Opinion dynamics with memory: how a society is shaped by its own past
aa r X i v : . [ phy s i c s . s o c - ph ] J u l Opinion dynamics with emergent collective memory:A society shaped by its own past
Gioia Boschi a , Chiara Cammarota a , Reimer Kühn a a Mathematics Department, King’s College London, Strand, London WC2R 2LS, UK
Abstract
In order to understand the development of common orientation of opinions in themodern world we propose a model of a society described as a large collection ofagents that exchange their expressed opinions under the influence of their mutualinteractions and external events. In particular we introduce an interaction biaswhich results in the emergence of a collective memory such that the society isable to store and recall information coming from several external signals. Ourmodel shows how the inner structure of the society and its future reactionsare shaped by its own history. We provide an analytical explanation of suchmechanism and we study the features of external influences with higher impacton the society. We show the emergent similarity between the reaction of asociety modelled in this way and the Hopfield-like mechanism of informationretrieval in Neural Networks.
Keywords:
Opinion dynamics , History of Agreement and Disagreement , External Information , Collective memory
1. Introduction
Contemporary human society lies under the effect of an almost fully ac-complished globalisation. International barriers have become more permeablethanks to the spread of English as universal language. Connections of individu-als develop across huge distances throughout the world thanks to the daily useof social media and the immediate coverage nowadays attained by informationmedia. These observations naturally support the picture of a global societydescribed as a large collective system involving strongly interacting degrees offreedom, represented by the individuals’ actions or opinions. With this picturein mind, scientists started to study the human society by means of a statisticalmechanics approach. Early results of this effort resulted in the first opinion
Email addresses: [email protected] (Gioia Boschi), [email protected] (Chiara Cammarota), [email protected] (ReimerKühn)
Preprint submitted to Physica A July 16, 2020 ynamics model proposed by a physicist [1] and the introduction of the Isingmodel to study consensus in societies [2, 3] . An important amount of workfollowed these first seminal articles, building a literature in which the pressureof the society is modelled by assimilative interactions that mimic the tendencyof people to imitation. The most widely studied models of a society of this kindcome from physics, i.e. the Voter model [4] the Ising model with its variants[5] already mentioned and the Majority rule model [6, 7]. Yet, despite manyother interesting features, it soon appeared evident that models only based onassimilative interactions describe a society in which full consensus is typicallyinevitable. To overcome this problem the concept of homophily [8, 9], that isthe tendency of people to interact more often or with grater intensity with sim-ilar others, has been introduced [10, 11, 12]. However, the fragmentation ofthe society into groups with different opinions obtained in models that includeboth assimilation and homophily was found to be unstable under the introduc-tion of noise [13, 14]. In fact an infinitesimal amount of noise it is seen toirremediably redirect the society to a state of full agreement. The idea thatantagonistic interactions [14, 15, 16, 17, 18, 19, 20, 21, 22, 23] and xenophobia(the phenomenon for which the larger the dissimilarity between two interact-ing individuals, the more they evaluate each other negatively) [16, 17, 23, 24]should be taken into account to resolve this issue has been developed only fairlyrecently. In the present work we build up on these observations and put themin conjunction with the traditional approach that assumes imitation tendenciesbetween individuals. We propose a model for opinion formation based on therule that individuals in agreement with each other tend to reinforce their mutualpositive influence, while individuals in disagreement will develop an antagonisticrelation based on the mistrust towards one another’s view. These tendencieswill be encoded in an interaction term that for each pair of individuals reflectsthe history of their agreement or disagreement at previous times. Past relevantinteractions are only those that lie within the finite range of agent’s memory.We note that the dynamics of the model we propose is strongly reminiscentof the dynamics of graded response neural networks [25, 26, 27] which havebeen used to describe associative memory. Indeed, the model discussed herewill, under suitable conditions, develop interactions of the Hopfield type [28].Models of society including Hopfield-like interactions have been already usedin social sciences in a few occasions [16, 17] to study consensus formation andopinion polarization. In these works Hopfield interactions are introduced andstudied in conjunction with a number of other elements like individual flexibility,broad-mindedness, and open-mindedness [16] or in more complicated networkstructures [17]. Moreover in all these cases the focus of the study was on thestationary state reached by a society with fixed interpersonal interactions andhow it is approached from a random initial condition. Our contribution willinstead focus on a model society whose internal interactions develop startingfrom historical interpersonal relationships. Yet, it will, under suitable condi-tions, spontaneously turn out to closely resemble a model society exhibitingHopfield-like couplings. We will develop a detailed analysis of the way the2opfield-like interactions develop and analyze the resulting collective behaviourof the system. To this aim we will focus on a society that is constantly underthe influence of new external events and we will pay particular attention to thereaction of the society to world-wide news, modelled as external fields appliedto it [29, 30]. We will study whether the society does develop self-maintainedcollective memories of certain news, and will therefore be irremediably shapedby them. In literature the concept of collective memory was first introducedin [31] and only recently scholars have focused on how collective memory is in-fluenced by media [32, 33]. In our model the possibility for external events toimpress the society will depend on a number of parameters including the extentof the news influence on single individuals and how frequently they are impact-ing the society. In particular our model shows that strongly impacting news, oreven just very frequent news, can change the internal structure of the societyin a drastic way and will determine its non linear collective response to futureexternal influences. The structure of the paper is as follows. In Section 2 we willintroduce the model and its main features, before entering into the descriptionof different scenarios corresponding to different kinds of external information.The results are presented in Section 3 and organized in the following sectionsstarting from the simplest scenario to more complicated ones. Given the grow-ing complexity of the problems studied, not all the cases considered can be fullysolved analytically. For each choice of the external stimuli we first derive all theanalytic predictions we have access to, then we complete the picture by showingsimulation results.
