The hipster eect:When anticonformists all look the same

Jonathan Touboul1, 2, 1The Mathematical Neuroscience Laboratory, CIRB / Colle`ge de France (CNRS UMR 7241,

INSERM U1050, UPMC ED 158, MEMOLIFE PSL*)2MYCENAE Team, INRIA Paris

(Dated: October 30, 2014)

In such dierent domains as statistical physics and spin glasses, neurosciences, social science,economics and finance, large ensemble of interacting individuals taking their decisions either inaccordance (mainstream) or against (hipsters) the majority are ubiquitous. Yet, trying hard to bedierent often ends up in hipsters consistently taking the same decisions, in other words all lookingalike. We resolve this apparent paradox studying a canonical model of statistical physics, enrichedby incorporating the delays necessary for information to be communicated. We show a generic phasetransition in the system: when hipsters are too slow in detecting the trends, they will keep makingthe same choices and therefore remain correlated as time goes by, while their trend evolves in timeas a periodic function. This is true as long as the majority of the population is made of hipsters.Otherwise, hipsters will be, again, largely aligned, towards a constant direction which is imposed bythe mainstream choices. Beyond the choice of the best suit to wear this winter, this study may haveimportant implications in understanding dynamics of inhibitory networks of the brain or investmentstrategies finance, or the understanding of emergent dynamics in social science, domains in whichdelays of communication and the geometry of the systems are prominent.

PACS numbers: 02.50.-r, 02.10.Yn, 05.40.-a, 87.18.Sn.

Hipsters avoid labels and being labeled. However, theyall dress the same and act the same and conform in theirnon-conformity. Doesnt the fact that there is a hip-ster look go against all hipster beliefs? This perspica-cious observation of the blogger Julia Plevin [1] in 2008proves true along the years, and 2014 hipsters all lookalike, although their look progressively evolves. The hip-ster eect is this non-concerted emergent collective phe-nomenon of looking alike trying to look dierent. Uncov-ering the structures behind this apparent paradox goesbeyond finding the best suit to wear this winter. Theycan have implications in deciphering collective phenom-ena in economics and finance, where individuals may findan interest in taking positions in opposition to the major-ity (for instance, selling stocks when others want to buy).Applications also extend to the case of neuronal networkswith inhibition, where neurons tend to fire when othersand silent, and reciprocally.

The question, as well as methods developed in the do-main, are evocative of a wide literature dealing with largesystems of interacting agents making random choiceswith probabilities depending on the choice of the ma-jority. Models were developed in such dierent domainsas the alignment of spins in magnets [2, 3], transmissionof electrical information in networks of neurons [4, 5],and choices in economics and social science [6]. In a gen-eral framework, we consider here populations made ofhipsters, or anticonformists, who take their decisions inopposition to the majority, and mainstream individualsthat tend to follow the majority. The main novelty isto take into account the time needed by each individualto feel the trend of the majority. These delays, relatedto the dynamics of the interactions and sometimes the

geometry of the system, are often neglected in physicsor social systems, but it is well known to computationalneuroscientists that the time taken by the conduction ofneuronal influx along the axons shape the collective dy-namics of coupled cells [7, 8]. In economics as well asin social science and opinion dynamics, the time takenby the information to be transmitted to the network, aswell as the relative influence of the past on the decisionstaken seem prominent: a given individual needs sometime to receive and take into account a decision of another individual. Moreover, specific individuals are moreinfluential than others, at least to the eyes of some. Thisheterogeneity in the way trends are perceived are centralin applications. However, our understanding of their im-pact on collective dynamics is still poor, and is essentialto understand the dynamics as we show here. We willconcentrate on a simple canonical model and show thatdepending on delay and heterogeneity distribution, qual-itative dynamics are substantially modified, and thesecan even lead to a dramatic synchronization of hipsterschoices.

We will investigate these questions in a generic modelof spin glasses. We consider n individuals, that randomlyswitch between two states {1, 1}, depending on theirhipster or mainstream nature and the majority trendthey feel. Individual i is randomly chosen to be hip-ster with probability q, or mainstream, and therefore thetype of individuals in the population is characterized bythe sequence of random variables ("i)i=1n 2 {1, 1}n,drawn prior to the evolution of the network and frozenduring time evolution (we choose by convention " = 1for hipsters). Each individual is defined by its currentstate si(t) and the network state is described by a vector

