Phase transitions and oscillations in a temporal bi-layer echo chambers model
BBifurcations and catastrophes in temporal bi-layer model of echo chambers andpolarisation (cid:32)Lukasz G. Gajewski ∗ and Julian Sienkiewicz Faculty of Physics, Warsaw University of Technology, Koszykowa 75, 00-662 Warszawa, Poland
Janusz A. Ho(cid:32)lyst
Faculty of Physics, Warsaw University of Technology, Koszykowa 75, 00-662 Warszawa, Poland andITMO University, Kronverkskiy Prospekt 49, St Petersburg, Russia 197101
Echo chambers and polarisation dynamics are as of late a very prominent topic in scientificcommunities around the world. As these phenomena directly affect our lives and seemingly moreand more as our societies and communication channels evolve it becomes ever so important for usto understand the intricacies opinion dynamics in modern era. We build upon an existing echochambers and polarisation model and extend it onto a bi-layer topology. This new topologicalcontext allows us to indicate possible consequences of interacting groups within this model. Fourdifferent cases are presented - symmetric negative and positive couplings, an asymmetric couplingand an external bias. We show both simulation results and mean field solutions for these scenariosoutlining the possible consequences of such dynamics in real world societies. Our predictions showthat there are conditions in which the system can reach states of neutral consensus, a polarisedconsensus, polarised opposition and even opinion oscillations. Transitions between these states interms of bifurcation theory are identified and analysed using a mean field model.
I. INTRODUCTION
It is well established that there is a lot that can be saidon how our societies form and function with techniquesand approaches familiar to physicists [1–5]. A particu-lar interest lately has been the dynamics of opinion for-mation, especially in the light of recently better studiedphenomena such as echo chambers [6–9] and misinfor-mation [10–15]. One of the major effects that seems tobe strongly connected with echo chambers and misinfor-mation is that of polarisation. While not every topic ispolarising [16, 17] many certainly can be [7, 18–25]. Itseems to have been recognised as dangerous to the stateof democracy around the world by the scientific commu-nity and the need for research in this very topic is ratherclear [26–32] especially in the light of a possible event ofdemocracy backsliding [33, 34].We find that it is also of interest to study the possibledynamics between two clearly defined groups as it oftencan be in politics (e.g. Democrat vs Republican in theUSA), topics (pro- or anti-) as well as has precedencein sociophysics [35–41]. In particular we felt inspiredby the work of Baumann et al. [6] where the authorsintroduce an echo chambers and polarisation model oncomplex networks. In this paper we modify said modelso that it operates on a bi-layer temporal network, as op-posed to a mono-layer, where each layer can represent aclearly defined group of individuals (agents). This trans-formation is directly driven by the fact that many systemdrastically change their physical properties (e.g., phasetransition type change) when considered on a duplex (bi-layer) topology [42, 43]. We show that several complex ∗ [email protected] behaviours can be acquired by simply changing the na-ture of the coupling between those layers. Let us under-line that the question of interacting layers is an extremelyvivid topic in the view of COVID-19 epidemic. Recentstudies point to a pivotal role played by risk perceptionlayer in the spreading of a disease [44] or explicitly theattitude toward vaccination [45]. In this scope examiningthe dynamics of two coupled opposite groups (e.g., pro-and anti-vaccination) seems to be highly relevant.Originally, in the work of Baumann et al., the systemconsists of N agents each with a real, continuous opinionvariable x i ( t ) ∈ R . The sign determines the nature ofopinion (for/against) while the value the conviction toit. The opinion dynamics is driven exclusively by theinteractions between agents and is described by a systemof coupled ordinary differential equations presented in [6]:˙ x i = − x i + K N (cid:88) j =1 A ij ( t ) tanh ( αx j ) , (1)where K > social interaction strength and α > A is an N × N adjacency matrix in anactivity-driven (AD) temporal network model [50–53](see Fig. 1 for our bi-layer interpretation). This is amodel where there is no statically set social network butin each time step an agent can become active with propen-sity a i ∈ [ (cid:15), F ( a ) = 1 − γ − (cid:15) − γ a − γ . (2) a r X i v : . [ phy s i c s . s o c - ph ] J a n FIG. 1. Illustration of the temporal bi-layer network model. At any given moment an agent from either group can get activatedand impose its influence upon other (red arrows) while in some cases this influence can be reciprocated (green arrows). Eacharrow is labelled with an appropriate social influence coefficient later on used in system of equations (3).