2. The model
The society that we consider is composed of a set of N agents, each poten-tially connected to all the other agents. With agent i we associate a continuouspreference field u i . Expressed opinions are given by nonlinear functions g ( u i ) of the preference field u i of agent i . We will take g ( u i ) to be of sigmoid form,implying that expressed opinions remain bounded. We will take the stochasticdynamics of the system to be of the form: ˙ u i = − u i + I i + N X j = i J ij g j + η i , (1)where we use the abbreviation g j = g ( u j ) and we dropped time dependencies,which in general pertain to all variables in the equation. Here J ij > repre-sents a mutually supportive interaction between the individuals i and j while J ij < indicates an antagonistic interaction between the same agents. Thequantity I i represents the mass media information as perceived by individuals,while − u i is a mean reversion term which entails that in absence of externalinfluences the preference field of each agent will fluctuate around zero. Thelast term η i is a white noise with Gaussian distribution with zero mean andfinite variance h η i ( t ) η j ( t ′ ) i = σ δ ij δ ( t − t ′ ) . The combined effect of individually3erceived external sources of information, I i , and the interaction with othersagents’ expressed opinions, g j , may act to drive the preference field of an agentaway from zero, and thus may favour the development of specific orientations.The heterogeneous external information I i is defined as I i = I ξ i separatingthe strength I of the signal from the variables ξ i encoding the local variability.Note that such variability might be genuine or arise as a result of individuallyvariable perception of an underling uniform message (which may be caused byidiosyncratic interpretations). We simply assume that ξ i can only be either +1 or − .Finally and most importantly the tendency of each person to agree or disagreewith others is based in our model on the memory of past history of agreementand disagreement with them. People which have a history of agreement in thepast, will be more likely to agree also in the future, and an analogous resultholds for disagreement. This feature represents the key ingredient of our model.More specifically we consider the recent history of agreement or disagreementto have a larger weight than the distant past to take into account how vivid theexperience of past interactions is. We will assume an exponentially weightedmemory and take interactions between agents at time t to be given by J ij ( t ) = J · γN Z t d s g i ( s ) g j ( s ) e − γ ( t − s ) (2)for some J > , assuming for simplicity the time scale τ γ = 1 /γ of the memoryto be uniform across agents. The normalization factor /N is introduced so thatthe interaction with other agents is not overwhelmingly dominant, but remainscomparable to the influence of external sources of information in the large N limit.In this way we have a society that uses the past history to interpret any in-stantaneous inputs that it receives. In particular agreement (disagreement) willbe perceived if the value of g i ( s ) g j ( s ) will be continuously positive (negative)on the time scale τ γ and will bias agents i and j toward future agreement (dis-agreement). When the history of past interaction is instead characterized by analternation of agreement and disagreement periods the agents will tend to beneutral with each other, J ij ∼ . This memory effect is particularly importantwhen studying the influence of the external information on the agents’ opin-ions. Note that the memory that appears in our model, being a memory of pastrelations, must not be confused with the memory of past actions or opinionsof single agents that has been more often considered in the literature of socialbehaviour [34, 35, 36]. The main ingredient of our model is the memory of past interactions, whichis associated with the time scale τ γ . The model also contemplates a secondtime scale which is the relaxation time of the individual preference u i . Thelatter has been set equal to 1, without loss of generality as in general all the4ther parameters and time can be expressed in terms of its unit. A third timescale ∆ should be also considered. It is the one associated with the durationof the exposure of the society to external stimuli. We will typically focus onthe regime τ γ ≫ ∆ ≫ corresponding to a fast adaptation of u to eventualexternal stimuli and a slow memory decay which will be responsible of the storingof previous opinion configurations in the memory of interpersonal relations.This process will describe how the whole society can be shaped by its pastby learning from patterns of opinions produced by sustained signals or seriesof repeated external stimuli. Among the different scenarios studied, we willdescribe the case of the arrival of different external stimuli represented by alocal field changing in time I ( t ) = I ξ µ ( t ) . In this expression p different randomchoices of ξ µ ( t ) = ( ξ µ ( t ) i ) ∈ {± } N are considered, one for each integer value in { ...p } that µ ( t ) assumes, each of them for a time ∆ . Each of these randomvectors represents the perceived piece of news that influence the society duringthe time ∆ and is later substituted by a different piece of news. Under theeffect of such external influences we expect that the society will likely developinteractions comparable to the classic Hopfield couplings [28] defined from acollection of p random patterns ξ µ , albeit rescaled by a factor /p , so in thelong time limit we expect J ij ≃ J H ij /p = 1 N p p X µ =1 ξ µi ξ µj . (3)In fact, if each strong signal clamps the expressed opinions towards the positionsthat it suggests ( g ( t ) = ξ µ ( t ) ) for a time ∆ ≪ τ γ , and the sequence µ ( t ) ofthe signals’ appearances is repeated many times in τ γ , the average over thepast history in Eq.(2) can be approximated by an average over the product of ξ µi and ξ µj as by definition of Hopfield couplings. In section 5 we will discussthe similarity between the generated interpersonal couplings in our modeledsociety and Hopfield couplings in more detail. For the moment it is interestingto note that, despite the similarity that emerges at first sight, the couplings inthe Hopfield model [28, 25] were taken to be fixed from the start, whereas inthe present case they evolve dynamically under the influence of external signalsand internal dynamics. It is only in the case of the special scenario describedabove that we will see the emergence of Hopfield type couplings. While a few of the simplest scenarios investigated in the present paper can bestudied analytically, we are in many cases forced to use numerical simulations.To perform these, we note that the couplings of Eq. (2) satisfy a dynamicalevolution equations: ˙ J ij ( t ) = γ (cid:20) J N g i ( t ) g j ( t ) − J ij ( t ) (cid:21) . (4)We use Euler integration to integrate Eq. (1) and Eq. (4); we found a step sized t = 0 . (with d η i ( t ) = σ √ dt ) sufficiently small for our purposes. There are5any parameters involved in the simulations, some are fixed in all the cases,some vary. The fixed parameters are listed here: we choose N = 100 for thenumber of agents and p = 3 for the number of different external signals. Al-though these numbers are small, they produce results that are representativeof the N → ∞ limit with p ≪ N (as we have checked by using other ( p, N ) combinations). Throughout this paper we used a low noise level, σ = 0 . ,to ensure that non-trivial collective states can emerge. All simulations startwith random initial conditions u i ∼ N (0 , σ / which would be the equilibriumdistribution in a non-interacting system without external signal. The other im-portant parameters that change from case to case are the time length of eachexternal signal ∆ , the amplitude I of the polarizing signal (apart from a fewexceptions taken to be I = 50 ), the strength of the interactions J and the timescale τ γ of the memory of past interactions.
3. Results Overview
Our model is constructed on the simple assumption that the mutual interac-tions between agents depend on their past history of agreement or disagreement.Our main result is that this creates a mechanism that allows a society to de-velop a collective memory of its past experiences. To study this mechanism inmore detail, we analyze different protocols of external influences that first trig-ger the different individuals’ opinions. The different scenarios presented rangefrom simple situations to more complex and realistic ones and are listed belowtogether with a list of the results obtained in each case.1. The external information I is heterogeneous but constant in time. We areable to treat the system analytically and predict its long time behaviour.Already in this simple setting the presence of the signal changes the waypeople interact and determines their future behaviour.2. The signal consists of a sequence of different (random) patterns, repeatedlypresented in a cyclic fashion. In the long run this creates a stable matrixof interactions between the agents which can be predicted analytically.By means of this interaction matrix the society develops a memory of theopinion patterns presented previously and it is found to be able to recalleach of them.3. A sequence of external stimuli is repeatedly presented in a cyclic fashionas before. However there are gaps between the presentation of successivesignals where the society is not exposed to an external stimulus and followsits own internal dynamics. In this scenario a critical ratio of patternsduration and length of the gaps without pattern presentation must beexceeded for the system to develop persistent memory of the signal. Weprovide an analytic treatment to predict this critical ratio. Interestinglywe also found that in the case of high-impact news that influence thesociety very frequently, the presentation time needed for the news to bememorized by the society is unexpectedly small.6he study of even more realistic situations such as sequences of external influ-ences with different impact on the society and random appearance in time willbe considered in a follow up paper.
