Emergence of simple characteristics for heterogeneous complex social agents
EEmergence of simple characteristics for heterogeneouscomplex social agents
Eric Bertin
Univ. Grenoble Alpes, CNRS, LIPHY, F-38000 Grenoble, France
Abstract
Models of interacting social agents often represent agents as very simple en-tities having a small number of degrees of freedom, as exemplified by binaryopinion models for instance. Understanding how such simple individual char-acteristics may emerge from potentially much more complex agents is thusa natural question. It has been proposed recently in [E. Bertin, P. Jensen,C. R. Phys. , 329 (2019)] that some types of interactions among agentswith many internal degrees of freedom may lead to a ‘simplification’ of agents,which are then effectively described by a small number of internal degreesof freedom. Here, we generalize the model to account for agents intrinsicheterogeneity. We find two different simplification regimes, one dominatedby interactions, where agents become simple and identical as in the homoge-neous model, and one where agents remain strongly heterogeneous althougheffectively having simple characteristics. Keywords:
Sociophysics, statistical physics, heterogeneity, phasetransition.
1. Introduction
When modeling complex systems, statistical physicists often posit thatthe interacting entities they consider have simple individual properties, andthat the possibly more complex behavior observed at a collective level, whenconsidering many such simple entities, is simply the result of interactions be-tween entities [1, 2, 3, 4]. The emergence of macroscopically different prop-erties from simpler interacting entities, often through a collective symmetrybreaking mechanism, has been emphasized long ago by P. W. Anderson in hisseminal paper ”More is Different” [5] as one of the key mechanisms at play
Preprint submitted to Elsevier July 3, 2020 a r X i v : . [ phy s i c s . s o c - ph ] J u l o account for the wealth of different objects or behaviors found in the realworld. In this paper, Anderson actually presents the statistical physics ap-proach as a generic and somewhat iterative method, to deal with the emergingcomplexity in a multilevel way, the outcome of a given level of analysis beingthe building blocks (i.e., the interacting entities) of the next level. The keyrole played by symmetry breaking phenomena in the emerging complexityat collective scale has been widely acknowledged in many different contexts(see, e.g., [6, 7, 2, 8] among many others). Interestingly, a perhaps less ex-plicit suggestion of Anderson’s paper is also to consider statistical physicsmodels where the interacting entities are not simple objects with a handfulof characteristics, but are already rather complex objects with many internaldegrees of freedom. Although it has been formulated almost fifty years ago,this suggestion looks very timely by now.In the last decades, statistical physics has gone beyond the equilibriumparadigm based on molecular entities, and has indeed started to consider as-semblies of macroscopic and potentially complex objects like grains of sand[9, 10], or active particles modeling for instance some types of bacteria, orself-phoretic colloids [8]. However, in these examples, macroscopic particlesmay in practice still be considered as simple particles, as their many internaldegrees of freedom can be subsumed into a small number of effective param-eters encoding their macroscopic, non-equilibrium character, like dissipationcoefficients [9, 10] or self-propulsion forces [8]. To find genuine examples ofassemblies of complex entities in a theoretical context, one may rather turn tothe field of population dynamics and evolution in theoretical biology, wherefor instance large populations of individuals characterized by a complicatedgenome evolve under some evolutionary rules [11, 12].When considering models of social systems, it would be necessary at firstsight to take into account the intrinsic complexity of human beings [1]. How-ever, this complexity is too extreme to be captured by any type of statisticalphysics models, so that statistical physicists have often considered very sim-ple models of social agents, retaining a small number of characteristics, forinstance a binary [13] or continuous [14] opinion. A natural question is thento understand how more complex agents could in some situations reducetheir intrinsic complexity to effectively appear as simple. This question canalready be addressed within the framework of statistical physics, becausethere is no need to model the full complexity of human beings to address atleast some aspects of this issue. A simple tentative answer has been givenin [15], by illustrating on a toy model how the simplification of agents with2any internal degrees of freedom may result from interactions among agents.This point of view is qualitatively consistent with the idea advocated by somesociologists that ‘the whole is less than the parts’ [16, 17], in the sense that,roughly speaking, human beings may leave aside part of their complexityto build a group. It has been argued in [15] that this point of view can bereconciled with the seemingly antagonist viewpoint of statistical physicistsaccording to which ‘the whole is more than the sum of the part’, due tocollective phenomena and symmetry breakings.Yet, the model introduced in [15] considered an assembly of identicalagents, while agents heterogeneity may be expected to be an important char-acteristics of human beings. In this note, we extend the model of [15] byincluding heterogeneity between agents. We show, using methods inspiredby the physics of glasses, that two different types of agents simplification canoccur in this model, one driven by interactions as in [15], and the other onedriven by heterogeneity.
