Phase transition from egalitarian to hierarchical societies driven by competition between cognitive and social constraints
PPhase transition from egalitarian to hierarchical societies driven bycompetition between cognitive and social constraints
Nestor Caticha ∗ and Rafael Calsaverini † Dept. de F´ısica Geral, Instituto de F´ısica,Universidade de S˜ao Paulo, 05508-090, S˜ao Paulo-SP, Brazil
Renato Vicente ‡ Dept. Matem´atica Aplicada, Instituto de Matem´atica e Estat´ıstica,Universidade de S˜ao Paulo, 05508-090, S˜ao Paulo-SP, Brazil (Dated: July 9, 2018)Empirical evidence suggests that social structure may have changed from hierarchical to egali-tarian and back along the evolutionary line of humans. We model a society subject to competingcognitive and social navigation constraints. The theory predicts that the degree of hierarchy de-creases with encephalization and increases with group size. Hence hominin groups may have beendriven from a phase with hierarchical order to a phase with egalitarian structures by the encephal-ization during the last two million years, and back to hierarchical due to fast demographical changesduring the Neolithic. The dynamics in the perceived social network shows evidence in the egal-itarian phase of the observed phenomenon of Reverse Dominance. The theory also predicts formodern hunter-gatherers in mild climates a trend towards an intermediate hierarchy degree and aphase transition for harder ecological conditions. In harsher climates societies would tend to bemore egalitarian if organized in small groups but more hierarchical if in large groups. The theoret-ical model permits organizing the available data in the cross-cultural record (Ethnographic Atlas,N=248 cultures) where the symmetry breaking transition can be clearly seen.
I. INTRODUCTION
Behavioral phylogenetics makes it plausible that the common ancestor of
Homo and
Pan genera had ahierarchical social structure [1–5]. Paleolithic humans with a foraging lifestyle, however, most likely had alargely egalitarian society and yet hierarchical structures became again common in the Neolithic period.Contemporary illiterate societies fill the ethological spectrum [6] from egalitarian to authoritarian anddespotic. This non-monotonic journey, a so called U-shaped trajectory, along the egalitarian-hierarchicalspectrum during human evolution, was stressed by Knauft [1] and has defied theory despite severalattempts of anthropological explanation [1]. Our approach to the study of social organization usestools of information theory and statistical mechanics. It is inspired in previous work by Terano et al. [7, 8] on a different problem, the emergence of money in a barter society as a consequence of limitedcognitive capacity. We model the perception by each agent of the social network of its society, taking intoaccount cognitive constraints and social navigation demands, which define the informational constraintsadequate to a probabilistic description. According to the model, whether an egalitarian-symmetric orhierarchical-broken symmetry state occurs depends on a scaling parameter which grows with cognitivecapacity and decreases with group size, modulated by a Lagrange multiplier which can be interpretedas an environmental pressure. The hypothesis that social perceptions mediate motivations and hencepossible behaviors (PMB hypothesis), permits making predictions about the effect of these variables inthe probable forms of social organization. Since there was a massive increase in encephalic mass in the last ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] a r X i v : . [ phy s i c s . s o c - ph ] A ug two million years, our theory expects a phase transition towards a more egalitarian social organizationto occur. As food producing and storage methods permitted the populational increase in the Neolithic,the scaling parameter decrease permitted a reversal to hierarchical structures.Furthermore, the same model makes predictions about a totally different empirical situation, dealingwith the influence of ecology on the expected hierarchy of modern human groups. The theory suggestsa form of looking at the available ethnographic data [9, 10] and allows a new interpretation of observedpatterns involving social structure, community size and environment in terms of a competition betweencognitive constraints and social navigation demands and a symmetry breaking phase transition. Thebifurcation suggested by the theory is seen in the ethnographic records. The empirical data can be foundin the Murdock’s Ethnographic Atlas [9], which resulted from the tour de force attempt to compile allethnographic available knowledge that can be represented in a quantitative form.We are studying the properties of finite size groups and so cannot take the thermodynamic limit. Thechanges in behavior are therefore not singular but we still find that the language of phase transitionsis adequate, after all from a technical point of view the infinite size limit is a tool to simplify themathematical treatment of very large systems. II. THE THEORY AND THE MODEL
In a group of n agents, each agent of the group will have a perceived social web of interactions rep-resented by a graph. An important distinction has to be made between the social network, that say anethnographer might describe and the perceived social network of a particular agent. Each vertex of agraph stands for a represented agent of the group. In this representation of the social web, undirectededges joining any pair of vertices might be present or not. An edge links two represented agents if theirsocial relation is known by the owner of the graph. Since inference depends on the available information,these graphs might differ from agent to agent. Call S i the representation of the social web by agent i ,given by a set { s ijk } of variables that can take values either zero or one. The indices i, j and k run from1 to n and every s ijk is symmetric with respect to interchange of the lower indices. If s ijk = 1 then agent i has knowledge of the social relation, the capacity to cooperate and form coalitions or the antagonismbetween agents j and k , while if s ijk = 0 this relation is unknown to agent i . Known alliances, feuds orneutral interactions are represented by s ijk = 1. The total number of memorized social relations is N icog = n (cid:88) j,k =1 s ijk / i . This is a cognitive contribution to thecost of a given representation S i .Now consider the agent i ’s social cost for not knowing a given social relationship, when s ijk = 0. Fortwo agents j and k there is either a bond or a path of bonds, joining intervening agents, connecting themso that their social relation can be estimated by agent i . We will assume that the representation is aconnected graph, what can be accomplished by defining the distance of two unconnected vertices to beinfinite. It is reasonable to assume that this lack of direct knowledge will imply in a social cost whichincreases with the length of the shortest path between the agents. This implements the idea that relyingon heuristics to infer the relationship between them (e.g., “a friend of an enemy is an enemy”, etc.) ismore amenable to errors as the number of intermediate agents grows.Call l ijk the social distance in the graph defined by the adjacency matrix S i . The λ − th power of S i , M i ( λ ) = [ S i ] λ permits verifying whether there is a path joining two agents, and the social distance is l ijk = min λ, such that M i ( λ ) jk > FIG. 1.