4. Learning from a persistent external signal
In this section we start illustrating the behaviour of the society describedby our model in a simple setting. Here we study its reaction to an externalinformation persistent in time and described by a signal I i = I ξ i where I isits strength and ξ i is a random variable taking values in {± } , which representsthe way in which the agent i perceives it. Note that the uniform perception I i = I ∀ i is a particular case of what discussed here. Our aim is to understandhow the society reacts to this signal and how the memory of the opinions inducedby it develops in this simple case before moving to more complicated settings.In order to do this, we will study the evolution in time of the agents’ preferencefield u i and of interaction couplings J ij between agents. Even in the simple caseof a constant signal, solving the equation for u i is not a trivial task, mainlybecause of the dependence of the couplings on the expressed opinions. Wewill see that the presence of the signal will induce the agents to change theiropinion, consequently modifying the relationship of agreement and disagreementbetween them. As a results of this change, the couplings J ij , which are nullat the beginning of the dynamics, will start to evolve and establish the newinteractions between the agents. This process allows the society to learn theopinion pattern ξ and to collectively retrieve it in the future.We can write down a formal solution of Eq. (1) with couplings defined in Eq.(2): u i ( t ) = u i (0) e − t + J Z t d s [ U i ( s ) + η i ( s )] e − ( t − s ) + I ξ i (1 − e − t ) , (5)in which U i ( s ) = γ Z s d s ′ e − γ ( s − s ′ ) g i ( s ′ ) q ( s, s ′ ) , (6)with q ( s, s ′ ) = 1 N X j g j ( s ) g j ( s ′ ) . (7)We note that, by appeal to the law of large numbers, the correlator q ( s, s ′ ) will be non-random in the large N limit. Noting further that, for γ ≪ , thefunction U i ( s ) are very slowly varying funcions of s , we see that: Z t d sU i ( s ) e − ( t − s ) ≃ U i ( t ) Z t d s e − ( t − s ) = U i ( t )(1 − e − t ) . (8)To proceed we adopt one further approximation to replace U i ( t ) in Eq. (8)by its noise average (indicated by h·i ), which is tantamount to replacing the g i ( s ′ ) in Eq. (6) by their noise average. While this replacement leads to an7nderestimation of the noise contribution in the evolution of the u i ( t ) , we foundthat the effect remains small in the parameter ranges considered. With thisapproximations the solution for u i becomes u i = u i (0) e − t + J h U i ( t ) i (1 − e − t ) + Z ∞ η i ( τ ) e − τ d τ + I ξ i (1 − e − t ) . (9)This means that u i will be a Gaussian process with mean h u i i = u i (0) e − t + J h U i ( t ) i (1 − e − t ) + I ξ i (1 − e − t ) , (10)covariance C ( t, t ′ ) = h ( u i ( t ) − h u i i ( t ))( u i ( t ′ ) − h u i i ( t ′ )) i = Z t Z t ′ d s d s ′ h η i ( s ) η i ( s ′ ) i e − ( t − s ) e − ( t ′ − s ′ ) = σ (cid:16) e −| t − t ′ | − e − ( t + t ′ ) (cid:17) , (11)and variance σ u ( t ) = C ( t, t ) = σ − e − t ) . (12)We are interested in studying the system in the long time limit t, t ′ → ∞ , wherethe u i -process become stationary, with the covariance of the u i only dependingon the time difference τ = | t − t ′ | = O (1) , i.e. C ( t, t ′ ) = C ( τ ) , so we will have lim t →∞ h U i ( t ) i = γ h g i i ˜ q ( γ ) (13)where ˜ q ( γ ) is the Laplace transform of q ( t − s ′ ) . The mean of u i , its covarianceand variance will thus become h u i i = γ h g i i ˜ q ( γ ) J + I ξ i (14) C ( τ ) = σ e − τ (15) σ u = C (0) = σ . (16)Now we realize that h u i i = ξ i h u i and h g i i = h g ( h u i i + σ u ζ i ) i ζ = ξ i h g i , with h·i ζ being the average taken over ζ ∼ N (0 , , are a consistent solution of Eq. (14).This allows us to take out the dependencies on i from the equations.If we now choose the expressed opinions to be the error functions of the prefer-ence fields, with g i = erf ( u i ) , we can exploit the properties of the error functionto obtain a self consistency equation for h u i . We will have h g i = erf h u i p σ u ! (17)8nd q ( τ ) = * erf ( h u i ∞ + σ u x ) erf h u i ∞ + ρ ( τ ) σ u x p − ρ ( τ )) σ u !+ x , (18)where h·i x stands for an average over a Normal random variable x and thecorrelation coefficient ρ ( τ ) is ρ ( τ ) = C ( τ ) σ u = e −| τ | . (19)(See [37] for further details on these last passages from a similar computationin a different context.)Gathering these results together we obtain a closed system of equations thatdescribe the preference field of the agents in our society under the effect of aconstant external signal in the stationary regime: h u i = I + J γ erf h u i p σ u ! ˜ q ( γ ) ,σ u = σ , ˜ q ( γ ) = Z ∞ d τ q ( τ ) e − γτ ,q ( τ ) = * erf ( h u i + σ u x ) erf h u i + ρ ( τ ) σ u x p − ρ ( τ )) σ u !+ x . In order to understand to what extent the external field has influenced thesociety we will focus on the overlap of the asymptotic system state with thepattern ξ : m ( t ) = 1 N X i ξ i g i ( t ) . (20)This quantity measures whether the expressed opinions in the society are similarto those induced by the signal ξ , i.e. m ∼ O (1) , or not. The overlap is self-averaging in the N → ∞ limit and its value at late times, t → ∞ , is given by m = 1 N X i ξ i h g i i = h g i , (21)which therefore satisfy the following equation m = erf h u i p σ u ! = erf I + J γ ˜ q ( γ ) m p σ u ! . (22)We chose to write the last equation in a self consistent form to show that undercertain conditions we can expect a non zero value of m even when the signalis finally removed, I = 0 . Indeed, for sufficiently large values of the product9 ˜ q ( γ ) in comparison with the noise contribution quantified by σ , the equationadmits a non zero solution. For example, if the society is exposed to a signal I = 1 for long times, given a J = 6 and γ = 10 − its overlap will be close to 1and will remain close to 1 when the signal is removed (we will see that in the sameconditions this value will match the results obtained with a simulated dynamics).In other words the society is potentially able to remember the opinions inducedby the signal even when it is removed, after having been exposed to such signalfor sufficiently long time. To shed more light on this mechanism we now focuson the evolution of the couplings and the value they reach in the stationaryregime for a society described by a choice of the parameters that admits a nonzero solution to Eq.(22). The upper panel of Fig. 1 shows how couplings evolvein a numerical simulation starting from a society without interactions betweenagents. The continuous curve shows the norm of the J ij matrix defined as | J | = sX ij J ij . (23)For sufficiently large I , a simple analytic argument allows us to give an accurateprediction of this growth. In the presence of a signal I i = I ξ i , preference fieldsrapidly orient towards the signal with large absolute values of u i and g i willtherefore soon be approximately equal to ξ i . By using this information we canintegrate Eq. (2) to obtain: J ij ( t ) ≈ J · γN Z t d s ξ i ξ j e − γ ( t − s ) = J N ξ i ξ j (1 − e − γt ) , (24)where the memory time scale appears explicitly. The average absolute value ofthis analytic prediction is represented also in units of J by the dashed line inthe upper panel of Fig.1 and nicely superimpose with the numerical results. Asthe system will quickly approach a stationary state, aligned with the externalsignal, interactions further stabilizing that the very same stationary state willbe established after a time set by the memory time scale, chosen to be τ γ = 10 in that simulation.Note that the sign of the predicted couplings is also peculiar. For long timesthe J ij approach the Mattis couplings [38]: lim t →∞ J ij ( t ) = J N ξ i ξ j ≡ J Mij (25)equivalent to the Hopfield couplings in Eq.(3) for p=1. It is well known thatin a system with pairwise interactions given by Mattis couplings (Eq.(25)) withsufficiently large amplitude, the variables spontaneously align in the directionsdefined by ξ . In our modelled society this would correspond to the fact thatthe opinion pattern can be spontaneously retrieved because the correspondingMattis couplings have been formed as a consequence of the memory of thesustained past opinion patterns induced by the exposure to the signal I i = I ξ i .10
10 20 30 40 5000.20.40.60.81
Figure 1: Simulated dynamics with a persistent external signal. Upper panel: results fromsimulations for | J | /J are compared with the analytical prediction (Eq. (24)) and are seen toapproach the asymptotic Mattis couplings (Eq. (25)). Lower panel: the overlap Q betweenthe simulated and the analytical J is compared with the overlap between the simulated J andthe Mattis couplings. In these simulations J = 6 , I = 1 and γ = 0 . .