2. Model
We introduce a model of complex agents generalizing the one introducedin [15] by now considering heterogeneous agents. The model is composed of N interacting agents having an internal state described by a configuration C i ∈ { , ..., H } , with i = 1 , . . . , N and where the number H of configurationsis large. We write for later convenience H in the form H = n M , where n is afixed integer number, and M is assumed to be large. This is typically the caseif the configuration C i is composed of M degrees of freedom, each of whichtaking n possible values. Each agent is endowed with a characteristic thatcan be either present or absent depending on the configuration C i . Intuitively,this characteristic could be a preference for a specific kind of music or tempofor the members of a vocal ensemble [17] for instance, or more generally berelated to a binary opinion [13]. This feature is conveniently encoded by avariable S i ( C i ) ∈ { , } : the characteristic is present when S i ( C i ) = 1, andabsent when S i ( C i ) = 0. The general idea is that the characteristic would bepresent only in a small number of internal states, so that it would typicallyremain unobserved except if there is a strong probability bias towards the fewconfigurations for which S i ( C i ) = 1. To implement this idea in practice in themodel, we assume that the characteristic is present in a single configuration,that we label C i = 1. 3e now need to define the dynamics of agents. Following standard prac-tice in the modeling of social agents [18], we assume that the dynamics isdriven by an individual utility function u i = u i ( C i |C j (cid:54) = i ) that accounts both forindividual preferences and for interactions with agents. An agent i stochas-tically changes configurations according to the following rule. Given thecurrent configuration C i , the new configuration C (cid:48) i is randomly chosen with aprobability rate given by the logit rule, W ( C (cid:48) |C ) = 11 + e − ∆ u i /T (1)with ∆ u i = u i ( C (cid:48) i ) − u i ( C i ) the variation of utility generated by the change ofconfiguration (note that configurations C j of the other agents j (cid:54) = i are keptfixed). The parameter T plays a role similar to temperature in statisticalphysics, and characterizes the degree of stochasticity in the decision rule.Our goal is to model heterogeneous agents that interact through theircharacteristic S i (their internal state is otherwise invisible to other agents).With this aim in mind, we choose the following form of the utility functionof agent i , u i ( C i |C j (cid:54) = i ) = U i ( C i ) + KN (cid:88) j ( (cid:54) = i ) S i ( C i ) S j ( C j ) , (2)where U i ( C i ) is the intrinsic (or idiosyncratic) utility of configuration C i foragent i , and K is the coupling constant characterizing the interaction withthe other agents. Note the 1 /N scaling of the interaction term, typical of fullyconnected models where all particles or agents interact with each other in asimilar way. This interaction terms was already present in the homogeneousmodel of Ref. [15]. We then model the heterogeneity of agents as a quenchedrandomness of the intrinsic utilities (which were absent from the model of[15]). More precisely, for all i = 1 , . . . , N and C i = 1 , . . . , H , the intrinsicutility U i ( C i ) is randomly drawn from a Gaussian distribution ρ ( U ), ρ ( U ) = 1 √ πM J e − U /MJ , (3)where we recall that M is defined by H = n M . The utilities U i ( C i ) do notchange in time.The utility variation ∆ u i can be reformutated as the variation ∆ E = ∆ u i of a global observable E that plays a role similar to the energy in physics (up4o a change of sign). Note that contrary to u i , the pseudo-energy E does notdepend on the agent i .Here, the function E takes the form E ( C , . . . , C N ) = N (cid:88) i =1 U i ( C i ) + K N (cid:88) i,j ( i (cid:54) = j ) S i ( C i ) S j ( C j ) . (4)The quantity E is thus different from the total utility (cid:80) i u i , due to the factor in the interaction term. Note also that the present model shares similaritieswith the Random Energy Model [19, 20, 21], as well as with the Ising model[7], the Potts model [22] or the Blume-Emery-Griffiths spin-1 model [23].However, it also exhibits important differences with each of these models.Given the property ∆ u