Each agent has a perceived social web of interactions represented by a graph. Each vertex of agraph stands for a represented agent of the group. An edge in this perceived web links two agents iftheir social relation is known by the owner of the graph. Using the methods of information theoryentropic inference, we attribute a probability P ( S i | I ) that agent i perceives a network S i , conditionalon any available information I .is the length of the smallest path of bonds linking j and k . We take the social cost of agent i of havinga representation S i to be just the distance averaged over all pairs of agents¯ L i = 2 n ( n − n (cid:88) j,k =1 l ijk . (2)The joint cognitive-social cost of the representation is defined as a sum of monotonic functions of N icog and ¯ L i and the simplest form is just C ( S i ) = N icog + α ¯ L i , (3)For high α , optimization is obtained by decreasing ¯ L i independently of N ie . For low α the number ofedges N icog has to be controlled, independently of ¯ L i . Hence α is a parameter of the theory that measuresthe relative importance of the social and cognitive components and can be interpreted as a measure ofthe cognitive capacity of the agent.We now attribute a probability P ( S i | I ) that agent i perceives a network S i = { s ijk } , conditional onany available information I , using the methods of information theory entropic inference. Call C = IE ( C )the expected value of C ( S i ) under P ( S i | I ): C = IE ( C ) = (cid:88) S i C P ( S i | I ) (4)Suppose that either C or equivalently the scale in which fluctuations of C ( S i ) above its minimum areimportant, are known. Or possibly we just know that such knowledge would be useful, but we have noaccess to their specific values at present. The procedure calls for the maximization of the entropy (seee.g [11]) subject to the known constraints, P ( S i | I ) = argmax P (cid:40) − (cid:88) S i P ( S i ) log P ( S i ) − λ (cid:34)(cid:88) S i P ( S i ) − (cid:35) − β (cid:34)(cid:88) S i P ( S i ) C ( S i ) − C (cid:35)(cid:41) (5)The result is the standard Boltzmann-Gibbs probability distribution P ( S i | I ) = 1 Z e − βC ( S i ) , (6)where β is the Lagrange multiplier conjugated variable to C and controls the scale in which the fluctuationsare important. The information content in β is equivalent to that in C . Low β values means that S i configurations of high joint cost will not be unlikely. For high β only configurations near the ground statewill be possible. We can interpret β as an ecological pressure, possibly correlated to a measure of theeffort to collect a minimum number of calories in one day. The normalization factor in equation 6, thepartition function Z , depends on the number of agents, α and β : Z = Z ( n, α, β ).In order to characterize the state of the system we need appropriate order parameters. In particularwe want to probe whether represented agents are considered symmetrically or if distinctions are made.It is useful to introduce the degree of vertex j in a graph S i , d ij = (cid:80) k s ijk , the number of edges emergingfrom the vertex or in the case of the represented social web, the number of memorized social relations ofan agent; as well as the maximum degree and the average d imax = max j d ij , d iavg = 1 n n (cid:88) j =1 d ij . Natural order parameters are the expectation values IE ( d max ) and IE ( d avg ) with respect to P ( S i | I ), theBoltzmann distribution in equation 6. III. METHODS
Several techniques can be used to obtain estimates of the order parameters and here we present resultsobtained employing numerical Monte Carlo methods. We first considered isolated agents and the MonteCarlo simulation of the Boltzmann distribution (eq 6). Then we considered 2-body interactions of the n agents, exchanging information about other pairs of individuals, through a mechanism that can becalled gossip. The effect of gossip is to generate highly correlated perceived social webs. The advantageof Monte Carlo methods is that a simple extension of the type of dynamics permits incorporating verysimply interactions like gossip.We now run the simulation for the n agents together. A parameter g (0 < g <