11o verify that this is the case we define the correlation Q = P ij J ij J ′ ij qP ij J ij P ij J ′ ij (26)that reveals the degree of alignment between two sets of couplings J = ( J ij ) and J ′ = ( J ′ ij ) , with Q = 1 implying perfect alignment and Q = 0 the absence ofany correlations. In the lower panel of Fig.1 we exhibit the correlation betweencouplings observed in a simulation with both the analytic prediction (Eq. (24))and the asymptotic Mattis couplings (Eq. (25)). It shows that interactions arevery quickly perfectly correlated with both the analytically predicted couplingsand the Mattis couplings, although their norm is initially smaller than | J M | (seecomparison with the absolute value of J M /J in the upper panel of Fig.1). Inconclusion, the memory of interpersonal relations developed in response of anexternal stimulus ξ produces in the society interactions of increasing strength,which are very quickly aligned with Mattis couplings corresponding to the ξ itself.For large enough strength of the couplings we can expect that the society willspontaneously polarize along ξ autonomously sustaining its memory even afterthe external signal is gone. To give evidence of this interesting phenomenon westudied different dynamics in which the external signal ξ is switched on at t = 0 and removed at a time t r . We then focus on the asymptotic average overlap for1. dynamics in which couplings keep evolving after t r (evolving J ij )2. dynamics in which couplings are fixed to the value they had at t r (frozen J ij ).The average overlap for both versions of the dynamics has been calculated asthe average of the simulated overlap at stationarity over the last 1000 units oftime and plotted in Fig. 2 as a function of t r .The dynamics with frozen J ij shows at which t r the couplings have grown largeenough for a spontaneous m to arise. We note here that this overlap m corre-sponds to the solution of the self-consistent Eq. (22) derived for the stationarysolution once the amplitude of the asymptotic couplings J ˜ q ( γ ) is replaced bythe amplitude of the frozen couplings | J ( t r ) | (also reported in Fig. 2) and I = 0 . According to this equation the critical value of the strength of Mattistype couplings needed for the model to exhibit spontaneous order with m > is | J c | = 0 . J and it corresponds to the | J | = (0 . ± . J reached at theminimum t r where m > in simulations with frozen J ij . We will come back tothis minimum time t r in a different context in Section 6.Interestingly the dynamics with evolving J ij is instead characterized by an asymptotic spontaneous overlap arising in a discontinuous way even before thatminimum t r . Indeed, at variance with the frozen case, the evolving couplingswill continue to grow after t r even in absence of the external signal as a re-sult of the interactions embedded in the system. If this residual reinforcementof the couplings is large enough a positive feedback loop will occur betweenthe evolving couplings and the degree of order supported by the interactions12 Figure 2: Simulated dynamics with a persistent external signal. The figure shows the valueof m at stationarity as a function of the time t r at which the signal is removed, for two kindof dynamics: 1) dynamics with evolving couplings after t r
2) dynamics with frozen couplingsafter t r . The dashed line represents the amplitude of couplings norm | J ( t r ) | at t r . Theparameters used are the same as Fig. 1. currently established in the society. The degree of order will be strengthenedby strong interactions, which will in turn grow further due to a higher degreeof persistence of the expressed opinion pattern. As soon as this self-sustainedmechanism takes place it leads to a strongly polarized society represented bythe large value of m in Fig. 2.In the following sections we will study more complicated cases in which theexternal signal is composed of a sequence of different stimuli both with andwithout interposed periods of complete absence of stimuli. The understandingof the system’s reaction to a single external stimulus gained in the present sec-tion as well as the quantities introduced here will be used in the next sectionsto understand if and how the society is able to learn and subsequently retrievethe different opinion patterns to which it has been exposed in the past.
5. Periodic external signal
As a second step in the exploration of our society’s behaviour we will changethe external information structure, passing from a constant signal I ξ to a signalthat changes in time. The aim is mimicking a sequence of different news orevents, labeled by the index µ ∈ { , . . . , p } , to which the society is exposed andreacts. As before, the variables ξ µ represent the way in which the populationreacts to a single piece of information, and are modelled as random variableswith entries ξ µi ∈ {± } . The different contributions I µ = I ξ µ will appear in anordered sequence, each switched on for a time span ∆ before being substituted13y the following piece of news in the sequence. This process is repeated in acyclic fashion. The resulting expression of the signal is thus I i ( t ) = I ξ µ ( t ) i , (27)with µ ( t ) = 1 + (cid:22) t ∆ (cid:23) mod p , (28)where ⌊·⌋ is the notation for the integer part. This series of repetitive signalsrepresents a simple but effective way to model series of events that are repeatedlyappearing in television or newspapers.As introduced in section 2.1, we expect that in the long run the exposure of thesociety to a sequence of signals defined in Eq. (27) and (28) will result in thedevelopment of couplings that are similar to the Hopfield couplings (Eq. (3)).Indeed as before, given a signal with large amplitude I , we can assume that theopinions g i become rapidly equal to the opinion pattern proposed every timewe have a signal spike, so we have that g ( t ) = ξ µ ( t ) . For this assumption toprovide an accurate approximation of the full dynamics, we also assume that ∆ ≫ allowing us indeed to neglect transient behaviour after the switches ofthe external signal. Using this approximation we can calculate (see AppendixA for details) the couplings J ij developed in the society at times t , which aremultiples of the presentation time p ∆ : J ij = J N ( e γ ∆ −
1) ( e − γt − − e γ ∆ p ) p X µ =1 ξ µi ξ µj e ( µ − γ (29)In the long time limit t → ∞ (thus N p → ∞ ) this tends to lim N p →∞ J ij ( t = N p ∆ p ) = J ij, ∞ ( p ) = J N ( e γ ∆ − e γ ∆ p − p X µ =1 ξ µi ξ µj e ( µ − γ . (30)Eq. (30) shows explicitly that the learning protocol allows the couplings toapproach in the long run a weighted version of the Hopfield couplings where eachpattern’s weight is a function of its presentation order ( µ ). This means that weexpect an uneven storing of the patterns: the pattern last seen is rememberedbest, while the memory of the previous ones decays exponentially on a timescale τ γ , in a similar way as in some generalized Hopfield models of forgetfulmemories [39, 40, 41]. Finally expanding equation (30) for small γ ∆ (manyrepetitions of news presented within a memory time) we obtain J ij, ∞ = J pN p X µ =1 ξ µi ξ µj + J pN p X µ =1 ξ µi ξ µj ( µ − γ + o ((∆ γ ) ) . (31)Note that the first term is proportional to the Hopfield couplings (see Eq. (3)),which can be thus thought as a zeroth order approximation to our couplings.14imilarly to what happens in Hopfield Neural Networks, our society will be ablein the long run to store and easily retrieve the opinion patterns ξ µ . The level ofretrieval of the society for each of the patterns µ can be evaluated using a setof overlaps m µ : m µ ( t ) = 1 N X i ξ µi g i ( t ) . (32)These overlaps will tell us if the system is aligned with one of the opinionconfigurations previously presented ( m µ = O (1) ) or not ( m µ = O (1 / √ N ) ). Thevalue of m µ in the long time limit can be obtained assuming the Gaussianity of u i (as done in the scenario of the previous section) and evaluating the average u i in the following way: we use the couplings evaluated in Eq. (30) to evaluatethe long term limit of Eq. (1). In the absence of a signal ( I i = 0 ∀ i ) we thusobtain u i ∼ N ( h u i i , σ u ) , with h u i i = X j J i,j, ∞ h g j i (33) σ u = σ / , (34)and h g i i = erf h u i i p σ u ! = erf ( e γ ∆ − e γ ∆ p − J P pµ =1 ξ µi m µ e ( µ − γ √ σ ! . (35)If different patterns have negligible mutual overlaps, such as for uncorrelatedpatterns with N P i ξ µi ξ νi ∼ √ N for µ = ν , the equation above can have a nontrivial solution for which the society aligns with exactly one of the patterns, ν .In this case m ≃ { , .. , m ν , ..., } and m ν ≃ erf (cid:18) m ν J ( e γ ∆ − e γ ∆ p − e ( ν − γ √ σ (cid:19) . (36)Note that under certain conditions the solution of Eq. (36) is non trivial andwill be larger for more recently presented patterns and smaller for older ones,meaning that, if remembered at all, recent pieces of news will be better recalledby the society.As a confirmation of this behaviour we simulated the dynamics of our societyuntil a time t r at which we froze the couplings. To check whether the societyhas developed a memory of the p external signals to which it has been exposed,after t r we apply each signal contribution again for a short time after which weremove it to observe the response of the society in terms of the overlaps m µ inabsence of it. As shown in Fig. 3, the society quickly reacts to each of the spikesafter t r = 6090 as the corresponding m µ (highlighted in Fig. 3 with differentcolours for different signal patterns) jumps to during the spike and relaxes15
500 1000 1500-0.200.20.40.60.815500 6000 6500 7000-0.200.20.40.60.81
Figure 3: Simulated dynamics with periodic external signal. Upper panel: Early dynamicsof the society. The overlaps with different patterns are represented by different colours andquickly reach m µ ≃ when their corresponding signal contribution ξ µ is on. Lower panel:after presenting many times the patterns in a cyclic fashion, we freeze the couplings at time t r = 6090 and we presented each of the patterns for a time very short compared to ∆ , afterwhich the signal is removed. The analytic predictions of the overlaps in absence of signal arecompared with the simulations. In this simulation we set I = 50 , J = 6 , γ = 10 − and ∆ = 70 . = 70 I = 1 I = 5 I = 50 Analytical m ± · − ± · − ± · − m ± · − ± · − ± · − m ± · − ± · − ± · − Table 1: The table shows the values of m µ obtained averaging m µ over 100 simulations withfixed ∆ for different values of I , against their analytical predictions. The control parameterof the simulations are J = 6 and γ = 10 − . to a non trivial value in absence of external signal. It remains stationary untilthe subsequent spike of a different pattern is applied. The expected stationarynon trivial overlap m µ obtained from Eq. (36) matches quite well with thesimulation results. To further confirm our findings we defined the quantities m µ as the average of m µ over 100 simulations and we compared them for differentstrength I of the signal applied during the dynamics, with their analyticalpredictions in Table 1. In this case to obtain m µ in each simulation we froze thecouplings at t r and we averaged the values of m µ on the last 2000 steps after thecorresponding subsequent signal spike. We note that predictions always slightlyoverestimate the simulation results. The two main reasons for this discrepancyare that in Eq. (9) we neglected the contributions of the fluctuations of g i ( t ) andthat the transients of the u i dynamics after every change of external signal wereneglected in the analytical evaluation of the couplings. Moreover the theoryworks better for higher I as the assumption we made that the opinions alignimmediately to the signal becomes more accurate for large signals. Finally wecan observe that predictions get worse for earlier external signals. Indeed theyare associated with smaller effective couplings in Eq. (1) and consequentlyoverlap solutions more susceptible to the neglected fluctuations.The possibility to store and retrieve all the presented external signals as shownin Fig. 3 is expected, given the similarity between the spontaneously formedinteraction couplings and classical Hopfield couplings (see Eq. (3)). For thesimulation reported in Fig. 3 we indeed find that interactions very soon alignalmost perfectly with corresponding Hopfield couplings as shown in the upperpanel of Fig. 4. The figure also shows that analytic predictions of J from Eq.(29) are very accurate as the corresponding Q ≃ at all times for J analytical.As discussed in the previous section, the possibility for the society to retrievethe pattern induced by an external signal requires sufficiently large couplingsto allow a non-zero solution of Eq. (36). In the bottom panel of Fig. 4 wecompare the norm of J (defined in Eq. (23)) obtained with simulations to itsanalytical prediction and the norm of the Hopfield couplings over p J H /p . Theevolution of the analytical curve predicts closely the evolution of the simulated | J | /J while | J H | /p overestimates the true value of | J | at the beginning of thedynamics. However at t r = 6090 the value of the simulated | J | has reached astationary value very close to | J H | /p . The society has been irremediably shapedby the opinion patterns ξ µ . 17 Figure 4: Simulations with periodic external signal. The upper panel shows the evolutionof the correlation Q between the simulated and the analytical J and between the simulated J and the Hopfield couplings J H /p . The lower panel shows the evolution of | J | against itsestimated analytical evolution and the norm of the Hopfield couplings over p | J H | /p in unitsof J . The control parameters are the same as figure 3. . Intermittent external signal In this section we study a still stylized but slightly more complicated sce-nario. We analyse the response of the society to intermittent external informa-tion. We keep the cyclic presentation mode described in the previous section.However, the different opinion patterns are no longer influencing the society forthe entire presentation period ∆ , but only for a fraction θ ∆ of each period,with θ < . The signal is absent for the remainder ∆ = (1 − θ )∆ of thepresentation period: I i ( t ) = ( I ξ µ ( t ) i n ∆ < t ≤ n ∆ + θ ∆ otherwise (37)with n ∈ N and µ ( t ) = 1 + (cid:22) t ∆ (cid:23) mod p . (38)In this way we represent a society hit by a periodic sequence of different strongstimuli, such as repetitive political propaganda or a series of shocking events (e.g.terrorist attacks) alternated with periods of absence of external information.Questions that arise in such a scenario are: Will the society be shaped by theseshocks? What is the smallest fraction θ of the time for which the system isexposed to external stimuli that still allows the society to spontaneously retainthe information presented?To evaluate the couplings in this case, we assume that during the time θ ∆ inwhich the signal I ξ µ is on, the preference fields immediately align and g = ξ µ ,which requires the signal strength I to be sufficiently large. As long as thesociety remains unable to retain the presented patterns, we find that, as soon asthe signal is removed, the preference fields very quickly decay to and remainsmall during the time ∆ in which the signal is off. Once the society hasbeen exposed to sufficiently many presentation cycles, couplings of sufficientstrength may have developed allowing the society to retain information aboutthe latest pattern presented, even when the signal is removed. When evaluatingthe couplings for this situation, we assume for simplicity that the system remainsnearly fully aligned with the previously presented pattern, g ≃ ξ µ , even after thethe signal is turned off. Figure 5 shows a simulation exhibiting a transition froman early time regime, where information is not retained after signal removal, toa late time regime, where the system remains aligned with a signal even at timeswhere the signal is switched off. In Fig. 6, we present a zoom into both theearly time and the late time regimes. Shaded rectangles in the figure representthe intervals θ ∆ in which an external signal is present. At early time when thesignal is switched off, the opinions take some time to disalign to it. This extratime, that we will indicate as θ ′ ∆ , is not easy to calculate, however we can givea rough estimation of it assuming that the preference field u i ( t ) freely decaysto zero when the signal is removed (see Appendix B). We define the time t ∗ ,in multiples of p ∆ , as the time of the last cycle of external stimuli that is stillinsufficient to create couplings of a strength needed for the society to retain19