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2s(t) 2 {1, 1}n, that switches randomly with a proba-bility varying with the mean-field trend mi(t) felt by i,which depends on (i) how the individual i sees its envi-ronment and (ii) on the history of the system. In detail,individual i assigns a fixed weight Jij 0 to individualj, and only sees its state after a delay ij , so that thetrend seen by individual i at time t is simply:

mi(t) =1

n

Xj

Jijsj(t ij)

We will make the assumption that for any fixed i, theweights and delays pairs (Jij , ij)j=1n are independentand identically distributed random variables (or with lawp"i,"j that only depend on the type of i and j), drawnprior to the stochastic evolution of the network, andfrozen along the evolution in time. They constitute arandom environment. From the modeling viewpoint, theweight Jij is large if j has a prominent impact on thechoice of i, and it is null if decisions of j do not impact i.An important example that we treat here is when bothdelays and weights depend on a hidden variable, whichis the relative location in space of the two individuals.Once a configuration is fixed, individuals evolve ac-

cording to a random Markov process. Given the states(t) of the network at time t, each individual makes theswitch si ! si as an inhomogeneous Poisson processwith rate '("imi(t) si) where ' is a non-decreasing sig-moid function centered at zero. In that model, if the statesi(t) is opposite to the felt trend mi(t), mainstream indi-vidual ("i = 1) have a higher switching rate, and hipstersa lower switching rate. The gain of the sigmoid '() isdirectly related to the level of noise. To fix ideas, wechose '(x) = 1+ tanh(x), where > 0 is called inversetemperature, and governs the sharpness of the rate func-tion: the larger , the sharper '(x) and therefore the lessrandom the transition.Before developing our theory, let us spend some time

describing the relationship between this model and moreclassical spin-glass systems. Beyond the presence of de-lays that are specific to the present model, we considerasymmetric interactions, meaning that the action of in-dividual i on j is of the same amplitude as the reciprocalaction of j on i. In that sense, our system is comparableto binary neuron models as introduced in early works inthe domain [9]. Another dierence appears in the way weincorporate the mainstream-hipster nature as a charac-teristic of each individual, which diers from works donein neuroscience or in the Sherrington-Kirkpatrick spinglass system [2, 10] in which interaction between i andj is generally assumed to have a random sign, which isindependent pair are positive to occur with positive ornegative amplitude depending on the pair (i, j) consid-ered.This model however remains simple enough to be com-

pletely solvable: one can find a closed-form solution forthe thermodynamic limit of the system, in terms of aself-consistent jump process with rate depending on the

statistics of the solution. Moreover, the average behavioris exactly reduced to a set of delayed dierential equa-tions, and therefore we will be able to use the bifurcationtheory developed in this context to uncover phase tran-sitions related to the delays distribution. This is howwe will be able to show rigorously how delays induce asynchronization of hipsters.The thermodynamic limit of the system can be de-

scribed as a jump process whose jump statistics dependon a self-consistent quantity. In detail, in the limitn ! 1, individuals behave independently (a propertysimilar to Boltzmanns molecular chaos, called propaga-tion of chaos property in mathematics [11]), and thereforethe jump rate s! s averages out to '(""(t)s) with

"(t) =X"0=1

q"0ZR2

jm"0(t u)dp","0(j, )

where m"(t) := E[s"(t)] is the averaged value (statisti-cal expectation) of individuals of type " at time t andq the proportion of conformists (q) and anticonformists(1 q) individuals. Heuristically, each individuals jumpintensity is the rate of one process, averaged statisti-cally and also averaged over all possible configurationsof weights Jij , delays ij and individual types. A rigor-ous mathematical proof can be done using the theory ofMcKean-Vlasov limit theorems for jump processes devel-oped in the 1990s [1214]. Specific care has to be takenin our case, since we deal with (i) delayed systems thatrequire to use infinite-dimensional state spaces of trajec-tories (s(u))u2[t,0], and (ii) random environments, butthe principle of the proof however remains identical.The mean-field equation is a priori complex: it is a

non-Markov process in the sense that the jump rate ofa given solution depends on the law of the solution andnot on the value of the process itself. However, thanks tothe simplicity of the model, we can characterize very pre-cisely the probability distribution of this process. Indeed,the thermodynamic limit is univocally described by thetwo jump rates "(t), that only depend on the knowledgeof the average state of individuals in the two populationsm". It is not hard to show that the latter variables aresolution to the dierential equation with distributed de-lays:

m"(t) = 2m"(t) + tanh(""(t))

.

These equations allow to analyze rigorously the systemand the role of dierent parameters. We concentrate ontwo simple situations in which the role of the dierentparameters are disentangled: (A) a case with constantcommunication and delay coecients independent of theindividual type, and (B) a case where delays and com-munication coecients are both dependent on a hiddenrandom parameter, the respective locations of the dier-ent individuals, treated for simplicity in a pure hipstersituation.