Once the agent is activated it makes m random connec-tions with other agents and as is standard in AD modelsthe connections are uniformly random. In [6] there is ad-ditionally an element of homophily as it is expected tobe necessary to create polarisation effects [54, 55], how-ever, since we will be considering a bi-layer model lateron this is not the case for us. A proper study of the effectsof the homophily in the from presented by Baumann etal. could turn out to be of interest yet we find it goingbeyond the scope of this paper.As the interactions in social media can often be asym-metric and so it is not always true that A ij = A ji . How- ever, in this model there is a mechanism of reciprocity where each agent j that has received a connection froman active agent i can reciprocate the connection withprobability r . II. MODEL DESCRIPTION
We modify the scenario described by Baumann et al.by considering a system of two (potentially) opposinggroups represented by layers - X and Y - such that N X of agents belong to group X and N Y to Y. With this (1)becomes: ˙ x i = − x i + K xx N X (cid:80) j A xxij ( t ) tanh( α xx x j ) + K xy N Y (cid:80) j A xyij ( t ) tanh( α xy y j ) + B ˙ y i = − y i + K yy N Y (cid:80) j A yyij ( t ) tanh( α yy y j ) + K yx N X (cid:80) j A yxij ( t ) tanh( α yx x j ) , (3)where B denotes an external bias that will function as areplacement for the second layer later in the paper.Let us further assume K xx = K yy = K , α xx = α yy = α xy = α yx = α , N X = N Y = N and both r and a tobe the same for both groups, within as well as without.Average activity is given by: (cid:104) a (cid:105) = 1 − γ − γ − (cid:15) − γ − (cid:15) − γ . (4)Similarly as in [6] we assume that processes relatedto topology changes as described by matrices A ij ( t ) aremuch faster than changes of opinions x i ( t ) and y i ( t ) andwe shall insert into (3) mean values of these matrices (cid:104) A ij ( t ) (cid:105) t,a = m (1 + r ) (cid:104) a (cid:105) . When K xy = K yx then the Jacobian of (3) calculated in the point x i = y i = 0 pos-sesses two special eigenvectors, e + = [1 , , ... , , T and e − = [1 , , ... − , − , − T and corresponding eigenval-ues λ + = cα [ K ( N x − /N x + K xy ] and λ − = cα [ K ( N x − /N x − K xy ].Then we can write mean field equations for the ex-pected values of opinions in X and Y .For simplicity let us set c = m (1 + r ) (cid:104) a (cid:105) and then, (cid:40) ˙ (cid:104) x (cid:105) = −(cid:104) x (cid:105) + Kc tanh( α (cid:104) x (cid:105) ) + K xy c tanh( α (cid:104) y (cid:105) ) + B ˙ (cid:104) y (cid:105) = −(cid:104) y (cid:105) + Kc tanh( α (cid:104) y (cid:105) ) + K yx c tanh( α (cid:104) x (cid:105) ) . (5)We show that in our bi-layer variant of the echo cham-bers and polarisation model [6], without the mechanismfor homphily, there are conditions in which the systemcan reach a state of neutral consensus, a polarised con-sensus [56, 57] (or radicalisation phase as named by Bau-mann et al.), an opposite polarisation (named simplypolarisation by Baumann et al.) or even opinion oscil-lations. Further on we provide agent based simulationsand detailed mathematical analysis verifying these pre-dictions. III. METHODOLOGY
All simulations were conducted, unless stated other-wise, with parameter values: network size N = 1000 , γ =2 . , (cid:15) = 0 . , m = 10 , r = 0 . , K = 1 , α = 1 , K xy = K yx = − c ≈ . dt = 0 .