500 1000 1500 2000 2500-0.200.20.40.60.81
Figure 5: Simulated dynamics with intermittent signal. Three patterns are presented ina cyclic fashion. Each pattern presentation for a time θ ∆ = 70 is followed by a period ∆ = 30 during which there is no external signal. Each time a pattern is presented thecorresponding overlap m µ very quickly approaches 1, and it decays to smaller values when thesignal is removed. At early times, couplings are still too small to retain previously presentedinformation and overlaps decay to small O (1 / √ N ) values when external signals are removed.However, after a time t ∗ ≃ the couplings are able to sustain the opinion patterns evenwhen the signal is removed. In this simulations I = 5 , J = 6 and γ = 10 − Figure 6: The figures represent a zoom into the early time regime (left panel), and into thelate time regime (right panel) of Fig. 5. The coloured rectangles represent the time periods inwhich a signal is switched on. After the signal removal at early times the value of m µ dropsto significantly lower values, whereas it remains much close to m µ = 1 at late times. t < t ∗ and for t > t ∗ under the simplifying assumptions made above (details of the calculations inAppendix C): J ij ( t < t ∗ ) = J N ( e γ ∆ ( θ + θ ′ ) − e − γt (1 − e γt )(1 − e γ ∆ p ) p X µ =1 ξ µi ξ µj e ∆ γ ( µ − (39) J ij ( t > t ∗ ) = J N e − γt (cid:18) ( e γ ∆ ( θ + θ ′ ) − e γ ∆ ) (1 − e γt ∗ )(1 − e γ ∆ p )+ ( e γ ∆ −
1) (1 − e γt )(1 − e γ ∆ p ) (cid:19) p X µ =1 ξ µi ξ µj e ( µ − γ (40)The couplings in Eq.s (39) and (40) are for simplicity evaluated only for integermultiples of ∆ . As shown in Fig. 7, the couplings thus predicted compareremarkably well with those evaluated in a numerical simulation of the dynamicsas presented in Fig. 6. However, given the approximations used in the estima-tion of θ ′ we cannot expect to have a perfect agreement between the analyticalprediction and the simulations (the limitations of our approach are discussed inAppendix D). In Fig. 7, the choice of parameters is such that the interactionsrapidly align with the Hopfield couplings, and their norm grows along the dy-namics eventually granting retrieval of the signal patterns after a finite time t ∗ .To determine t ∗ , we use these equations to obtain an expression for the cou-plings J at time t ∗ + ( θ + θ ′ )∆ (see Appendix C for details of the calculation)and use these in the self-consistency equation for m (also derived in the sameappendix) m = erf m J (1 − e − γ ∆ ( θ + θ ′ ) ) p (1 + σ ) (1 − e − γ ( t ∗ +∆ p ) )(1 − e − γ ∆ p ) ! . (41)The value of t ∗ is then obtained by requiring that Eq. (41) has a non trivialsolution, which gives t ∗ = − ∆ p " γ ∆ p log − p (1 + σ ) J (1 − e − γ ∆ ( θ + θ ′ ) ) √ π − e − γ ∆ p ) ! . (42)Note that t ∗ increases with decreasing θ , and it will eventually diverge (and afinite t ∗ will cease to exist) as θ is decreased below θ min θ min = − γ ∆ log − p π (1 + σ )2 J (1 − e − γ ∆ p ) ! − θ ′ (43) = − γ ∆ log − p π (1 + σ )2 J (1 − e − γ ∆ p ) ! − log (cid:18) I (1 − e − ∆ θ )0 . (cid:19) SimulationsAnalyticHopfield
Figure 7: The upper panel shows the evolution in time of the correlation Q between thesimulated J , its analytical prediction and the Hopfield couplings over p , J H /p . The lowerpanel compares the norm of J/J from simulations to its analytical prediction and to thenorm of J H /p . The control parameters are the same as Fig. 5. θ ′ is evaluated in Appendix B. The solution of this equation can be found nu-merically and the resulting behaviour in function of / ∆ is shown in Fig. 8.The condition θ > θ min thus guarantees the existence of a finite time t ∗ atwhich persistent memory starts to form, and at least one of the patterns storedcan be recovered . The existence of a minimum value of θ required for thesociety to be able to spontaneously retrieve the information contained in thesignals presented earlier is of immediate practical relevance. For advertisementcampaigns, for instance, it defines the minimum fraction of time needed for arepeatedly presented signal to permanently impress the audience as a collectivebody. In the domain of news, it would, for instance allow to assess, whether ornot repeated news items might leave a subtle persistent trace in the society andproduce collective responses otherwise unpredictable.To verify our predictions of θ min we simulated dynamics with external signalof different amplitude I until stationary interaction couplings are reached. Wethen froze the couplings and counted how many times the society recovers atleast one of the patterns after a signal spike. Recovery is reached when thecorresponding overlap in absence of external signal satisfies the threshold con-dition m µ > . . Such recovery threshold has been chosen significantly higherthan the overlap ( ∼ / √ N = 0 . ) expected if the system state is uncorrelatedwith the pattern, but not too high in order not to exclude recovery with a low O (1) overlap given the system parameters. We finally estimate θ min from sim-ulations as the smallest θ for which at least half of trial runs of the dynamicsshow such retrieval behaviour. In Fig. 8 we plot θ min for different values of I as a function of the inverse total presentation time / ∆ alongside the analyticprediction θ min obtained above.As clearly visible in the figure, for large ∆ the analytic curve gives a goodprediction of the simulation results for all the signal strengths, confirming thatthere is not significant dependence on I in this regime. It is interesting to no-tice that in this regime the value of ( θ min + θ ′ )∆ corresponds to the minimumtime needed by the society to embed a single pattern presented with a persistentsignal (the minimum t r mentioned in section 5): θ min + θ ′ ≃ − γ ∆ log − p π (1 + σ )2 J ! (44)assuming J > p π (1 + σ ) / . This means that at the beginning of the dynam-ics the society is able to maintain its orientation towards the very first patternseen, that will be the one most easily remembered.Interestingly at small ∆ a much stronger dependence on I develops. Theanalytical curves qualitatively capture the trend of the numerical ones but over- For θ > θ min at least one pattern will be recovered, but this does not guarantee that thefirst pattern of the cyclically repeated sequence is among those recovered. This includes also mixture states with m = ( m , m , m ) , where m = 0 , m = 0 and m = 0 , which are also encountered in some instances. Figure 8: Simulated relation between the minimum θ necessary for the recovery of at leastone pattern and the inverse of the time ∆ compared with their analytical estimation for γ = 10 − and different signal strengths. estimate their true value. This discrepancy can be related to the approximationsused in the estimate of θ ′ . When solving the dynamics in the fraction of time θ ′ ∆ we neglect the interactions between agents. In doing this we underestimatethe partial memory that has started to form in the society, and consequentlyoverestimate the θ min needed for recovery.Lastly but very importantly we note that both numerical results and analyticestimations show a decrease in θ min at small ∆ which is more pronouncedwhen I is larger. This phenomenon is a direct consequence of the fact that θ ′ increases with I , therefore for larger I the term ∆ θ ′ allows recovery withsmaller θ min. In applications the fact that θ min reaches very small values forsmall ∆ and large I would suggest to invest in short but frequent and high-impact advertisements. Similarly we can conclude that shocking events suchas terror attacks, if repeated on short periods, might leave deep traces in thesociety despite being very localised in time.
