Nash equilibrium fixed point trast with these, the mechanism we have proposed in with measurable level of social (cultural) coherence. In this paper suggests a spontaneous ordering of the men- our model with random preferences this coherence man- talrepresentations,whichemergeswhenslow changes in ifested itself as a spontaneous ordering of the individual somerelevantvariables(cognitiveabilities,socialinterac- concept (sub)spaces, and was quantified by an adequate tions) reach a critical level. In this respect the “culture- order parameter. We demonstrated that as the funda- less state” of the society, originally rooted in the hetero- mental cognitive and social parameters change, cultural geneity of its members in individual preferences, needs, 8 experiencesandtheextremesimplicityofthementalrep- this iteration step we calculate the updated (final) order resentationstheypossess,becomesunstableatabifurca- parameter as tion point, beyond which the collective social dynamics K drives the system into a drastically different, culturally K Ofin = γ γ = 1+εO . (A.5) ordered fixed point. * iν ◦ iν+ D l+1 It would be interesting to go beyond mean field, and νX=1 investigate the model’s dynamics in more realistic so- which defines O . The disordered state is stable and l+1 cial networks, where the fixed point may possess a more the perturbation dies away if O 0 as l ; oth- l || || → → ∞ complex internal structure. A non-mean-field-like so- erwise it is unstable. cial structure is expected to give rise to subcultures or In fact each step of the dynamics defines a mapping cultural domains, where individuals are highly coher- fromO to O . This mapping canbe approximatedas l l+1 ent within a domain, but the average cultural subspace linear if ε is small. O can be decomposed into D D l × the domain represents may be rather different from that “eigenmatrices”eachofwhichtransformsintoaconstant of another domain, as occurs usually in real societies. (theeigenvalue)timesitselfunderthemapping. Thedis- The type of the occurring dynamic transitions may also orderedstatelosesstabilitywhenthemaximaleigenvalue change character. An evaluation of how a potential becomesgreaterthanone. Thush canbefoundbyseek- c structure in the attribute space or non-linearity in the ing the solution of the equation world/mental representations influence the results could also be interesting. Other intriguing problems involve Ol+1 =Ol. (A.6) the social dynamics itself, namely how the model should Let us introduce the orthonormal eigenvectors of the be extended tocope withintrinsically dynamicalaspects Wishart matrix W0 of culture such as fads and fashions, where the social at- i tractorismuchratheralimitcycleorachaoticattractor. W0φ =λ φ , n=1,...,D, (A.7) i in in in with λ ... λ . When ε is small we can use first i1 iD ≥ ≥ Acknowledgments order perturbation theory to obtain the concept vectors (eigenvectors) in Eq. (A.4). This yields TheauthorsacknowledgesupportfromINSEADFoun- D [O ] dation and the Hungarian Scientific Research Found γ =φ +εh l νm φ + (ε2) (A.8) (OTKA) under grantNos. F31949,T43330and T47003. iν iν λiν λim im O m6=ν − X Substituting this into Eq. (A.5) we obtain APPENDIX K K D Ofin = φ φ +εh Inthedisorderedphasetheorderparameterisdiagonal * iν ◦ iν+ * νX=1 I νX=1mX=1 [O ] O = K1. (A.1) λ l νλm (φiν ◦φim+φim◦φiν) +O(ε2). D iν − im (cid:29)I (A.9) In order to assess the stability of this phase assume that in step l of the iterated dynamics the actual order pa- Toevaluatethe termswecanusethefactthattheeigen- rameter differs from this by a small perturbation values and eigenvectors of the random matrix ensemble W0canbeconsideredindependent,andthustheaverage i K ofa productcontainingboth decouples. Remarkingthat Oini = 1+εO , (A.2) l D 1 φiνdφi′ν′d′ I = δii′δνν′δdd′, (A.10) where ε is infinitesimal. In the next step, using Oini, we h i D should first create the agent specific context matrices we obtain that the first term is simply (K/D)1, and the second term defines O as W =W0+hOini, (A.3) l+1 i i D and diagonalize these to obtain the actual (orthogonal) [Ol+1]dd′ =h Add′;mm′[Ol]mm′ (A.11) concept vectors m,m′=1 X where the mapping operator A is W γ =η γ , n=1,...,D. (A.4) i in in in K 1 Wderh,enthtehaegeeingtenkveaelpusesCoηninceaprtes snor=ted1,i.n..d,eKsc.enTdoinfigniosrh- Add′;mm′ = D2 ξν (δdmδd′m′ +δdm′δd′m) (A.12) ν=1 X 9 with In this expression the eigenvalues can be interpreted as coordinates of particles put in an external potential V D 1 and interacting through a 2D Coulomb interaction. For ξ = (A.13) ν λ λ large D and X, the average position of particles (eigen- * iν im+ mX6=ν − I values)followsfromasaddle-point approximationleading to the equation Using the fact that both O and O is symmetric this l l+1 reduces to min(D,X) 1 K ∂ V = ′ . (A.20) 2h λn λ λ [Ol+1]dd′ = D2 ξν[Ol]dd′, (A.14) mX=1 n− m ν=1 X Using the explicit form of V in Eq. (A.19) this reads showingthatthe(normalized)eigenmatricesofthe mapping are the D(D+1)/2 independent (unit) matrix ele- min(D,X) X D 1 1 ments, and the common eigenvalue is D(D+1)/2 times 1 | − |− =2 ′ . (A.21) degenerate. Invirtue ofEq.