1) measures the intensityof information exchange through gossip. Choose an agent i and pair ( j, k ), independently of anything else,uniformly at random. With probability 1 − g , a MC Metropolis update is performed on the bond of thepair ( j, k ) . Let ¯ s ijk = 1 − s ijk be the complementary value of the bond variable s ijk . Also independentlyand uniformly at random, with probability g another agent is chosen, call it l . Its corresponding edge s ljk is copied to ¯ s ijk . Let ¯ C be the joint cognitive-social cost with the bond s ijk replaced by ¯ s ijk . Withprobability min (cid:8) exp( − β ( ¯ C − C )) , (cid:9) let the change of s ijk by ¯ s ijk be accepted. Otherwise s ijk is keptfixed.The step performed with probability 1 − g simulates the update of the social web representation byindependent observations, learning new relations and forgetting about previously known relations. Thegossip step, done with probability g , simulates the exchange of information where agent l tells and agent i learns or forgets something about the relation of agents j and k . Gossip can be introduced by moreelaborate schemes but this is sufficient for our modelling purposes.After all i = 1 , ...N have been considered, a MC sweep has been completed. The α range 4 . ≤ α ≤ β range 0 < β ≤
20 was divided into 200 intervals. Thevalues of n varied from 7 to 15. The number of degrees of freedom is n n ( n − = O ( n ). We run a MCsimulation for each fixed n and for each pair of α and β . One to two million MC steps were made forthermalization, and then data about the order parameters was collected every 4 MC steps for aroundfour million MCS. The results for the order parameters are shown in figures 2 and 3 are discussed below. FIG. 2.
Top: Monte Carlo estimates of the maximum degree and the average degree ( IE ( d avg ) / ( n − IE ( d max ) / ( n − z = 2 α/n ( n −
1) for β = 10. The shaded areas are bands at ± H = 2(1 − D ), with D = IE ( d avg ) /IE ( d max ). Roughly three regimes can be seen: forvery large z , H goes to zero (symmetric phase), all agents are equal. For very small z , H is ≈
2, thebroken symmetry phase, where a particular agent occupies the central position of the web. Anintermediate z transition region shows intermediate values of H . All agents are statistically but notstrictly equals, some occupy, but only temporarily, in the stochastic dynamics a more central position.The different curves are for different values of n (= 10 , , ...
15) , note the almost exact colapse whenplotted as a function of z . The insets are typical realizations of the inferred web of social interactions byan agent at that z position. Bonds are only drawn if the bond variable is one.The average length of the distances in a given graph S i has to be measured in each Metropolis step.It was obtained using Dijkstra’s algorithm to calculate every pair distance. IV. RESULTSA. Joint dependence on cognition and band size
An interesting result is that, given β , to a good approximation the properties of the system do notdepend separately on α and n but on the ratio z = αn ( n − , which can be thought of as a measure of theeffective cognitive capacity per dyadic relation on a social landscape. FIG. 3.
Left, Hierarchical-Egalitarian phase transition: phase diagram in the plane of specificcognitive capacity per dyadic relation z = αn ( n − and the inverse ecological pressure β − . The colorcode represents a measure of the social hierarchy measure or symmetry breaking parameter H = 2 − IE ( d avg ) /IE ( d max ). The red region is where the symmetry is broken ( H near 2) and themaximum degree IE ( d max ) is much larger than the mean IE ( d avg ). The blue region is the unbrokensymmetry phase, H ≈
0. Right, same but for constant α in the log n, β plane. The white lines dividethe phase diagram into regions that can be used for the comparison to the Ethnographic Atlas data.The dark red region is where the symmetry is broken and the maximum degree is much larger than themean. Dark blue is the symmetrical or egalitarian representation region. FIG. 4.
