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Null models for Dunbar’s circles Manuel Jiménez-Mart´ın,1,∗ Ignacio Tamarit,2 Javier Rodr´ıguez-Laguna,1 and Elka Korutcheva1 1Dpto. F´ısica Fundamental, UNED, C/ Senda del Rey 9, 28040, Madrid, Spain 2Universidad Carlos III de Madrid, Grupo Interdisciplinar de Sistemas Complejos (GISC), Departamento de Matemáticas, 28911 Leganés, Madrid, Spain (Dated: February 14, 2017) Anindividual’ssocialgroupmayberepresentedbyhisego-network,formedbythelinksbetween the individual and his acquaintances. Ego-networks present an internal structure of increasingly large nested layers of decreasing relationship intensity, whose size exhibits a precise scaling ratio. Startingfromthenotionoflimitedsocialbandwidth,andassumingfixedcostsforthelinksineach layer, we propose two statistical models that generate the observed hierarchical social structure. Boththemicro-canonicalandgrand-canonicalensemblesareequivalentinthethermodynamiclimit, wherethespecificlayersizesarePoissonvariables. Inoursetting,wefurthershowthatthenumber ofconfigurationsexhibitingsuchhierarchicalorganizationisdominant. Thisresultsuggeststhat,if we assume the existence of layers demanding different amounts of resources, the observed internal 7 structure of ego-networks is indeed a natural outcome to expect. We also find that, under certain 1 conditions,equispacedlayercostsarenecessarytoobtainaconstantgroupsizescaling. Finally,we 0 compare the models with an empirical social network. 2 b I. INTRODUCTION been reported. In each case, the scaling relationship be- e tween consecutive groups holds. This hierarchical struc- F The computational capacity to store and manage ture appears to be a fundamental organizational princi- 2 ple of human groups, and has been confirmed in online 1 an ever-changing social network is thought to depend games [12], online social networks[13, 14] and telephone roughly on neocortical size, which evolved driven by the call detail records [15]. ] need of managing increasingly large social groups [1]. h This is the statement of the far reaching social brain hy- p - pothesis [2], which links brain volume in humans, pri- c mates and other mammals with the size of their social o groups. Forhumans,theDunbar’snumberconstitutesan s . upperlimitof∼150forthesocialgroupsize. Thatis,the While several null models suitable for weighted social s c total number of active relationships that we can main- networks have been proposed recently [16–18], no such i tain at any given time; a cognitive constraint that seems models have yet been proposed to analyze the observed s y to operate also in virtual environments, such as Twitter ego-networkhierarchicalstructure. Inthispaper,wepro- h [3]. Certainly,tomonitorandhandlesocialtiescomesat pose a micro-canonical and grand-canonical ego-network p a cost, it takes time to cultivate these relationships [4], statisticalensemblesthatreproducesqualitativelythehi- [ and the process is limited by cognitive constraints, such erarchical structure, and is able to fit experimental data 2 as memory and mentalising skills [5]. Current sociologi- successfully. We simplify the problem by assigning the v calstudiesrelyonlargedigitaldatasetsinordertobuild tiesineachlayerconstantcostsandpostulateanabstract 8 weighted networks of human relationships, where link social capital [19], or resource, that is spent in placing 2 weightsencodingactor-to-actorinteractionfrequencyare thelinksintothedifferentlayers. Then,byfixingtheac- 4 used as a proxy for emotional closeness [6, 7]. In this tors total degree and resource, the hierarchical structure 7 framework, an individual’s social group is equivalent to emerges spontaneously. These ensembles are unbiased 0 itssetofneighbors,whichisoftencalleditsego-network. null models for ego-networks, which offer a parsimonious . 