i = ∆ E , the dynamics defined by the transitionrate Eq. (1) obeys the detailed balance property in terms of the equilibriumdistribution P ( C , . . . , C N ) = 1 Z e βE ( C ,..., C N ) , (5)with β ≡ /T , and where Z is a normalization constant. Now that we de-termined the equilibrium distribution of the model, our goal is to investigateits phase diagram to assess the effect of the competition between agents het-erogeneity (due to their quenched intrinsic utility) and collective effects thatcould arise from interactions. As recalled in the introduction, the homoge-neous version of the model, studied in [15], exhibits a transition driven byinteractions between a phase where agents essentially visit all their internalstates and thus have no strongly preferred configurations, and an orderedstate where all agents ‘standardize’ in the same configuration C such that S i = 1 (i.e., the characteristic of the agent becomes visible). In the presentgeneralization of the model, we wish to explore whether the standardizedstate survives the heterogeneity of agents intrinsic utility. Indeed, the inter-nal state C such that S i = 1 may have a lower intrinsic utility U i ( C ) thanother more favored configurations of agent i , and this may prevent a commonstandardization of all agents in the same state.To investigate this issue, we introduce the order parameter q defined as q = 1 N N (cid:88) i =1 S i . (6)If the number N of agents is large, the pseudo-energy E can be expressed in5erms of the order parameter q as E = N (cid:88) i =1 U i + 12 N Kq . (7)We now wish to determine the distribution P ( q ) of the order parameter q .This distribution is obtained by summing the joint distribution P ( C , . . . , C N )over all configurations ( C , . . . , C N ) having a given value of q , namely P ( q ) = (cid:88) C ,..., C N P ( C , . . . , C N ) δ (cid:16) q ( C , . . . , C N ) − q (cid:17) (8)with δ the Kronecker delta. For a given value of q , there are qN agentsin configuration C i = 1 and (1 − q ) N agents in any of the other n M − S q,N a subset with qN elements of the set[1 , N ]. With these notations, P ( q ) can be written as P ( q ) = 1 Z (cid:88) S q,N (cid:89) i ∈S q,N e βU i (1) (cid:89) i/ ∈S q,N n M (cid:88) C i =2 e βU i ( C i ) e βNKq , (9)where in the second product, the index i is implicitly restricted to the in-terval [1 , N ]. The distribution P ( q ) defined in Eq. (8) actually depends onthe specific realization of the random utilities U i ( C i ), so that P ( q ) shouldin principle be eventually averaged over these random utilities. However,to make calculations easier, we do not compute explicitly the average over U i ( C i ), but rather use heuristic arguments to evaluate the typical values ofthe random quantities appearing in Eq. (9). Such an estimate is expected tobe sufficient to determine the leading exponential behavior P ( q ) ∼ e − Nf ( q ) ofthe distribution P ( q ). The first product between brackets in Eq. (9) can berewritten as the exponential of (cid:80) i ∈S q,N βU i (1). The latter sum is (up to thefactor β ) a sum of independent and identically distributed Gaussian randomvariables drawn from the distribution Eq. (3). The sum is thus also Gaussiandistributed, with zero mean and variance qN m ( βJ ) . It follows that (cid:89) i ∈S q,N e βU i (1) ∼ e O ( √ N ) , (10)and this product can thus be neglected (i.e., replaced by 1) when looking forthe behavior of P ( q ) at exponential order in N .6he key point in order to determine P ( q ) explicitly is now to evaluatethe typical value Z typ of the sum Z i = n M (cid:88) C i =2 e βU i ( C i ) , (11)where we recall that each U i ( C i ) is an independent random variable drawnfrom the distribution ρ ( U ) given in Eq. (3). Once this estimate is known,the distribution P ( q ) can be approximated as P ( q ) ∼ Z (cid:18) NqN (cid:19) Z (1 − q ) N typ e βNKq . (12)The quantity Z i has the same form as the partition function of the RandomEnergy Model (REM) [19, 20], and we can thus borrow some methods fromthe REM to evaluate it. A standard approach to study the REM is to evaluatethe density of states of a typical realization of the disorder. It is convenientat this stage to define the rescaled variable u = U/M . Denoting as n ( u ) thedensity of state of a given realization, the average (cid:104) n ( u ) (cid:105) over the disorder isgiven by (cid:104) n ( u ) (cid:105) = n M ˜ ρ ( u ) ∼ e M (ln n − u /J ) (13)to exponential order in M , having defined ˜ ρ ( u ) = M ρ ( M u ). Hence if ln n − u /J >