The effect of gossip inside the hierarchical phase. Continuous curves: The entropy (essentiallya measure of the width (eq. 7)) of the distribution of central agents decreases with the increase ofgossip, meaning that there is a particular agent that preferentially occupies the centers of the stars.Dashed lines: Another way of seeing the same ordering: the frequency of the most frequent centralagent in their representations.We start by considering the case with no gossip g = 0 where agents process information in a decou-pled way. Larger g gives similar results for the individual web representations but they are no longerindependent and correlations of the webs appear. Figure 2 shows the results of Monte Carlo estimates ofthe order parameters, the expected values of d max and d avg , respectively IE ( d avg ) and IE ( d max ). Theseare plotted as a function of the scaling variable z = 2 α/n ( n − H = 2 − D as a function z for β fixed, where D = IE ( d avg ) /IE ( d max ). For H = 0 the typical graph is the totally symmetric graph, while for H = 2 the typical graph is the star.Since n is finite, H can’t be 2. The maximum value is H max = 2(1 − /n + 1 /n ) ≈
2. Three differentregimes can be identified: low, intermediate and high H regions. In figure 3 (left) we show H as a heatmap in the z − β − plane. The three phases can be seen again. An intermediate fluid phase has theshape of a wedge that decreases in width as ecological pressure increases. B. Gossip and shared perception
Figure 4 shows how frequent is the most frequent central agent as a function of the level of gossip g .Let P ( c = j | i ) be the probability that for the social web representation of agent i the central elementis j . The spreading of the probability distribution can be measured by the ratio of its entropy to themaximum possible value s cf = ¯ S log n = − n log n n (cid:88) i,j =1 P ( c = j | i ) log P ( c = j | i ) (7)The results indicate that large correlation occurs when gossip dynamics dominates independent dynamics,starting around g ≈ . FIG. 5.
The probability of acceptance of changes P ( acc | zβ ) from the Monte Carlo simulationsmeasures the tolerance to changes in the social web representations. As a function of z , for differentvalues of β ( β = 2 ., . and 10) For high pressure, or harsher environments, ( β = 10) changes that wouldpermit upstarts to be different from other agents are not tolerated for large z . This is analogous tocounter dominance behavior theory [12]. Changes of the central agent of the star topology are unlikelyto be accepted for small z . For milder pressures changes are more easily accepted. Tolerance ismeasured by the Monte Carlo acceptance probabilityAgain we stress the hypothesis that the likelihood of an agent in tolerating inequalities is associated tothe perceived inequalities of its social web representation. The three regimes will have strong influencein the possibilities of social organization of the group. In the region where H is close to zero, theinterpretation is that no inequalities can be tolerated. These would represent large fluctuations on thecognitive-social cost and the combination of cognitive resources and band size given by z is large enoughto permit a representation web given by a full graph.The intermediate wedge region could be interpreted as the “Big Man” society, where some inequalityis possible, but is not solidified and these temporarily more central figures can be though of as “firstamong equals” and their position is liable to changes. Since the wedge decreases for increasing pressure,for extreme ecological pressure, a Big Man organization is not possible. Either there is a stable centralfigure, e.g. a chief, or symmetry among members of the band.The lower left hand part of figure 3 (left) is where the symmetry breakdown of the web representationpermits the emergence of tolerance towards inequalities. The exchange of information about the socialwebs leads to the choice of an almost unique and stable central agent for all agents. This would allow thecreation of a society where authority is stable and social egalitarianism is lost. In figure 5 we show theprobability of a change in the cognitive representation webs as a function of z , measured by the MonteCarlo (Metropolis) acceptance rate. Only in the intermediate Big Man fluid region a significant rate ofchanges is acceptable. In both hierarchical and egalitarian phases, the dynamics turns out to be veryconservative and change is rare, maintaining status quo for very long times. This prediction of the theoryis in accordance to what is expected from anthropology’s Reverse Dominance theory [12]. C. Knauft’s U-shape
FIG. 6.
Schematic (inverted-) U-shape trajectory for the specific cognitive capacity z ∝ α/n ( n −
1) asa function of time (thick curve). The higher the value of z , the more symmetrical or egalitarian thesociety will be. This is just a representation of externally caused changes in the cognitive capacity α ( t )and the mean size of social groups n ( t ) as a function of time. The thin lines are the shadows projectedonto the respective planes. The surface is z ( α, n ) ∝ αn ( n − . The contours are drawn for constant z values.The fact that the phase diagram can be drawn using the combination z = 2 α/n ( n −
1) immediatelysuggests a scenario that accommodates the U-shape dynamics along the egalitarian-hierarchical spectrum.The schematic drawing in figure 6 shows the curve z = z ( α ( t ) , n ( t )) in a parametric representation usingsome rough measure of time as the parameter. We use a simple model of the growth of the cognitivecapacity α in an evolutionary time scale and the fast increase in band sizes n in the transition to theneolithic. For viewing purposes only we use different time scales along the trajectory so that the shapeis clearly a nice inverted U, otherwise it would be very skewed, since it takes around 7 million years togo up from hierarchical to egalitarian and few thousands years to go down back to hierarchical. It startswith low z around 7 Mya, in the hierarchical region of left hand side of the phase diagram of figure 3.It slowly grows, reaching a peak of z in an egalitarian region due to increased encephalization. Finallyit goes back to the region of low z due to increase in band size, in the hierarchical phase of figure 3. Ofcourse the specific details of such trajectory would depend on many other conditions, but this furnishesa plausible qualitative scenario for the evolution of z . V. ETHNOGRAPHIC DATA AND THEORETICAL PREDICTIONS
FIG. 7.