1 Ego-networksareinternallyhighlystructuredandtheir explanation for the nested structure. They can also be 0 links can be sorted by their weights [8]. Moreover, links used to generate synthetic data with the desired layer 7 canbeclusteredintogroupsofincreasingnumberoflinks scaling, as well as for hypothesis testing against more 1 : anddecreasingemotionalcloseness[9]. Theselayersform complex models. The formalism for a single actor is a v a nested hierarchy, where the cumulative sizes of consec- micro-canonical ensemble, which is presented in section i X utive groups follow a preferred scaling ratio of approxi- II. Here we also prove that, in our setting, equispaced mately 1/3, forming a sequence of typical group sizes of costs are a condition for a constant group size scaling in r a 5, 15, 50 and 150, which are sometimes called Dun- the outer layers. Individuals within groups of many ac- bar’s circles. A smaller inner layer of size 1.5 [10], and tors with given average degree and resource are modeled two larger groups of sizes ∼500 and ∼1500 [11], have also with a grand-canonical ensemble, which is presented in sectionIII.Bothensemblesarefoundtobeequivalenton the thermodynamic limit. We fit and compare the models to an empirical dataset in section IV. Finally, section ∗Electronicaddress: manuel.jimenez@fisfun.uned.es V is dedicated to the discussion. 2 II. MICRO-CANONICAL ENSEMBLE FOR After some algebra we arrive to DUNBAR’S CIRCLES (cid:90) (cid:88)(cid:89)e(hr+λ+µsr)kr Ω= D[λ,µ]k!e−(λk+µs) (6a) Let us assume that we have a total social resource, kr! s, that we can employ to establish ties or relationships (cid:90) {kr} r of different kind with our k acquaintances. We consider = D[λ,µ] exp(cid:8)lnk!−(λk+µs)+(cid:80) ehr+λ+µsr(cid:9), r r =1,...,K different relationship layers with respective costs, s ∈R, with 0≤s ≤s and sorted such that s > (6b) r r r s . Wedefinethelayer-degree,k ,asthenumberofties r+1 r where D[λ,µ] = dλdµ/4π2. We can approximate the ofcosts . TheDunbar’scirclesareinclusivegroupingsof r integral by steepest descent leading to decreasingemotionalcloseness,hencethegroupatlevelj includes all layers of cost ≥s . The group-sizes are thus defined as nr = (cid:80)rl=1kl. Wer consider that the totality lnΩ≈lnk!−(λk+µs)+(cid:80)rehr+λ+µsr, (7) of our resource, s, is spent in the process. This can be where the values of λ, µ are those maximizing the inte- expressed by the constraints grand, and can be obtained by solving the saddle point equations (cid:88) k = k , (1a) r (cid:88) r ∂ lnΩ| =0 =⇒ k = eλ+µsr, (8) (cid:88) λ hr=0 s= srkr. (1b) r (cid:88) r ∂ lnΩ| =0 =⇒ s= s eλ+µsr. (9) µ hr=0 r r Our system can take any state, {k }, that verifies the r constraints Eqs. (1). The problem is equivalent to that Let us define the parameters x = eλ and y = eµ. The of distributing k particles among K energy levels {sr} above expressions allow us to identify the layer-degree in a quantum micro-canonical ensemble [20]. The micro- cumulants, consistently with Eqs. (3), (4) and (5). canonical ensemble assigns an homogeneous probability distribution on the configuration space, hence the prob- (cid:104)k (cid:105) = xysr, (10) r lem is solved if we are able to compute the volume of σ2 = (cid:104)k (cid:105), (11) the configuration space, Ω. We may consider each of the kr r σ = 0. (12) kr ties belonging to the layer of cost sr as either being kr,kl distinguishable or not. This choice will define the degen- Hence, the layer-degrees are uncorrelated Poisson vari- eracy of the configuration, and results in two different ables which are, however, related through the parameter partitionfunctions,ΩandΩu,respectively. Letustackle y. Wecanidentifyysr astherelativeweightofeachlayer here the distinguishable case, Ω, as each of our k ties when writing the average link weight representsasocialrelationshipwithanindividual,which atgerivmeedsniasitnnidntghsuoeicsaihapalpbcelonendevinxetniAttii.oesnsb.yTahlmeocsatlcaulllaktnioonwsnfolergΩalusayrse- s¯= ks = (cid:80)(cid:80)rrsyrysrsr. (13) Thus,yrelatestheexpectedlayer-degreescalingwiththe Ω= (cid:88) (cid:81)k!k !e(cid:80)rhrkrδ(s−(cid:80)rsrkr)δ(k−(cid:80)rkr). difference of the link costs: (cid:104)kr+1(cid:105)/(cid:104)kr(cid:105) = ysr+1−sr. On {kr} r r (2) tthhee tootthaelrdheagnrdee,,xxi=s ak/vo(cid:80)lumyestrr.ic parameter which fixes r The sum carries over k values from 0 to ∞, and the r configurations not verifying the constraints Eqs. (1) are suppressed