7. Conclusions
News of disruptive events in history such as terror attacks often appear tochange the behaviour of a society and will influence how people will react tonews of future events of a similar kind. In this work we introduce a simple modelof opinion dynamics which includes otherwise well-studied phenomena such ashomophily [8, 9, 10, 11, 12] (the tendency of people to interact more oftenwith others who share similar opinions) and xenophobia [16, 17, 23, 24] (thetendency to adopt opinions different different from those of people with whomthere has been disagreement in the past). The model includes dynamically24volving couplings, which effectively record an exponentially weighted historyof co-expressed opinions between any pair of agents in the system. We showhow this mechanism allows a society do develop a collective memory of externalinformation it had previously been exposed to, allowing it to spontaneouslyretrieve such information in the future when briefly triggered by exposure tothat information.We study the emergence of this type of collective memory both analytically andby way of simulations for three stylized scenarios representing different historiesof exposure to external information: (i) information consisting of a persistentsignal, (ii) information consisting of repeated presentations of a set of differentsignal patterns, and (iii) information consisting of repeated presentations of aset of different signal patterns separated by periods of absence of any signal. Inthe first scenario, the external information does not change in time; the societyaligns to the signal and — after a sufficiently long time of exposure to the signal— will remember it in the future even when the signal is removed. In the secondscenario, the society is exposed to a series of signals, corresponding to differentnews. If these news are repeated for a sufficient number of times, the society isable to remember all of them, and recall them in the future when triggered by abrief spike of the same information. This can be true also in the third scenario,in which the different news are are interspersed with phases of absence of anysignal. The determining factor here is the relative length of the periods of signalpresentation and signal removal. We were able to compute the critical minimalratio of presentation time and signal removal time that allows the society todevelop a persistent memory of the sequence of news and thereby to remainaligned to external information even when the signal is removed. Moreover wedemonstrated that even very short signals, if sufficiently strong and repeatedsufficiently often, can guarantee the spontaneous retrieval of their information.In the three scenarios analysed, polarizing signals presented to the society wereable to deeply change the collective behaviour of the society described by ourmodel, in the sense that persistent memory of past events emerged which causesit to react differently in the future. The condition for this to occur is alwaysthat the bare coupling constant J in the model is sufficiently large comparedto the noise level of the dynamics. Thus our model is able to capture, how thecollective behaviour of a society can be strongly influenced by its past events.A follow up work will concern the study of the model under more realisticassumptions about signal structures, including presentation of news items inrandom order and presentation of news with different signal intensities. Aninteresting further generalization that could be considered to make the modelmore realistic is to define interactions depending on multidimensional (ratherthan binary scalar) opinions so as to represent the effect that individuals interactin ways which depend on judgments about a collection of topics. In this settingthe evolving interactions based on past interpersonal agreement or disagreementon the entire set of topics would correlate the agents’ response to the differenttopic in a non trivial way. 25 cknowledgements The authors acknowledge funding by the Engineering and Physical Sci-ences Research Council (EPSRC) through the Centre for Doctoral Trainingin Cross Disciplinary Approaches to Non-Equilibrium Systems (CANES, GrantNr. EP/L015854/1). The authors would like to thank Nishanth Sastry for hiscontribution at the initial stage of this project and Jean-Philippe Bouchaud,Ton Coolen, Imre Kondor and Francesca Tria for interesting discussions.
Appendix A. Analytical prediction of J ij for a repetitive signal We want to give an analytical estimation of J ij for a signal defined in Eqs.(27) and (28). For simplicity, we will evaluate the couplings only at the end ofeach pattern presentation period (i.e., for times t which are integer multiplesof ∆ ). We thus define J ij ( ν, t ) as the coupling at time t = N p p ∆ + ν ∆ , inwhich N p is the number of complete cycles of p patterns and ν < p the numberof additional patterns seen in the final (possibly incomplete) cycle presented upto time t . If we assume that as soon as the signal is switched on, the expressedopinions g will for each µ presented align with the pattern ξ µ , then J ij ( ν, t ) willtake the form J ij ( ν, t ) = J N γ Z t d se − γ ( t − s ) g i ( s ) g j ( s )= J N γ " ξ i ξ j Z ∆ d se − γ ( t − s ) + ... + ξ pi ξ pj Z p ∆ ( p − d se − γ ( t − s ) ! + ξ i ξ j Z ( p +1)∆ p ∆ d se − γ ( t − s ) + ... + ξ pi ξ pj Z p ∆ (2 p − d se − γ ( t − s ) ! + ... + ξ i ξ j Z ( N p p +1)∆ N p p ∆ d se − γ ( t − s ) + ... + ξ νi ξ νj Z t =( N p p + ν )∆ ( N p p + ν − d se − γ ( t − s ) ! . (A.1)Using Z a +∆ a e γs d s = Z ∆ e γx + γa dx = ( e γ ∆ − e γa γ , (A.2)26e obtain J ij = J N e − γt ( e γ ∆ − h(cid:16) ξ i ξ j + ... + ξ pi ξ pj e γ ( p − (cid:17) + (cid:16) ξ i ξ j e γp ∆ + ... + ξ pi ξ pj e γ (2 p − (cid:17) + ... + (cid:16) ξ i ξ j e γN p p ∆ + ... + ξ νi ξ νj e γ ( N p p + ν − (cid:17)i . (A.3)Now we can group the terms and using t = N p p ∆ + ν ∆ we have J ij ( ν, t ) = J N ( e γ ∆ − e − γt p X µ =1 ξ µi ξ µj e ( µ − γ ( t − ν ∆ ) / ( p ∆ ) − X k =0 e γ ∆ pk + ν X µ =1 ξ µi ξ µj e − ( ν − µ +1)∆ γ ! = J N ( e γ ∆ − ( e − γt − e − γν ∆ ) )(1 − e γ ∆ p ) p X µ =1 ξ µi ξ µj e ( µ − γ + ν X µ =1 ξ µi ξ µj e − ( ν − µ +1)∆ γ ! . (A.4) Appendix B. θ ′ estimation for intermittent signal We would like to have an estimate of the the time needed by the expressedopinions to disalign with the signal when this is removed. Given that we arenot able to evaluate the integral in Eq. (2) during the decaying transient of g i ( t ) , we can estimate it considering | g i ( s ) g j ( s ) | = 1 for a time ( θ + θ ′ )∆ and 0for the remaining time ∆ (1 − θ − θ ′ ) . The time θ ′ ∆ can be estimated as thetime in which | g i ( s ) g j ( s ) | falls to half of its value at the removal of the signal(that is approximately 1). In this way the overestimation implied by assuming | g i ( s ) g j ( s ) | = 1 for s ≤ θ ′ ∆ is compensated by the underestimation made byassuming | g i ( s ) g j ( s ) | = 0 at subsequent times. We can give a rough estimateof g i during the decay assuming that the signal is switched on at time 0 and assoon as it is removed the preference fields follow: u ( t ) = h u (∆ θ ) i e − t . (B.1)where we recall that u = ξ i u i and u i ( θ ∆ ) is the solution of