3In order to characterize the phase transitions in thesystem, we investigate the linear stability of the disor-dered solution m" = 0, which depends on the spectrumof the linearized operator given by the solutions of thedispersion relationship

" = 2" 2"X"0=1

q"0ZR2

je"0dp","0(j, ). (1)

Let us start by dealing with situation (A) where p","0 =J, . The variable z = qm+1+(1q)m1, the total trendover the whole population, satisfies the equation:

z = 2z + (2q 1) tanh(Jz(t ))

and from now on consider without loss of generalityJ = 1. The linearized equations around the disorderedequilibrium (z = 0) greatly simplify, and it is then easy tocharacterize stability by finding the characteristic rootsof the system:

= 2(1 + (2q 1)e )For = 0, 0 is stable for 1 + (2q 1) > 0 and un-stable otherwise. This implies that populations in whichanticonformists are majoritary (q > 1/2) never find con-sensus, while populations dominated by conformists canfind a consensus, but at suciently small temperature.In detail, consensus are found for larger than a criticalvalue c(q) that increases with the proportion of hipsters

c(q) =1

1 2q .

Below the noise level, the disordered state z = 0 loosesstability and a state with non-zero trend is found. Heuris-tically, as long as anticonformists are majoritary, theywill compensate instantaneously any alignment of themainsteam individuals, and therefore prevent any mag-netization to emerge. But when there is a majority ofmainstream individuals, a trend may emerge if the levelof randomness in their choices is small enough. Hipsterswill then consistently oppose to this trend, creating aclear non-trivial hipster trend. The fact that the levelof noise at which this equilibrium emerges is lower thanthat of a pure ferromagnetic spin glass system can be in-terpreted as the fact that, from a microscopic viewpoint,the systematic furstration and misalignment of hipstersresults in an increased eective temperature. Precisely atthe critical transition q = 1/2, very complex phenomenaappear, where populations of anticonformists and hip-sters align transiently, in a non-periodic manner, beforeswitching at random times. A typical example is plottedin Fig. 1 (a).Instead, we shall concentrate of the role of the delays.

For < c(q), it is easy to see that 0 is the unique stablesolution. Indeed, shall there exist such (, ) for which0 is unstable, we would then have characteristic roots = a+ ib with positive real parts, i.e. such that:

a = 2 + 2(1 2q)ea cos(b)

Oscillations

(A)

0.604CD

E

0 50 1001

0

1

individu

altre

nd

0 50 1001

0

1

0 50 1001

0

1

time time time

(B)

(C) (D) (E)

FIG. 1. (A) Delay-induce Hopf bifurcation in the plane (, )with = (1 2q). (B-E) simulations of the discrete systemfor n = 5000, = 2. (B) q = 12 : phase transition. (C-E): q = 1 (fully anti-conformist system) and dierent delays = 0.5 (C), 0.7 (D) and 1.5 (E) respectively. Top row: timeevolution of all particles as a function of time, bottom row:empirical (blue) and theoretical (red) total trend.

but |2(2q1)ea cos(b)| < 2 hence this is impossible.However, at low temperature ( > c(q)) a destabi-

lization may occur. Shall this happen, the characteristicroots will cross the imaginary axis, and this is only pos-sible for purely imaginary eigenvalues of the linearizedoperator. Algebraic manipulations yield to the fact thatHopf bifurcations occur along the following curve in theparameter space:

= + arctan(

p(2q 1)22 1) + 2k

2p(2q 1)22 1 k 2 .

A non-trivial solution therefore emerges, which oscillatesbetween positive and negative values. The individualsremain synchronized, even if their orientation is not sta-tionary, but switches very regularly, in a periodic manner,between positive and negative.Heuristically, this oscillatory phenomenon arises from

the slowness of the information transmission. Indeed,during the evolution of the network, fluctuations of thetrend will tend to be amplified by the delay mechanism.Indeed, a random imbalance will be detected after sometime and all anticonformist individuals will tend to dis-align to this trend, regardless of the fact that an increas-ing proportion of them do and therefore yield a clear biastowards the opposite trend. This will be detected at latertimes, leading to a reciprocal switch, and these oscilla-tions will periodically repeat. Despite their eorts, atall times, anticonformists fail being disaligned with themajority.