by the deltas. We have also introduced the Constant group size scaling condition auxiliary fields h , which allow us to compute the cumu- r lants of the layer-degrees as Theexpectedgroup-sizescaling,(cid:104)n /n (cid:105),definesthe r r+1 hierarchical social structure. This quantity is the quo- (cid:104)k (cid:105) = ∂ lnΩ({k })| , (3) tient of two functions of the configuration variables and r hr r hr=0 σ2 = ∂2 lnΩ({k })| , (4) cannot be calculated analytically. Instead, we must ap- kr hr r hr=0 proximate it in terms of (cid:104)k (cid:105), (cid:104)k2(cid:105) and (cid:104)k k (cid:105), as ex- σ = ∂2 lnΩ({k })| . (5) r r r s kr,kl hr,hl r hr=0 plainedintheappendixB.Theensembleaverageisgiven by In order to compute the partition function we use an (cid:28) (cid:29) integral representation of the Dirac deltas, which imply n (cid:104)n (cid:105) r = r (1+(cid:15) ), (14) the introduction of two auxiliary parameters λ and µ. n (cid:104)n (cid:105) r+1 r+1 r+1 3 where (cid:15) is a second order correction term, which can where D({k })= (cid:0)N(cid:1)k!/(cid:81) k ! is the degeneracy of the r+1 r k r r be expressed as configuration {k }, which counts all possible ways of se- r (cid:80) lecting k = k links among N actors, and all ways (cid:15) = (cid:104)nr(cid:105)(cid:104)kr2+1(cid:105)−(cid:104)kr+1(cid:105)(cid:104)n2r(cid:105) = (cid:104)kr+1(cid:105)2−(cid:104)kr+1(cid:105)(cid:104)nr(cid:105). of assigning k dristringuishable links into layers of degree r+1 (cid:104)n (cid:105)(cid:104)n (cid:105)2 (cid:104)n (cid:105)2 k . Finally, Z and H are the partition function and cost r r+1 r+1 r (15) function, respectively. Therightmostexpressionwasobtainedbyusingtheiden- tity for Poisson variables: (cid:104)k2(cid:105)=(cid:104)k (cid:105)+(cid:104)k (cid:105)2. (cid:88) r r r Z = D({k })eH({kr}), (21) Let us now consider the scaling of two consecutive r group pairings (cid:104)n /n (cid:105) and (cid:104)n /n (cid:105). We will con- {kr} r+1 r r r−1 (cid:80) sider the implications of having a constant group-size H({kr})=γk+νs+ rhrkr, (22) scaling, such as observed in empirical relationship networks. Imposing (cid:104)n /n (cid:105)=(cid:104)n /n (cid:105) we get where γ and ν are the Lagrange multipliers and will act r+1 r r r−1 as fitting parameters in order to enforce the constraints (cid:104)n2r(cid:105)=(cid:104)nr+1(cid:105)(cid:104)nr−1(cid:105)Rr. (16) Eqs. (19). We have also included the auxiliary fields h for convenience. The maximum entropy method has The correction factor, R = (1+(cid:15) )/(1+(cid:15) ), tends r r r r+1 beenappliedtoformulatealargenumberofcomplexnet- to 1 provided that (cid:104)n (cid:105) (cid:29) 1. Indeed, this is a good r work models with prescribed features, known generally approximation for the outer layers, as shown in the inset as exponential random graphs [16, 18, 22–24]. Our ap- of Fig. 2. Considering R ≈ 1, simple manipulations r proach here seeks instead a probability distribution for lead to ego-network configurations {k }. All the information of r (cid:28)n (cid:29) (cid:104)k (cid:105) our system, including the cumulants of the layer degrees r−1 ≈ r =ysr−sr+1. (17) and group sizes, as well as their marginal distributions, n (cid:104)k (cid:105) r r+1 are recovered from the partition function, which can be This result states that, in the micro-canonical ensemble, calculated analytically. a constant group-size scaling in the outer layers is possible only if the cost difference between them is constant. Z = (cid:88)(cid:18)N(cid:19)(cid:81)k! e(cid:80)r(γ+νsr+hr)kr (23a) k k ! {kr} r r s −s =∆, (for r s.t. n (cid:29)1). (18) r r+1 r (cid:88)N (cid:18)N(cid:19) (cid:88) (cid:89)e(γ+νsr+hr)kr = k! (23b) k k ! r III. GRAND-CANONICAL ENSEMBLE FOR k=0 {kr|k} r DUNBAR’S CIRCLES (cid:88)N (cid:18)N(cid:19)(cid:32)(cid:88) (cid:33)k (cid:32) (cid:88) (cid:33)N = eγ+νsr+hr = 1+ eγ+νsr+hr . k The micro-canonical ensemble considers constant de- k=0 r r (23c) gree, k, and resource, s. However, the degree and resource do vary across individuals in real datasets. We (cid:80) canformulateagrand-canonicalensemble[20],consisting where the symbol {kr|k} on the second