Eq. (1) just beforethe signal is removed. Here we also assume that the interactions between theagents and the noise are neglected, such that the preference field freely decayto 0. Using this, we can calculate the time t = θ ′ ∆ for which | g i ( t ) g j ( t ) | = 0 . that is: θ ′ ≃ log( h u (∆ θ ) i / . . (B.2)27n order to find u i ( θ ∆ ) we use Eq. (9) and we assume that the opinions are 0before the signal is presented and they align to it as soon as it is switched onwith g i ∼ ξ µi . As a consequence in Eq. (9) we will have u i (0) = 0 and h U i (∆ θ ) i = γN Z ∆ θ d s e − γ (∆ θ − s ) g i ( s ) X j g j ( s ) g j (∆ θ )= ξ i m ( t )(1 − e − γ ∆ θ ) , (B.3)which in the limit ∆ ≪ τ γ which we consider in this paper gives: |h U i (∆ θ ) i| = | m ( t ) γ ∆ θ | ≪ (B.4)and so it is negligible respect to the other terms in the equation. The finalestimation of h u (∆ θ ) i will thus be: h u (∆ θ ) i = I (1 − e − ∆ θ ) , (B.5)that inserted in Eq. (B.2) gives: θ ′ = 1∆ log (cid:18) I (1 − e − ∆ θ )0 . (cid:19) . (B.6)We want also to remark that neglecting the terms in Eq. (B.4) from the calcu-lations does not influence significantly the final result. In fact if we substitute m ≃ in Eq. (B.4) and we use this to estimate θ ′ the results that we obtainonce the other parameters inserted the results do not differ substantially whencompared to those obtained by neglecting this subdominant contribution. Appendix C. J ij for intermittent signal Given a signal of the form in Eq. (37) and (38) we want to calculate thecouplings J ij in our model. In order to do this we will assume that at thebeginning of the dynamics, as soon as soon as each signal contribution µ isswitched on, the preference field aligns to it with g i = ξ µi . The opinions willremain aligned to the signal for a time ∆ ( θ + θ ′ ) where θ ∆ is the time thesignal is actually on and θ ′ ∆ is the additional time the opinions remain alignedto the signal during the decay of the preference field g i .Let us define t = N p ∆ p , with N p as the total number of complete cyclesof p pattern presentations seen at time t . Following the same reasoning of28ppendix A we will calculate J at time t as: J ij = J γN Z t e − γ ( t − s ) g i ( s ) g j ( s )= J γN ξ i ξ j Z ∆ ( θ + θ ′ )0 e − γ ( t − s ) ds + ... + ξ pi ξ pj Z ∆ ( p − θ + θ ′ )∆ ( p − e − γ ( t − s ) ds ! + ... + ξ i ξ j Z ∆ (( N p − p + θ + θ ′ )( N p − p ∆ e − γ ( t − s ) ds + ... + ξ pi ξ pj Z ∆ ( N p ( p − θ + θ ′ )∆ N p ( p − e − γ ( t − s ) ds ! (C.1)Now using Z a +∆ ( θ + θ ′ ) a e γs ds = (cid:16) e γ ∆ ( θ + θ ′ ) − (cid:17) e γa γ , (C.2)we get J ij = J N (cid:16) e γ ∆ ( θ + θ ′ ) − (cid:17) e − γt p X µ =1 ξ µi ξ µj e ∆ γ ( µ − N p − X k =0 e ∆ γpk (C.3)and finally exploiting N p − X k =0 e ∆ γpk = (1 − e γ ∆ pN p )(1 − e γ ∆ p ) = (1 − e γt )(1 − e γ ∆ p ) (C.4)we obtain J ij = J N ( e γ ∆ ( θ + θ ′ ) −
1) ( e − γt − − e γ ∆ p ) p X µ =1 ξ µi ξ µj e ∆ γ ( µ − . (C.5)If the amplitude J of the couplings is too small for the given noise level ofthe dynamics, the society may never be able to retrieve any of the informationit was previously exposed to, in which case the above expression holds for all t .Otherwise, if J is large enough, for some θ the society is able to retrieve thepattern 1 at time t = t ∗ + ∆ ( θ + θ ′ ) after its presentation, with t ∗ = N ∗ p ∆ p .For this to happen we thus need a non-trivial solution of: m = erf m J (1 − e − γ ∆ ( θ + θ ′ ) ) p (1 + σ ) ( e − γ ( t ∗ +∆ p ) − e − γ ∆ p − ! , (C.6)29btained using J at t = t ∗ + ( θ + θ ′ )∆ : J ij = J N (1 − e − γ ∆ ( θ + θ ′ ) ) (cid:20) ( e − γt ∗ − − e γ ∆ p ) p X µ =1 ξ µi ξ µj e ∆ γ ( µ − ++ ξ i ξ j (cid:21) . (C.7)We thus need J (1 − e − γ ∆ ( θ + θ ′ ) ) p (1 + σ ) (1 − e − γ ( t ∗ +∆ p ) )(1 − e − γ ∆ p ) 2 √ π ≥ (C.8)which is possible after a time t ∗ = − ∆ p " γ ∆ p log − p (1 + σ ) J (1 − e − γ ∆ ( θ + θ ′ ) ) √ π − e − γ ∆ p ) ! . (C.9)In order for this time to be finite θ should be larger than a certain thresholdthat is calculated in the main text. For time larger than t ∗ we thus have thatthe opinions remain aligned with the signal even when this is removed, so thecouplings J can be calculated considering g i = ξ µi for the whole time interval ∆ . This means that to calculate J at a time t = N p ∆ p > t ∗ we need to addterms to the sum of integrals in Eq. (C.1), which are integrals of the kind ofEq. (C.2) albeit with the upper limit replaced by a + ∆ . This results in thefollowing couplings: J ij = J N e − γt ( e γ ∆ ( θ + θ ′ ) − p X µ =1 ξ µi ξ µj e ( µ − γ N ∗ p − X k =0 e γ ∆ pk + ( e γ ∆ − p X µ =1 ξ µi ξ µj e ( µ − γ N p − X k = N ∗ p e γ ∆ pk (C.10) = J N e − γt p X µ =1 ξ µi ξ µj e ( µ − γ " ( e γ ∆ ( θ + θ ′ ) −
1) (1 − e γ ∆ pN ∗ p )(1 − e γ ∆ p )+ ( e γ ∆ −
1) ( e γ ∆ pN ∗ p − e γ ∆ pN p )(1 − e γ ∆ p ) (C.11)For simplicity in this appendix we calculated J ij only for discrete times multipleof p ∆ . However we can remark that it is possible to calculate them for anytime, following the same reasoning used here. Appendix D. The role of signal strength for intermittent signals
The analytically predicted couplings J do not always give the true norm | J | of the couplings in our model, given that the approximations made in the cal-culations are of limited validity. In Fig. D.9 we exhibit the dynamical evolution30 Figure D.9: Intermittent signal dynamics. The figure shows an example of the evolution of | J | /J in time for J analytical and from simulations and different signal strengths given threefixed patterns. The control parameters are γ = 10 − , J = 6 , ∆ = 50 and ∆ = 37 . . of | J | /J for intermittent signals for a variety of signal strengths and observethat while the analytic theory does predict the true evolution reasonably wellfor small values I of the signal strength, the analytical prediction fails qualita-tively for very large values of I . The different curves in the figure correspondto dynamics with parameters other than the signal strength I identical for allcurves. The analytical prediction appears to work better for simulations withsmaller I . This is due to approximations used in the estimation of the time θ ′ ∆ needed by the preference fields to decay to 0 when the signal is removedafter a pattern presentation. Our estimation does in fact neglect the effect ofthe couplings during the decay. For I = 1 the time θ ′ ∆ is not large enough toallow the couplings to grow much and neglecting their contribution in our cal-culation does not effect significantly the prediction of | J | . For I = 50 , the time ∆ θ ′ is large enough to allow the coupling to grow to sufficiently large values topermit eventual spontaneous recall, so neglecting them results in a substantialerror in the prediction of | J | . In fact, in the case of I = 50 , after a time t ∗ (approximately equal to the time of the steep increase of | J | /J in the figure)the couplings have become sufficiently strong to sustain the opinion patternswhen the signal is removed, kickstarting the positive feedback-loop which even-tually results in the society being capable of nearly perfect spontaneous patternretrieval — a behaviour very different from the one predicted analytically forthe given parameters (for which a finite t ∗ does not exist).31 eferenceseferences