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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
arXiv:
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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.
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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.
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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.
1 2 3 4 5 6 70.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80.14
0.16
0.18
0.2
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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