Can a signature of this competition between cognitive and social navigation constraints be seen todayfor modern humans? A clear theoretical prediction about the dependence of social stratification on eco-logical pressure β and group size ( z = z ( n ), α fixed) can be confronted to data from the ethnographicrecord. The prediction is divided into two parts. First, for very mild climates intermediate social struc-tures are expected, but, as climates of increasing harshness are considered, different social organizationswill occur. Second, this difference depends on group size. Cultures organized in small groups will bemore egalitarian, those in large groups more hierarchical.Using Murdock’s Ethnographic Atlas ([9]) we see in figure 8 that this prediction is indeed borne out by0 FIG. 8.
The bifurcation signature of the phase transition. For mild climates the expected hierarchieschange little with group size. For harsh climates the expected hierarchy is larger for large groups andsmaller for smaller groups. Top Left Ethnographic Atlas data. Top Right: theory. Bottom: Thedifference in expected hierarchy ∆ H between large and small groups decreases for milder climates.Continuous line: EA data, Dashed line: theory. Harsh climates are Tundra (northern areas), Northernconiferous forest, High plateau steppe, Desert (including arctic). Mild: Temperate forest,Temperategrasslands, Mediterranean, Oases and certain restricted river valleys.the data. These are not predictions about a specific group becoming more or less hierarchical as climatechanges. These are predictions about the expectations we should have about hierarchical organization asdifferent climates and group sizes are considered. Changes in a particular group would not be so easilyobserved, since the shift of perception of social webs will have an influence on motivations. See [13] for adescription of a system in the process of transition. How changing motivations lead to cultural changesand influence social organizations is outside the scope of the present theory.From the Ethnographic Atlas we extracted the relevant variables: data for social stratification h ,climate c and group size s . Each variable range is divided into three regimes, low, intermediate and high(0 , , P ( h | cs ) = P ( hcs ) /P ( cs ). The empiricalexpected hierarchy value H emp = (cid:80) h hP ( h | cs ) for each possible combination of climate and size, isshown in the left panel of Figure 8. The climate variables describe the type of environment but we need aquantitative description of the climate instead of just its name. We decided that a reasonable conversioncould be done by using the idea of Net Primary Production [15] which is a measure of the amount ofcalories per day that can be extracted from the environment and therefore correlates negatively to theecological pressure β .1To obtain the analogous theoretical predictions, the same is done by dividing the range of theoreticalparameters into three intervals as well and calculating the same quantity from the theoretical results, seefigure 3 (right). The theoretical expected value of H thy = 2 − IE ( d avg ) /IE ( d max ) is shown in the rightpanel of Figure 8 .The qualitative agreement between theory and empirical record supports that our methodology iscapable of suggesting new ways of looking at the available ethnographic records, which can now comeunder scrutiny by the community of quantitative ethnography dealing with cross-cultural studies.The fact that inequality rises with group size and that there are ecological factors involved, has beenpreviously considered [5, 16–20], but not how the rise is modulated by ecological pressure nor the hypoth-esis that this is due to the competition of cognitive and social navigation needs and therefore the influenceof climatic pressure on hierarchy can be reversed by demographics. This papers are of a theoretical naturein the spirit of the social sciences. Mathematical attempts at modeling are typically absent and at mostshow the results of regressions between pairs of variables extracted from the ethnographic data. VI. DISCUSSION
Our Statistical Mechanics approach, based on entropic inference through maximum entropy meth-ods, is a methodological approach to the mathematical-physics modeling of systems that incorporatesconditioning factors, in this case demographic, ecological, social and cognitive.Our main hypothesis is that a social-cognitive cost is relevant to characterize probabilistically theperceived social webs. The introduction of the conjugated parameter β , with the same informationalcontent of the average cost is an unavoidable theoretical consequence. It controls the size of fluctuationsabove the minimum possible value of the cost, prompting its interpretation as a pressure. Gossip, ametaphor for information exchange, correlates the perceived webs. The cognitive capacity and the size ofthe group combine into a variable z , the specific cognitive capacity, and the perceived social state can bedescribed in a space of just two dimensions ( z, β ). External to the model, the dynamics of encephalizationand band size, determine the historical evolution of z leading to a scenario for non-monotonic hierarchicalchange [1]. Further changes in z could occur, e.g. due to technological advances which translate intomore