line denotes on N individuals, or egos, each of which having different sums over configurations {kr} with total degree k. As degree, k(j), and resource, s(j), but with given mean de- usual, derivatives of the logarithm of the partition func- gree and resource. That is, the constraints Eqs. (1) are tion give the ensemble cumulants. Analogous to Eqs. verified only on average (10), (11) and (12), we can compute the layer degrees cumulants N (cid:104)k(cid:105)=(cid:88)(cid:16)(cid:80)rkr(j)(cid:17)P({kr}), (19a) (cid:104)k (cid:105)= ∂ lnZ | = Nzr , (24) j=1 r hr hr=0 1+(cid:80) z s s N (cid:80) (cid:104)s(cid:105)=(cid:88)j=1(cid:16)(cid:80)rsrkr(j)(cid:17)P({kr}), (19b) σk2r = ∂h2rlnZ |hr=0 = 11++(cid:80)s(cid:54)=srzszs(cid:104)kr(cid:105), (25) (cid:104)k (cid:105)(cid:104)k (cid:105) The least biased distribution P({kr}) verifying the con- σkr,ks = ∂h2r,hslnZ |hr=0 =− rN s , (26) straintsEqs. (19)canbecalculatedfollowingamaximum entropy principle [21]. Maximizing the distribution entropy, S = −(cid:80) P({k })lnP({k }), subject to the where we have defined zr = eγ+νsr. Hence, in the {kr} r r grand-canonicalensemblethelayerdegreesarecorrelated constraints Eqs. (19) plus normalization, we obtain a and not poissonian. We will prove the equivalence of Gibbs distribution themicro-canonicalandgrand-canonicalensembleinthe 1 thermodynamiclimitN →∞lateron. Letusfirstwrite, P({kr})= ZD({kr})eH({kr}), (20) however,thesaddlepointequations,usedtofixthekand 4 (cid:68) (cid:69) s ensemble averages. global stats. layers s (cid:104)k (cid:105) σ (cid:104)kr−1(cid:105) (cid:104)n (cid:105) nr−1 r r kr (cid:104)kr(cid:105) r nr (cid:80) z N 84 r=1 5 2.93 3.49 2.93 (cid:104)k(cid:105)=∂ lnZ | =N r r , (27) γ hr=0 1+(cid:80) z (cid:104)k(cid:105) 73.63 r=2 4 3.68 3.30 0.80 6.61 0.39 r r (cid:80) s z σk 17.17 r=3 3 9.50 6.04 0.39 16.11 0.40 (cid:104)s(cid:105)=∂νlnZ |hr=0=N1+r(cid:80)r zr , (28) (cid:104)s(cid:105) 145.45 r=4 2 30.07 16.78 0.32 46.18 0.37 r r σ 42.71 r=5 1 27.45 15.81 1.10 73.63 0.61 s In our maximum entropy setting, Eqs. (27) and (28) are Table I: Global statistics and layer structure of the Reci- solvedforγ andν inordertoobtaintheparametervalues procity Survey dataset network [25]. that fix the desired (cid:104)k(cid:105) and (cid:104)s(cid:105). Furtherderivationsrecoverthesubsequentk andscu- mulants. For instance, the variances are It is trivial to see that Eqs. (24), (25) and (26) reduce (cid:104)k(cid:105) to the micro-canonical cumulants, Eqs. (10), (11) and σk2 =∂γ2lnZ |hr=0= 1+(cid:80)rzr, (29) (w1e2)c,awnhseene Ntha→t E∞qs..M(2o7r)eoavnedr,(n2o8t)icriendgutcheatto(cid:80)threzmr →icro0-, (cid:104)s(cid:105) σ2 =∂2lnZ | = . (30) canonical saddle point equations (8) and (9) by identify- s ν hr=0 1+(cid:80)rzr ingNeγ withx,andeνsr withysr. Thus,theequivalence of both ensembles in the thermodynamic limit is made Finally, we can write the configuration probability func- manifest. Hence, sampling from a grand-canonical en- tion, which may be sampled with Monte Carlo methods. sembleinthethermodynamiclimitcorrespondstodraw- ingindependentrandomPoissonvariablesfromthelayer (cid:18)N(cid:19) k! (cid:81) zkr degree marginal distributions. P({k })= · r r . (31) r k (cid:81)rkr! (1+(cid:80)rzr)N IV. FIT TO EMPIRICAL SOCIAL NETWORK Ensemble equivalence in the thermodynamic limit The statistical ensembles presented above may func- Let us study the thermodynamic limit for the grand- tion as null models for ego-networks, providing a bench- canonical ensemble, that is when the number of ego- mark against which to test more complicated features. networks N → ∞ while keeping (cid:104)k(cid:105) and (cid:104)s(cid:105) constant. In order to accept an ensemble as a good model for so- From Eq. (27) we can write (cid:80) z = (cid:104)k(cid:105)/(N − (cid:104)k(cid:105)). cial structure, the model should meet two demands: (i) r r This in turn, allows us to rewrite the partition function It should generate ego-network instances with k and s as valuessimilartotheempiricalones(similarmacrostate); and (ii) Those instances should present a nested layer (cid:18) (cid:104)k(cid:105)/N (cid:19)N (cid:89) structure. The