0, corresponding to | u | < u = J √ ln n , the average density of state (cid:104) n ( u ) (cid:105) is exponentially large with M , and for a typical sample n ( u ) ≈ (cid:104) n ( u ) (cid:105) .By contrast, when | u | > u , the average density of state is exponentially smallwith M , meaning that in most realizations there are actually no states with | u | > u (the exponentially small value of the average density of states comesfrom very rare and atypical realizations having a few states in this range).Therefore, to describe a typical realization, one can in practice consider thatthe density of state is equal to zero for | u | > u , and equal to (cid:104) n ( u ) (cid:105) for | u | < u . We can thus evaluate Z i as Z i = (cid:88) C i e βU i ( C i ) ≈ (cid:114) MπJ (cid:90) u − u du e M (ln n − u /J + βu ) ≡ Z typ . (14)The integral in Eq. (14) can be evaluated by a saddle-point calculation inthe large M limit. Defining g ( u ) = ln n − u /J + βu , Z typ is given forlarge M by Z typ ∼ e Mg max , where g max is the maximum value of g ( u ) over the7nterval [ − u , u ]. Let us first look for the maximum of g ( u ) over the wholereal axis. Defining u ∗ such that g (cid:48) ( u ∗ ) = 0, we find u ∗ = βJ . Recallingthat β = 1 /T , and defining T g = J √ ln n , (15)we find that − u < u ∗ < u for T > T g , whereas u ∗ ≥ u for T ≤ T g . Hence g max = g ( u ∗ ) for T > T g , and g max = g ( u ) for T ≤ T g , taking into accountthe fact that g ( u ) is monotonously increasing for u < u ∗ . We thus obtain Z typ ≈ (cid:40) e M (ln n + β J ) if T > T g ,e MβJ √ ln n if T ≤ T g . (16)Using Eqs. (12) and (16), we obtain after expanding the factorials thanks tothe Stirling formula that P ( q ) takes a large deviation form P ( q ) ∼ e Nf ( q ) . Inphysical terms, f ( q ) may be thought of as (the opposite of) a free energy. Theexplicit expression of f ( q ) depends on the temperature range. For T > T g , f ( q ) reads f ( q ) = − q ln q − (1 − q ) ln(1 − q ) + (1 − q ) M (cid:20) ln n + J T (cid:21) + 12 T Kq , (17)whereas for T ≤ T g , f ( q ) = − q ln q − (1 − q ) ln(1 − q ) + (1 − q ) M JT (cid:112) ln n + 12 T Kq . (18)Note that we did not take into account in f ( q ) the contribution coming fromthe normalization factor Z , as this would simply add a constant to f ( q ).Inspired by Ref. [15], where interesting results were obtained for a couplingconstant K ∼ M , we assume in what follows that K = kM , (19)and take the reduced constant k as the relevant control parameter in themodel (on top of temperature T ). For large M , the expression of f ( q ) thensimplifies to f ( q ) = (cid:40) MT (cid:104) (1 − q ) (cid:16) T ln n + J T (cid:17) + kq (cid:105) if T > T g , MT (cid:2) (1 − q ) J √ ln n + kq (cid:3) if T ≤ T g . (20)8e first observe that in this large- M approximation, f ( q ) is a convex functionof q for all values of temperature T , so that the maximum of f ( q ) over theinterval 0 ≤ q ≤ f (0) or f (1). The most probable state is foundto be q = 0 for k < k c ( T ) and q = 1 for k > k c ( T ), where the critical line k c ( T ) is defined as (with k = 2 J √ ln n ) k c ( T ) = (cid:40) k (cid:16) TT g + T g T (cid:17) if T > T g ,k if T ≤ T g . (21)The curve k c ( T ) thus separates the ( k, T ) phase diagram into two regions, aregion with q = 0 at low coupling and a region with q = 1 at high coupling.The corresponding phase diagram is plotted in Fig. 1. Note that for J = 0(i.e., in the absence of disorder), the glassy region in the phase diagramdisappears, and one recovers the phase transition at T c = k/ (2 ln n ) betweena high-temperature phase with q = 0 and a low-temperature phase with q = 1found in the homogeneous model [15].Besides, we have also seen that a change of behavior occurs at T = T g .For T > T g , the agents dynamically visit a large number of configurations,while for T < T g their dynamics becomes essentially frozen, and only fewconfigurations have a significant probability to be visited. In other words,for T < T g agents become ‘stuck’ in a small number of configurations havingthe highest utility. In the context of the Random Energy Model for