effective information processing and better social navigation. Also an effective reduction of ecologicalpressure, following enhanced productivity can occur. Then a more egalitarian perception of the socialweb will follow. The PMB hypothesis predicts that motivations and behaviors will change, but the theorydoes not go into the area of predicting how behaviors change, nor what institutions will emerge in orderto permit such behaviors, nor the time scales of these changes. Our approach to the transition fromhierarchical to egalitarian and back dispenses the issue of whether the hierarchical type of behavior laydormant (Rodseth in [1]) and remained present throughout the Pleistocene or whether the resurgence wasdue to convergent evolution (e.g. [21]). It can be turned on or off by the joint effects of cognitive resources,social demands, ecology and demography. These transitions resemble the freezing or evaporation of waterby changing pressure or temperature. The possibility of being solid ice is not dormant in water when itis heated up. At least that is not a useful metaphor.We can speculate that the time spent in the large z egalitarian phase promoted conditions for thefixation of altruistic genes and the emergence of the ”do unto others” ideas since all are equal under therepresentation web. It is hard to imagine the fixation of altruistic behavior which arises from punishmentand collaboration [22–24] in other than the symmetric phase, but this should be amenable to modelconstruction and analytic studies.This simple model and the particular function we have used to represent the cognitive-social cost arefar from complete. We don’t claim specific numerical validation by confrontation with empirical data,in any other way than just a qualitative one. More sophisticated forms of coalitions, other than dyadicpairing, should lead to increased richness of the phase diagram, without disrupting the rough overallpicture. We have also avoided considering gender issues. Rampant sexual inequalities can exist in anegalitarian organization of males. Nevertheless, if competition between cognitive constraints and socialnavigation needs indeed occur, then phase transitions from egalitarian to hierarchical perception follows2from general arguments. It has been argued [25] that “in the history of the human species, there is nomore significant transition than the emergence and institutionalization of inequality.” We expect thatthese methods, which unify the theoretical analysis of the empirical facts behind the scenario for the U-shape dynamics and the conditions that influence the transformation of perception of social organization,will stimulate the use of information theory methods in the analysis of empirical research in cross-culturalstudies. Acknowledgments:
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I. APPENDIXA. Theory: Conditional Probabilities and order parameters
The phase diagram in the β − z is shown right figure in panel 3. We divided the ranges of β and z into three regions each: harsh, intermediate and mild climates and small medium and large groupsrespectively. The phase diagram is thus divided into 9 regions. The regions are chosen essentially so thatall points in the β − z space in the harsh-large region are of the same color (blue). The same is done forthe region of harsh-small (all red) and for mild-small and mild-large. The white lines show a reasonablechoice of what is meant by large, intermediate and small both for β and n . A reasonable choice for thevalues separating the three regions are β − HI = 0 .
15 and β − IM = 0 .
35. For β − > β − IM , climate c is mild.For β − HI < β − < β − IM , climate c is intermediate, and β − < β − HI , c is harsh.For z = αn ( n − the borders are set at z = 0 . z = 1 . > z > z , s = 1 small. For z > z > z , s = 2 intermediate. For 0 < z < z , s = 3 large. Then we consider the order parameter D ( s, c ) ¯ D ( s, c ) = (cid:82) β ∈ c (cid:82) z ∈ s Ddzdβ (cid:82) β ∈ c (cid:82) z ∈ s dzdβ , (1)where D = IE ( d avg ) IE ( d max ) . and the theoretical hierarchical order parameter that can be compared to the datais H thy = 2 − D ( s, c ). B. Data: Source
Data was obtained from [9], the Ethnographic Atlas (EA) “a database on 1167 societies coded byGeorge P. Murdock and published in 29 successive installments in the journal ETHNOLOGY, 1962-1980”, available for download from the site of Douglas R. White http://eclectic.ss.uci.edu/ drwhite/worldcul/world.htm
We used the file
EthnographicAtlasWCRevisedByWorldCultures.sav
The relevant variables for our study are s , h and c , which stand for size category, hierarchy categoryand climate category. All variables can take integer values 1, 2 or 3. They are obtained by grouping theEA variables into three groups: ↓ Category, Value → TABLE I.
EA variables and categories
C. Data: Conditional Probabilities and order parameters
This values are obtained by grouping the relevant variables of the EA according to tables 1-3SM below,into three categories. The results are presented in table 4SM below. We extract the numbers of cultures4
Number of cultures in Number of cultures in Number of cultures in ↓ Stratification Climate Small groups Medium groups Large groups1 1 12 1 01 2 43 40 21 3 4 3 02 1 7 5 02 2 11 25 32 3 4 3 63 1 0 1 33 2 8 23 313 3 2 6 5
TABLE II.