(A.6)this eigenvaluedefines − λn λn λm m=1 − the critical social coupling X When the number of zero eigenvalues is taken into ac- D2 count correctly, we obtain for ξ defined in Eq. (A.13) h = . (A.15) ν c 2 K ξ ν=1 ν 1 The only remaining quPestion is how to evaluate the 2ξν =1 (X D 1) , (A.22) − − − λ denominator. The theory of random matrices[22] gives (cid:28) ν(cid:29) a solution. Recall that W W0 belongs to the real ≡ i and thus using Eq. (A.15) Wishart ensemble as it is the outer product of a D X × real Gaussian matrix with its transpose D2 h = . (A.23) c W =ΩΩT, (A.16) K−(X −D−1) Kν=11/λν D E where Ω is the D X matrix constructedfromthe pref- P × It is interesting that the correction in the denominator erence vectors ω(x) as columns. The joint probability changes sign when X =D+1. densityofeigenvaluesforrealWishartmatricesis[20,23] In the general case a closed form approximation can be obtained for small K. The limit spectral density of min(D,X) 1 1 the nonzero eigenvalues of Wishart matrices is given by P( λ ) d λ = exp λ { } { } −2 d· the Marcenko-Pasturlaw [24] Z d=1 X min(D,X)  min(D,X)  1 (λ λ )(λ λ) λ(|X−D|−1)/2 (λ λ )d λ ρ(λ) = − min max− ; d n− m { } 2πmin(D,X)r λ2 d=1 n>m Y Y 2 (A.17) λ = √D √X , min − (cid:16) (cid:17)2 forthemin(D,X)nonzeroeigenvalues,andthereareD λ = √D+√X , (A.24) − max X zero eigenvalues if D > X. The partition function Z (cid:16) (cid:17) is a normalization constant. It is illuminating to write whichhasafinitesupportλ (λ ,λ )andasquare- this in the Coulomb gas representation [23] min max ∈ root singularity as λ λ . If D,X is large and max → K <<min(D,X), the K largesteigenvaluesareallclose min(D,X) min(D,X) 1 to λ . Thus in this limit P( λ ) = exp V(λ )+ ln λ λ max d n m { } − | − | Z d=1 n>m X Y K  (A.18) 1 K , (A.25) * λν+≈ λmax ν=1 with X which using Eq. (A.23) in the limit K << D,X 1 X D 1 → ∞ V(λ)= λ | − |− lnλ. (A.19) gives Eq. (28) in the text. 2 − 2 [1] S. Mithen, The Prehistory of the Mind (Thames and [2] R. G. Klein and B. Edgar, The dawn of human culture Hudson,London, 1996). 10 (John Wiley and Sons,2002). [16] S. R.White, Phys. Rev.Lett. 69, 2863 (1992). [3] H.A.Simon,Models of bounded rationality, Vol. 3: Em- [17] P.G.deGennes,ThePhysicsofLiquidCrystals (Oxford pirically grounded economic reason (MIT Press, 1997). University Press, 1974). [4] A. K. Romney, J. P. Boyd, C. C. Moore, W. H. [18] J. Hofbauer and K. Sigmund, Evolutionary Games Batchelder,andT.J.Brazill,Proc.Natl.Acad.Sci.USA and Population Dynamics (Cambridge University Press, 93, 4699 (1996). 1998). [5] R. Aunger, ed., Darwinizing culture: The status of [19] J. Wishart, Biometrica A 20, 32 (1928). memetics as a science (Oxford University Press, 2001). [20] A. Edelman, Ph.D. thesis, MIT (1989), downloadable [6] R.Axelrod, J. Conflict Res. 41, 203 (1997). from http://www-math.mit.edu/˜edelman. [7] R. Axelrod, The complexity of cooperation (Princeton [21] P. F. Baldi and K. Hornik, IEEE T. Neural Networ. 6, UniversityPress, 1997). 873 (1995). [8] G. Deffuant, D. Neau, F. Amblard, and G. Weisbuch, [22] M. L. Mehta, Random matrices (Academic Press, New Advancesin Complex Systems 3, 87 (2000). York, 1991). [9] C.Castellano,M.Marsili,andA.Vespignani,Phys.Rev. [23] J. Shen,Linear Algebra Appl.326, 1 (2001). Lett.85, 3536 (2000). [24] V. A. Marcenko and L. A. Pastur, Math. USSR-Sb 1, [10] K.Klemm,V.M.Eguiluz,R.Toral, andM.SanMiguel, 457 (1967). Phys.Rev.E 67, 045101(R) (2003). [25] Y.-C. Ding, H.-C. Chi, D. L. Grady, A. Morishima, and [11] G. Weisbuch,Eur. Phys. J. B 38, 339 (2004). R.K.Moyzis,Proc.Natl.Acad.Sci.USA99,309(2002). [12] J. Conlisk, Journal of Economic Literature 34, 669 [26] J. B. Carroll, ed., Language, Thought and Reality: Se- (1996). lected Writings of Benjamin Lee Whorf (MIT Press, [13] D. Fudenberg and J. Tirole, Game Theory (MIT Press, Cambridge, 1956). Cambridge, 1991). [27] P. E. Ross, Sci. Am. 290, 24 (2004). [14] L. Mér˝o, Ways of Thinking: The Limits of Ratio- [28] The idea that concepts one uses influence her thinking nal Thought and Artificial Intelligence (WorldScientific, and decisions dates back to E. Sapir and B. L. Whorf 1990). (Sapir-Whorf hypothesis) [26]. See also Ref. [27] for a [15] H. A. Simon, and W. G. Chase, Cognitive Psycology 4, more recent perspective on theissue. 55 (1973).