assumption of constant layer costs makes Z = 1+ −−−−→e(cid:104)k(cid:105) = e(cid:104)kr(cid:105). (32) 1−(cid:104)k(cid:105)/N N→∞ the ensembles specially suited to model data from sur- r veys, where the ties weights are chosen from prede- The expected layer degrees can be expressed as (cid:104)k (cid:105) = fined discrete scores or categories. We have fitted the r z (N −(cid:104)k(cid:105)) ∼= z N, for which it is needed that z → 0 grand-canonicalensembletotheReciprocitySurvey(RS) r r r as N → ∞. Then, the configuration probability distri- dataset[25]. Inthisexperiment,atotalofN =84under- bution Eq. (31), reduces to graduate students were asked to score their relationship with each of the other participants in a scale from 0 to P({kr})= (NN−−kNk)!!(cid:89)(cid:104)kkrr(cid:105)!kre−(cid:104)kr(cid:105) −N−→−−∞→(cid:89)pr(kr), 5in,cwrehaesrieng0dmeegarneet noof-frreileantdiosnhsiph.ipW, aendha1vetoco5nrseidpereresednttehde r r zero weight as a no-link. Thus, the allowed layer costs (33) are{s ,s ,s ,s ,s }={5,4,3,2,1},whichverifytheeq- wheretheprefactortendsto1asN →∞,forfinitek. In- 1 2 3 4 5 uispaced layer cost condition, Eq. (18). The global and deed, usingStirling’sapproximationandtheexponential hierarchicalstructureoftheempiricalnetworkissumma- limit rized in Table I. The individual ego-networks show the N−kN! (cid:18) N (cid:19)N−k expected Dunbar’s structure, albeit with some remarks: ∼=e−k −−−−→1. (34) By design, the total active network is incomplete as the (N −k)! N −k N→∞ maximum possible degree is limited by the total number ofexperimentparticipants,k =N−1. Consequently, Thus,thelayerdegreemarginaldistributionsarePoisson max the outer layer degree, (cid:104)k (cid:105), departs from the expected distributions. 5 scaling. However,alltheparticipantsbelongtothesame eNzr course and live in the same campus, hence we can ex- p (k )=P(k |N,z )= (Nz )kr. (35) r r r r k ! r pect that a significant fraction of their actual social net- r 5 varianceoftheouterlayerdegreesinthemicro-canonical 300 0 ensembleisslightlylargerthanitsgrand-canonicalcoun- 10 terpart. Hence, we will refer subsequently to the en- 250 lnP(k,s) semble, withnodistinctionbetweenmicro-canonicaland 20 grand-canonical. ObservingtheRSlayerdistributionswe 200 30 findthattheempiricalaveragesofthethreefirstlayerslie s150 40 within one standard deviation of the ensemble expected values,bothfordegrees,k ,andgroupsizes,n . Onthe 50 r r 100 otherhand,theoutermostlayersdegreessufferfromfinite 60 sizeeffects. The discrepancybetween ensembleanddata 50 70 is however corrected for the accumulated variables, n , r which follow closely the ensemble scaling trend. Thus, 0 80 0 10 20 30 40 50 60 70 80 condition (ii) is verified as well. In the thermodynamic k limit we can identify the micro-canonical scaling y with the exponential of the grand-canonical parameter ν. Re- Figure 1: Contour plot for the joint log-density of states in markably, the obtained value of eν =0.56 is comparable the k-s space for the grand-canonical ensemble with parameter values γ = 1.8858 and ν = −0.5842. The red diamonds with the empirical group size scaling for the outer lay- correspond to the dataset individual ego-networks and the ers,(cid:104)n4/n5(cid:105)=0.61. Indeed,theapproximationfromEq. red cross marks the empirical averages. The lines s=k and (17) is better for the larger outer layers, where the cor- s=5k delimit the configuration minimum and maximum al- rection factor, R , tends to 1, as shown on the inset of r lowed total weight, respectively. Fig. 2. work is captured by the experiment. Indeed, the inner V. DISCUSSION groups show an almost constant ratio of approximately 0.4, which is consistent with the ratio of ∼1/3 reported We have proposed two statistical ensembles as null by larger scale studies [10, 13–15]. The degree distribu- models for the hierarchical structure of social networks. tion is peaked close to the number of participants, with Themicro-canonicalensemblemodelsasingleactorwith low variance. The weights distribution, however, is more fixeddegreeandtotalsocialresource. Theensemblegen- spread-out. eratestheobservednestedstructureofincreasinglylarge