Number of Cultures in the different categories in the EA. N ( s, h, c ) with a given set of values ( s, h, c ) and the marginal numbers N ( s, c ) of cultures with a given pairof values of ( s, c ) independently of h . These are related by N ( s, c ) = (cid:80) h =1 , , N ( s, h, c ). The conditionalprobabilities are P ( h | sc ) = N ( s, h, c ) N ( s, c ) , (2)of a culture having a given class stratification , given its climate and group size.Then we calculate the average hierarchy of the cultures with the same values of n and c , that is, thatbelong to the same size and climate categories. We calculate the empirical average hierarchies conditionalon size and climate ¯ H = IE ( h − | sc ) = (cid:88) h =1 , , ( h − P ( h | sc ) , (3)which satisfies 0 ≤ ¯ H ≤ σ EA = IE (( h − ¯ h ) | nc ) = (cid:88) h =1 , , ( h − ¯ h ) P ( h | nc ) , (4)can be calculated to define error bars.5 D. Numerical results ↓ Climate, Group size → Small Medium Large groupsHarsh .37 1. 2.Intermediate .44 .81 1.81Mild .80 1.25 1.45
TABLE III.
The results for the empirical stratification ¯ H ↓ Climate, Group size → Small Medium Large groupsHarsh .01 .64 1.58Intermediate .13 .56 1.25Mild .39 .58 .77
TABLE IV.
The results for the theoretical prediction H T E. Ethnographic data
N Code Description v31. Size category681 0 Missing data (code .) 0118 1 Fewer than 50 1107 2 50-99 1104 3 100-199 283 4 200-399 260 5 400-1000 216 6 1,000 without any town of more than 5,000 336 7 Towns of 5,000-50,000 (one or more) 362 8 Cities of more than 50,000 (one or more) 3
TABLE V.
Variable v31 of the Ethnographic Atlas: Mean Size of Local Communities.
N Code Description v66. Hierarchy category182 0 Missing data (code .) 0533 1 Absence among freemen (O.) 1206 2 Wealth distinctions (W.) 239 3 Elite (based on control of land or other resources (E.) 2228 4 Dual (hereditary aristocracy) (D.) 379 5 Complex (social classes) (C.) 3
TABLE VI.
Variable v66 of the Ethnographic Atlas: Class Stratification.7
N Code Description v95. Climate category869 0 Not coded 03 51 Desert (including arctic) 111 23 Tundra (northern areas) 121 36 Northern coniferous forest 18 44 High plateau steppe 15 65 Oases and certain restricted river valleys 137 52 Desert grasses and shrubs 216 56 Temperate woodland 224 74 Sub-tropical bush 227 78 Sub-tropical rain forest 264 84 Tropical grassland 214 87 Monsoon forest 2113 88 Tropical rain forest 225 54 Temperate grasslands 319 46 Temperate forest (mostly mountainous) 311 55 Mediterranean (dry, deciduous, and evergreen forests) 3
TABLE VII. v95 Climate: Primary Environment. Group 1 is formed by NPP up to 350 gC/m /year Group 2: between 350 gC/m /year and 600 gC/m /year . Group 3: large than 600 gC/m /year .8 F. Cultures: Size, Class Stratification , Climate
Table 8
List of all cultures with available information in all three categories