4To conclude with, we shall concentrate on a more re-alistic situation (B). We now consider that the environ-ment variables (Jij , ij) have a pure geometric depen-dence: these are deterministic functions of the dissimilar-ity between the individual i and j, for instance a physicalor functional distance between them. In that setting, werandomness depends on a hidden variable ri, which is as-sumed for instance to take values on a compact set chosento be the one-dimensional circle of length a, S1a, and as-sume that this correspond to the location of individualsin a physical space. In that setting, individuals commu-nicate after a time proportional to the distance betweenthen added to a constant delay 0 corresponding to thetransmission of information ij = 0+|rij | =: T (rij) withrij is the distance between i and j (on the circle). Thedistribution of the distance can be computed in closed-form: it has linearly decaying slope d(r) =

2a 2ra2

dr.

Coecients Jij take into account the fact that distantindividuals have a smaller probability to communicate.The probability that two individuals at a distance r com-municate with each other is assumed to decays with aprofile (r), and the communication strength is assumedconstant equal J > 0 [15]. In other words, Jij = Jijwith ij a Bernoulli random variable of parameter (rij).from which we find the probabilities of the pairs (Jij , ij)given by the density dp(j, ) =

RS1 {j= (r),=T (r)}(r)dr.

This allows to compute the linearized operator, and findthe Hopf bifurcation curve in the space of delays and sizea. In detail, for (r) = er, the eigenvalues of thelinearized operators are solutions of the dispersion rela-tionship:

= 21 J

Z a0

e(+)r2

a 2r

a2

dr

and therefore Hopf bifurcations arise only if one can findparameters of the model, and a positive quantity ! > 0,satisfying the relationship:

i! = 2+ 2Ja( + i!)

1 1

a( + i!)+

ea(+i!)

a( + i!)

ei!s .

(2)This equation cannot be solved in closed form as in theprevious case, but however it is easy to express the locusof the Hopf bifurcation in the parameters space (a, 0) asa parametric curve, and therefore access with arbitraryprecision to the Hopf bifurcation in the plane (see Fig. 2).This curve has a very interesting, non-monotonic

shape. It shows that there is an optimal spatial extensionof the hipster population most favorable for synchroniza-tion: populations spreading on too small or too largeintervals will not synchronize, and there exists a specificlength interval in which hipsters synchronize. This ef-fect is actually the result of two competing mechanisms:increasing the size of the interval makes the average de-lay increase (as a/2), but the variance of the delays in-creases as well, which reduces the coherence of the signalreceived, and may make synchronization harder.

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0.16

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0.22

0.24

0.26

A

B

C

A

B

C

Oscillations

Oscillations

Hopf

Hopf

a

a

time

FIG. 2. Space-dependent delays and connectivity: bifurca-tions as a function of the length a of the interval on whichhipsters communicate. Parameters p = 4, = 0.3, s = 0.2,length of the interval: (A) a = 0.1 and (C): a = 3, no syn-chronization, (B): a = 1, synchronization. Simulation of theMarkov chain with N = 1000 together with the computedtrend below (computed averaged, plotted against a back-ground with color proportional to the trend).

We therefore showed that, in contrast to cooperativesystems, populations of individuals that take decision inopposition to the majority undergo phase transitions tooscillatory synchronized states if we take into account thedelays in the communication between these individuals.This study opens the way to the understanding of syn-chronization and correlations in other statistical models,such as those developed in finance, in which case specula-tors may make profit when taking decisions in oppositionto the majority in stock exchange. This problem has beenthe subject of intensive researches around the so-calledminority games (see the book [16] presenting motivationsand models), which our system is a particular case of.The analysis of the relatively simple model allowed togo very far in the understanding of the concurrent role ofnoise, delays and proportions of hipsters and mainstreamindividuals in this emergence of synchronization amonghipsters. Interestingly, synchronization may depend onthe precise shape of the distribution of the delays: forsynchronization to emerge, one needs both sucientlylong delays and sucient coherence (small standard de-viation of the delays). This yielded the unexpected phe-nomenon that synchronization among hipsters dependson the distribution, in space, of each individuals, whenthe delays are function of the distance between two in-dividuals. Along the way, we uncovered several pointsthat are well worth studying in depth. For instance, thebehavior of a system with an equal proportion of hipstersand mainstreams appears to be a singular phase transi-tion in which the whole population tends to randomlyswitch between dierent trends, and would be very inter-esting to further characterize.

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91, 679 (1998).[15] This is equivalent to an attenuation of the signal with

the distance between the two individuals, i.e. to considercommunication strength equal to J (r).

[16] D. Challet, M. Marsili, and Y.-C. Zhang, OUP Cata-logue (2013).