We have fitted the grand-canonical ensemble to the layers of decreasing link weight. Moreover we show that, observed data, by substituting the mean values of (cid:104)k(cid:105)= at least for the outer layers, a constant group size scal- 73.63 and (cid:104)s(cid:105) = 145.45, along with N = 84; into Eqs. ing between consecutive group pairings is possible only (27) and (28). Solving the saddle point equations nu- if the difference between costs of consecutive layers is merically, we obtained the parameter values γ = 1.8858 constant. The grand-canonical ensemble models an indi- and ν = −0.5842. The resulting distribution, along vidual’sego-networkwithinasocialgroupwithgivenav- with the data are shown in Figure 1. We have em- erage degree and social resource. In the thermodynamic ployed a Wang-Landau algorithm [26, 27] in order to limit,thatis,whenthenumberofactorsislarge,bothen- compute the joint density of states in the k-s space (the sembles are equivalent and their layer degrees are uncor- macrostate space). This function is defined as P(k,s)= related Poisson variables, which are related through the (cid:80) (cid:80) (cid:80) {kr}P({kr})δ(k− rkr)δ(s− rkrsr),whereP({kr}) group size scaling, y. Interestingly, a recent paper pro- is the grand-canonical probability distribution from Eq. viding more evidence on Dunbar’s theory further shows (31). We have made available online a python imple- the good fit of Poisson distributions to the layer-specific mentation of the algorithm [33]. As can be seen on the degree distributions [28]. In the case of the dataset ana- figure, the presence of various outliers with very low k lyzed, we observe that for typical values of social band- and s displaces the averages from the bulk of the distri- width or resource, (cid:104)s(cid:105), and number of ties,(cid:104)k(cid:105), the con- bution. Otherthanthat,theprobabilitydistributionisa figurationspaceregionwiththemaximumnumberofmi- well behaved unimodal function, hence fulfilling our first crostates is found around layer degree values {k } that r demand, (i). verify the scaling relationship. Next, let us compare the RS layer structure with the The proposed null models succeed at modeling sur- layer structure generated by both the micro-canonical vey social data, where the available categories could be and grand-canonical ensemble. Figure 2 shows the em- directly used as representations of layers. However, in pirical layer degree and layer group sizes distributions largerscalestudiesofego-networksocialstructure,layers along with the ensemble averages. It is interesting how, are not given a priori, but rather inferred from the in- despite the small size of the RS network, already the teraction patterns. The link weights represent frequency grand-canonical ensemble is barely distinguishable from of interaction which acts as a proxy for emotional close- the micro-canonical ensemble. Both ensembles average ness, and ties are clustered into discrete groups accord- layerdegreesandgroupsizesareequivalent,andonlythe ingtotheseweights[10,13–15]. Itisimportanttoassert 6 80 names: the support clique of size ∼5, the sympathy group of size ∼15, the affinity group of size ∼50, and the total 1.2 R active network whose size equals the Dunbar’s number, 1.1 r 60 ∼150 [9]. Assigning those layers a constant cost is not 1.0 only a convenient simplification, but has also a rather 0.9 kr 2 3 4 40 natural interpretation. Indeed the very existence of the layers would implicitly define different types of relationships for the ego. We simply consider that relationships 20 within a given layer are similar precisely because they have the same cost. Then, the problem we focused on 0 wastocomputeinhowmanydifferentwaysagivennum- berof actors canbe distributedin an ego’snetworkwith 100 theabovementionedrestrictions. Inoursetting,oncethe layers, the total degree and the total social resource are 80 fixed,thehierarchicalstructureemergesinanaturalway, as the number of possible configurations with few strong 60 r and many weak links is much higher than configurations n 40 made of only strong or weak links. This result