Culture Size(v31) Stratification(v66) Climate(v95)1 !KUNG 1 1 22 ILA 2 2 23 NYORO 2 3 24 AMBA 2 1 25 KPE 1 2 26 FON 3 3 27 KISSI 2 1 28 BAMBARA 3 3 29 YATENGA 3 3 210 KATAB 2 1 211 KONSO 3 2 312 SOMALI 1 2 213 WOLOF 3 3 214 TEDA 1 3 315 BARABRA 1 2 116 GHEG 2 1 117 NEWENGLAN 3 3 218 DUTCH 3 3 219 SERBS 3 3 220 SYRIANS 3 2 321 SINDHI 3 2 222 KAZAK 1 3 323 GILYAK 1 1 124 YAKUT 1 2 125 KOREANS 3 3 226 LOLO 2 3 327 ABOR 2 2 228 CHENCHU 1 1 229 TAMIL 3 3 230 ANDAMANES 1 1 231 MERINA 3 3 232 GARO 2 2 233 LAMET 1 2 234 MNONGGAR 2 2 235 ATAYAL 2 1 236 SAGADA 3 2 237 JAVANESE 3 3 238 MACASSARE 2 3 239 ARANDA 1 1 240 KAPAUKU 1 2 2 Culture Size(v31) Stratification(v66) Climate(v95)41 WANTOAT 1 1 242 TRUKESE 2 1 243 TROBRIAND 2 3 244 SAMOANS 1 3 245 TIKOPIA 2 3 246 NABESNA 1 1 147 TAREUMIUT 2 2 148 TWANA 1 2 149 NOMLAKI 2 2 350 TENINO 2 2 151 OJIBWA 1 1 152 HURON 2 2 153 HANO 2 1 254 CUNA 1 2 255 WARRAU 1 1 256 MUNDURUCU 1 1 257 SIRIONO 1 1 258 TUCUNA 2 1 259 INCA 3 3 160 YAHGAN 1 1 161 MATACO 1 1 262 TRUMAI 1 1 263 DOROBO 1 1 264 NAMA 2 2 265 LOZI 1 3 266 BEMBA 2 3 267 KUBA 2 3 268 CHAGGA 2 3 269 KIKUYU 2 2 270 FANG 1 2 271 ASHANTI 3 3 272 DOGON 2 2 273 TALLENSI 2 2 274 TIV 2 1 275 AZANDE 2 3 276 MASAI 1 1 277 TIGRINYA 3 3 278 SONGHAI 3 3 279 SIWANS 3 2 380 EGYPTIANS 3 3 3 Culture Size(v31) Stratification(v66) Climate(v95)81 RIFFIANS 3 2 382 ROMANS 3 3 383 IRISH 3 3 284 LAPPS 1 2 185 HUTSUL 3 2 386 PATHAN 2 3 287 KHALKA 1 3 288 CHUKCHEE 1 2 189 YURAK 1 2 190 MIAO 2 1 291 BURUSHO 2 3 192 LEPCHA 2 2 393 BENGALI 3 3 294 MARIAGOND 1 2 295 TODA 1 1 296 TANALA 2 3 297 VEDDA 1 1 298 BURMESE 3 3 299 SEMANG 1 1 2100 ANNAMESE 3 3 2101 IFUGAO 2 2 2102 SUBANUN 1 1 2103 BALINESE 2 3 2104 ALORESE 2 2 2105 MURNGIN 1 1 2106 TIWI 2 1 2107 WOGEO 1 1 2108 MAJURO 2 3 2109 IFALUK 1 1 2110 KURTATCHI 2 3 2111 LESU 2 1 2112 BUNLAP 1 2 2113 LAU 1 2 2114 PUKAPUKAN 2 1 2115 MAORI 2 3 3116 MARQUESAN 1 3 2117 COPPERESK 1 1 1118 KASKA 1 1 1119 YUROK 1 2 3120 TUBATULAB 1 1 2 Culture Size(v31) Stratification(v66) Climate(v95)121 HAVASUPAI 2 1 2122 SANPOIL 1 1 3123 OMAHA 1 1 3124 CREEK 2 1 3125 NAVAHO 2 1 2126 ZUNI 3 1 2127 AZTEC 3 3 3128 BARAMACAR 1 1 2129 TAPIRAPE 2 1 2130 JIVARO 1 1 1131 YAGUA 1 1 2132 AYMARA 2 2 1133 CAYAPA 2 1 2134 MAPUCHE 1 2 3135 BACAIRI 1 1 2136 NAMBICUAR 1 1 2137 AWEIKOMA 1 1 2138 RAMCOCAME 2 1 2139 MBUTI 2 1 2140 MBUNDU 2 3 2141 VENDA 2 3 3142 NYAKYUSA 2 1 2143 MENDE 2 3 2144 YORUBA 3 3 2145 BIRIFOR 2 1 2146 MAMBILA 2 1 3147 MARGI 2 1 2148 MAMVU 1 1 2149 SHILLUK 2 3 2150 LANGO 2 2 2151 IRAQW 2 2 2152 MZAB 3 3 3153 KABYLE 2 1 2154 TRISTAN 2 1 3155 WALLOONS 3 3 2156 CZECHS 3 3 2157 HEBREWS 3 3 3158 HAZARA 2 2 2159 KORYAK 2 2 1160 YUKAGHIR 1 1 1 Culture Size(v31) Stratification(v66) Climate(v95)161 JAPANESE 3 3 2162 MINCHINES 3 3 2163 TIBETANS 3 3 1164 COORG 2 3 2165 KERALA 3 3 2166 NICOBARES 1 1 2167 SINHALESE 3 3 2168 KACHIN 2 3 3169 PURUM 1 2 2170 CAMBODIAN 3 3 2171 HANUNOO 2 1 2172 DUSUN 2 2 2173 DIERI 1 1 2174 KARIERA 1 1 2175 KERAKI 1 1 2176 PONAPEANS 1 3 2177 YAPESE 1 3 2178 ULAWANS 2 1 2179 NASKAPI 1 1 1180 EYAK 1 2 1181 ATSUGEWI 1 2 3182 MIAMI 2 1 2183 CHEROKEE 2 1 2184 DELAWARE 2 1 2185 MARICOPA 1 1 2186 TAOS 2 1 2187 HUICHOL 2 2 2188 CHOCO 1 1 2189 CARINYA 2 1 2190 GUAHIBO 1 1 2191 CUBEO 1 1 2192 TUNEBO 2 1 2193 ONA 1 1 1194 CHOROTI 1 1 2195 CAMAYURA 2 1 2196 BOTOCUDO 1 1 2197 SOTHO 2 3 3198 YAO 2 1 2199 YOMBE 2 3 2200 GANDA 3 3 23