glasses,the temperature T g corresponds to the glass transition.Hence there are actually three different regions in the phase diagramshown in Fig. 1. For T > T g and k < k c ( T ), agents have no preferred con-figurations and visit many different configurations over time. For T < T g and k < k c ( T g ), each agent spends a lot of time in a small set of preferredconfigurations. In other words agents look simple, but they remain differentone from the other. This regime is dominated by agents heterogeneity, andthere is on average no macroscopic overlap between agents configurations( q = 0). In the last region k > k c ( T ), the coupling between agents dom-inates over agents heterogeneity, and all agents are essentially in the sameconfiguration, leading to a strong overlap ( q = 1) and to the emergence of acommon characteristic, a phenomenon that has been called standardizationin [15]. 9 T g T0k k Simple homogeneous agents (q = 1)Simple heterogeneousagents (q = 0) Complex heterogeneousagents (q = 0)
Figure 1: Phase diagram of the model in the ( k, T )-plane (reduced coupling constant k = K/M versus temperature), showing the three different regions separated by the critical line k c ( T ) (full line) and the ‘glass’ temperature T g (dashed vertical line). For k < k c ( T ), agentsconfigurations do not overlap ( q = 0), while for k > k c ( T ) agents become standardized byinteractions ( q = 1). The q = 0 area is subdivided into high- and low-temperature regions.For T > T g , agents may be in any of their internal configurations, while for T < T g , theyare dynamically blocked in the few configurations with the highest intrinsic utility. In thislatter case, agents appear simple but remain heterogeneous.
3. Conclusion
We have extended here the model proposed in [15], where agents havemany internal configurations, but can select a specific configuration thanksto interactions, leading to simplified, or standardized agents. While Ref. [15]focused on homogeneous agents, we have in the present work extended themodel to account for agents heterogeneity, introduced through random id-iosyncratic utilities associated with each configurations of each agent. Het-erogeneity is introduced in a minimal way, directly inspired from the RandomEnergy Model for glasses [19]. Including heterogeneity in the model leads tothe onset of a new phase at low temperature and small coupling, when het-erogeneity dominates over interactions among agents. In physical terms, thisnew phase shares similarity with a glass phase. The main difference here isthat we consider not a single glassy system as in physics, but a large assemblyof interacting agents, each of which experiencing an internal glass transition.10n this glassy phase, agents are stuck in a small number of possible internalconfigurations, those with maximal idiosyncratic utility. In spite of the indi-vidual simplification of agents, the assembly remains heterogeneous, becauseall agents are in different configurations and do not share common charac-teristics. Increasing the coupling strength, interactions eventually dominateover heterogeneity, effectively leading to simple agents all occupying the sameconfiguration and sharing the same characteristic. The latter phase is similarto the low temperature phase found in the homogeneous model of [15], whereagents have been called standardized. Finally, at high temperature, agentskeep their internal complexity and do not significantly feel interactions orheterogeneity, again similarly to the homogeneous model [15].A further step in the study of this model would to study possible collectiveeffects that could arise once agents are simplified. This has been done in thehomogeneous model by assuming that standardized agents could be in oneof two distinct states, corresponding for instance to two different opinions(or to a spin in physical terms). Interactions between these binary degrees offreedom may then lead, at low temperature, to a collectively ordered state.It would be interesting to see in more details how agents heterogeneity couldpossibly modify this simple picture. In addition, the role of a lower connectiv-ity (in the present work, agents interact with all other agents) would clearlydeserve to be investigated, as connectivity is known in many situations tomodify critical properties [7].