suggests that the observed hierarchical structure could be a con- 20 sequence of the existence of an internal discrete catego- rization in which individuals organize different types of 0 relationships. However, the reason why those categories 1 2 3 4 5 or layers would exist in the first place remains unknown r and shall be further explored. Another interesting ques- tion that arises naturally from our work is how a given Figure 2: Layer degrees (top) and layer group sizes (bot- number of ego-networks, as the ones generated by our tom)distributions. TheempiricalRSdistributionsarerepre- models, can be linked together in a way consistent with sented by the black box-plots, where the box comprises the theglobalstructureobservedinrealsocialnetworks[29– secondandthirdquartiles,separatedbythemedianline. The 31]. Solving this puzzle would allow us to generate null whiskers extend to the full distribution domain, and the av- models for societies compatible with the observed regu- erages are represented by the black diamonds. The dashed larities in the ego-networks. colored lines join the corresponding ensemble averages, (cid:104)k (cid:105) r (blue, top), and (cid:104)n (cid:105) (red, bottom) from a grand-canonical r (light-colored)andamicrocanonicalensembles(dark-colored) which lay practically identical values. One standard devia- Acknowledgments tion from the averages are represented by the shaded regions for the grand-canonical ensemble, and by the dotted lines M. J.-M. would like to thank Ignacio Morer, and pro- for the microcanonical. The parameter values are N = 83, fessorsConradPérezVicenteandAlbertD´ıaz-Guilerafor γ = 1.8858 and ν = −0.5842; and x = 61.76, y = 0.56 their inspiring work and encouraging conversations. I. forthegrand-canonicalandmicrocanonicalensemble,respec- T.wouldliketothankAngelSánchezandRobinDunbar tively. Finally, the inset shows the numerical value of the forinspiringconversationsandacknowledgespartialsup- micro-canonical correction factor, R = (1+(cid:15) )/(1+(cid:15) ), r r r+1 portingfromtheEUthroughFET-OpenProjectIBSEN which tends to 1 for the outer layers. and from Fundación BBVA through project DUNDIG. J. R.-L. acknowledges partial supporting from MINECO through grant FIS2015-69167-C2-1. that we do not identify our link weights with interaction frequency. Rather, we introduce the abstract notion of social resource, which can be spent in assigning discrete Appendix A: Non-degenerate micro-canonical weights to the social ties. In order to justify this con- ensemble structionitcanbearguedthatalimitedsocialbandwidth may arise both from biological or temporal constraints, In a non-degenerate micro-canonical ensemble, the since maintaining a social relationship is costly [1, 3]. identity of the alter nodes are interchangeable and con- Nevertheless our models are agnostic regarding what the figuration states, {k }, are counted once. Thus, the par- nature of that cost can be in either psychological or so- r tition function is ciological terms. Another key ingredient of our models isstrtehsse tdhiasctreitteisnnaotutroeuorfinthteenltaioynerst.o nIteiitsheimr jpuosrttiafyntnotor Ωu = (cid:88)e(cid:80)rhrkrδ(s−(cid:80)rsrkr)δ(k−(cid:80)rkr). (A1) provide any sort of explanation to their existence. We {kr} rather rely on the existing literature, where these lay- Again, the Dirac deltas can be expressed in its integral ers have been consistently identified and given specific form. The exponent factors over the k ’s, hence product r 7 and sum can be exchanged and the geometric sums over Inthenon-degenerateensemble,theinterpretationofpa- k values can be solved. rameter y as the group size scaling does not hold. r (cid:90) (cid:88)(cid:89) Ωu = D[λ,µ]e−(λk+µs) e(hr+λ+µsr)kr (A2a) {kr} r (cid:90) = D[λ,µ] exp(cid:8)−(λk+µs)−(cid:80) ln(cid:2)1−ehr+λ+µsr(cid:3)(cid:9). r (A2b) Appendix B: Cumulants of a quotient function Then, approximating the integral by steepest descent, we arrive to lnΩu ≈−(λk+µs)−(cid:88)ln(cid:2)1−ehr+λ+µsr(cid:3). (A3) The cumulants of a quotient function r(x,y) = x/y cannot be obtained directly from the partition function. r However,theycanbeapproximatedbyexpandingr(x,y) After renaming x = eλ and y = eµ, the saddle point around the expected values, (cid:104)x(cid:105) and (cid:104)y(cid:105), as done in ref- equations read erence [32]. The mean and variance up to second order are given by (cid:88) xysr (cid:88) xysr k = , s= s . (A4) 1−xysr r1−xysr r r (cid:104)x(cid:105)(cid:20) (cid:104)y2(cid:105) (cid:104)xy(cid:105) (cid:21) Finally, the layer degree cumulants are (cid:104)r(cid:105)= 1+ − , (B1) (cid:104)y(cid:105) (cid:104)y(cid:105)2 (cid:104)x(cid:105)(cid:104)y(cid:105) (cid:104)k (cid:105) = xysr , (A5) (cid:104)x(cid:105)2 (cid:20)(cid:104)x2(cid:105) (cid:104)y2(cid:105) 2(cid:104)xy(cid:105)(cid:21) r 1−xysr σr2 = (cid:104)y(cid:105)2 (cid:104)x(cid:105)2 + (cid:104)y(cid:105)2 − (cid:104)x(cid:105)(cid:104)y(cid:105) . (B2) xysr σ2 = , (A6) kr (1−xysr)2 σ = 0. (A7) kr,kl References nationalconferenceonand2012internationalconfernece on social computing (SocialCom) (IEEE, 2012), pp. 31– 40. [1] R.Dunbar,JournalofHumanEvolution22,469 (1992), [14] V. Arnaboldi, M. Conti, A. Passarella, and F. Pezzoni, ISSN 0047-2484. inINFOCOM,2013ProceedingsIEEE (IEEE,2013),pp. [2] R. Dunbar, Brain 9, 178 (1998). 3459–3464. [3] B. Gonçalves, N. Perra, and A. Vespignani, PLoS ONE [15] P.M.Carron,K.Kaski,andR.Dunbar,SocialNetworks 6, 1 (2011). 47, 151 (2016), ISSN 0378-8733. [4] G.Miritello,R.Lara,M.Cebrian,andE.Moro,Scientific [16] O. Sagarra, C. J. Pérez Vicente, and A. D´ıaz-Guilera, Reports 3, 1950 EP (2013). Phys. Rev. E 88, 062806 (2013). [5] J.StillerandR.Dunbar,SocialNetworks29,93 (2007), [17] T. Squartini, F. Picciolo, F. Ruzzenenti, and D. Gar- ISSN 0378-8733. laschelli, Scientific Reports 3, e1003374 (2013). [6] S. G. B. Roberts and R. I. M. Dunbar, Evolution and [18] O. Sagarra, C. J. Pérez Vicente, and A. D´ıaz-Guilera, Human Behavior 32, 186 (2011), ISSN 1090-5138. Phys. Rev. E 92, 052816 (2015). [7] V.Arnaboldi,A.Guazzini,andA.Passarella,Computer [19] A. Antonioni, S. Bullock, and M. Tomassini (2014). Communications 36, 1130 (2013), ISSN 0140-3664. [20] M. Kardar, Statistical physics of particles (Cambridge [8] J. Sarama¨ki, E. A. Leicht, E. Lo´pez, S. G. B. Roberts, University Press, 2007). F. Reed-Tsochas, and R. I. M. Dunbar, Proceedings of [21] E. T. Jaynes and G. L. Bretthorst, eds., Probability the- the National Academy of Sciences 111, 942 (2014). ory : the logic of science (Cambridge University Press, [9] W.-X.Zhou,D.Sornette,R.A.Hill,andR.I.M.Dunbar, Cambridge, UK, New York, 2003), ISBN 0-521-59271-2. ProceedingsoftheRoyalSocietyofLondonB:Biological [22] J. Park and M. E. J. Newman, Phys. Rev. E 70, 066117 Sciences 272, 439 (2005), ISSN 0962-8452. (2004). [10] R. Dunbar, V. Arnaboldi, M. Conti, and A. Passarella, [23] D. Garlaschelli and M. I. Loffredo, Phys. Rev. E 78, Social Networks 43, 39 (2015), ISSN 0378-8733. 015101 (2008). [11] R. I. M. Dunbar, Behavioral and Brain Sciences 16, [24] K. Anand and G. Bianconi, Phys. Rev. E 80, 045102 681–694 (1993). (2009). [12] B.Fuchs,D.Sornette,andS.Thurner,ScientificReports [25] A.Almaatouq,L.Radaelli,A.Pentland,andE.Shmueli, 4, 6526 EP (2014), article. PLoS ONE 11, 1 (2016). [13] V.Arnaboldi,M.Conti,A.Passarella,andF.Pezzoni,in [26] F. Wang and D. P. Landau, Phys. Rev. Lett. 86, 2050 Privacy, security, risk and trust (PASSAT), 2012 inter- 8 (2001). [31] N. Z. Gong and W. Xu, Social Network Analysis and [27] R. Fischer, J. C. Leit∼ao, T. P. Peixoto, and E. G. Alt- Mining 4, 184 (2014), ISSN 1869-5469. mann, Phys. Rev. Lett. 115, 188701 (2015). [32] O. Sagarra, F. Font-Clos, C. J. Pérez-Vicente, and [28] T.Kordsmeyer,P.MacCarron,andR.Dunbar,Current A. D´ıaz-Guilera, EPL (Europhysics Letters) 107, 38002 Anthropology 58, 000 (2017). (2014). [29] M. S. Granovetter, American Journal of Sociology 78, [33] It is straightforward to generalize a Wang-Landau algo- 1360 (1973). rithmtocomputeajointbivariatedensityofstatesinan [30] J.-P. Onnela, J. Sarama¨ki, J. Hyvo¨nen, G. Szabo´, integervaluedconfigurationspace.Thecodeinpythonis D.Lazer,K.Kaski,J.Kertész,andA.-L.Baraba´si,Pro- available at https://manu-jimenez.github.io/blog/ ceedings of the National Academy of Sciences 104, 7332 (2007).