Acknowledgment
The author is grateful to Pablo Jensen for a critical reading of the manuscriptand interesting comments.
References [1] C. Castellano, S. Fortunato, V. Loreto, Statistical physics of socialdynamics, Rev Mod Phys 81 (2009) 591.[2] J.-P. Bouchaud, M. M´ezard, J. Dalibard (Eds.), Complex systems, El-sevier, 2007.[3] A. Barrat, M. Barth´elemy, A. Vespignani, Dynamical Processes on Com-plex Networks, Cambridge University Press, 2008.114] J.-P. Bouchaud, Crises and collective socio-economic phenomena: simplemodels and challenges, J. Stat. Phys. 151 (2013) 567.[5] P. W. Anderson, More is different, Science 177 (1972) 393.[6] P. M. Chaikin, T. C. Lubensky, Principles of condensed matter physics,Cambridge University Press, 1995.[7] M. Le Bellac, Quantum and statistical field theory, Oxford UniversityPress, 1992.[8] M. C. Marchetti, J.-F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost,M. Rao, R. A. Simha, Hydrodynamics of soft active matter, Reviews ofModern Physics 85 (2013) 1143.[9] P. G. de Gennes, Granular matter: a tentative view, Rev. Mod. Phys.71 (1999) S374.[10] A. Puglisi, Transport and fluctuations in granular fluids, Springer, 2015.[11] B. Drossel, Biological evolution and statistical physics, Adv. Phys. 50(2001) 209.[12] G. Sella, A. E. Hirsh, The application of statistical physics to evolution-ary biology, Proc. Nat. Acad. Sci. USA 102 (2005) 9541.[13] K. Sznajd-Weron, J. Snajd, Opinion evolution in closed community, Int.J. Mod. Phys. C 11 (2000) 1157.[14] G. Deffuant, D. Neau, F. Amblard, G. Weisbuch, Mixing beliefs amonginteracting agents, Advances in Complex Systems 3 (2001) 87.[15] E. Bertin, P. Jensen, In social complex systems, the whole can be moreor less than (the sum of) the parts, C. R. Physique 20 (2019) 329.[16] B. Latour, P. Jensen, T. Venturini, S. Grauwin, D. Boullier, The wholeis always smaller than its parts, a digital test of gabriel tardes monads,The British Journal of Sociology 63 (2012) 590.[17] P. Jensen, The politics of physicists’ social models, C. R. Physique 20(2019) 380. 1218] D. Phan, M. B. Gordon, J.-P. Nadal, Social interactions in economic the-ory: an insight from statistical mechanics, in: J.-P. Nadal, P. Bourgine(Eds.), Cognitive Economics, Springer, 2004, p. 335.[19] B. Derrida, Random-energy model: Limit of a family of disorderedmodels, Phys. Rev. Lett. 45 (1980) 79.[20] B. Derrida, Random-energy model: An exactly solvable model of disor-dered systems, Phys. Rev. B 24 (1981) 2613.[21] J.-P. Bouchaud, M. M´ezard, Universality classes for extreme-valuestatistics, J. Phys. A: Math. Gen. 30 (1997) 7997.[22] F. Y. Wu, The potts model, Rev. Mod. Phys. 54 (1982) 235.[23] M. Blume, V. J. Emery, R. B. Griffiths, Ising model for the λ transitionand phase separation in he -he4