05. The temporal adjacency matrix A ij iscomputed at each integration step. Mean field equationswhere no analytical solution was possible were integratedusing an embedded Runge-Kutta 5(4)[5, 58]. Followingthe rationale in [6, 59] the AD network is updated on eachintegration step as to separate the timescales of connec-tions and opinion dynamics. IV. RESULTS
In this section we present the results of agent basedsimulations and the mean field approximation to the fourscenarios described before. The scenarios are symmetricand negative coupling of the opposing groups, asymmet-ric coupling, single group system under an influence ofthe second layer treated as an external bias and weakpositive coupling between the layers.
A. Opposite polarisation
We present here the case of the symmetrically and neg-atively coupled opposing layers. We arbitrarily choosegroups to start with all its agents with opinion +1 (X)and with -1 (Y) in Fig. 2, however, the results do notdepend on this choice, see Fig. 4b. With the use of amean field theory we expect a phase transition from aneutral consensus - where both groups converge at zero- to a polarised state where the layers remain in theirrespective opinions in opposition to one another, as thecontrol value cα is increased. We choose not to use onesingle control parameter as the behaviour of the systemslightly changes depending on whether we modulate c or α .We arrive at that prediction as follows. The Jacobian time o p i n i o n (a) XYX-MFY-MF0 25 50 75 100 time o p i n i o n (b) FIG. 2. Example trajectories of the groups’ average opin-ions as they change in time. Dashed lines represent agentbased simulations and there are 20 independent realisationsshown. Solid lines are the result of the mean field approxima-tion (-MF). The top panel - (a) - shows the behaviour belowthe critical value with α = 0 .
84 - both groups converge on aneutral opinion while the bottom panel - (b) - above it with α = 4 . matrix of (5) is J | (cid:104) x (cid:105) = (cid:104) y (cid:105) =0 = (cid:20) − cαK cαK xy cαK yx − cαK (cid:21) , (6)from which we get both eigenvalues as: λ , = cαK ∓ cα (cid:112) K xy K yx − . (7)When K xy = K yx then eigenvalues λ , reduce to λ + , − λ + , − = cα ( K ± K xy ) − , (8)calculated directly from the agent based model (5) in thelimit N → ∞ and in such a case corresponding eigenvec-tors of Jacobian (6) are e + = [1 , T and e − = [1 , − T .In general the product K xy K yx can be positive or neg-ative; if it is positive then either K xy > ∧ K yx > c o p i n i o n (a) x i , y i MF c o p i n i o n (b) FIG. 3. Phase transition (a pitchfork bifurcation) from the symmetric consensus to the opposite polarisation of opinions indifferent layers X and Y under different parameter modulation in both agent based simulations and mean field approximation.The transition takes place at the point ( K − K xy ) cα = 1. The left panel - (a) - shows the transition as we increase α and keep c ≈ .
306 while the right panel - (b) - shows what occurs when we keep α = 1 and change c by increasing the parameter m .The agent based results are averaged over 20 independent realisations with a 95% confidence interval present in the form ofthe error bands. Asymptotic behaviours observed at both panels for cα (cid:29) | K xy | | K yx | (a) MF 123 | K xy | | K yx | (b) FIG. 4. The coupling parameters phase space ( | K xy | − | K yx | in a form of a heat map, where the colour represents (cid:104)| opinion |(cid:105) ,with a visible transition from neutral consensus to polarisation. MF is Eq. (9). Left panel - (a) - shows results for initialconditions in an already polarised and opposed state. In the right panel - (b) - the initial conditions for all agents were drawnrandomly from a uniform distribution ( − ,
1) showing that this result does not depend on initial conditions. the system falls into what was described by Baumannet al. (unless a weak coupling is introduced as shownlater on) or K xy < ∧ K yx < λ max = λ − changes sign. Since the eigenvector e − is asymmetrical thus the case λ max > x = y = 0 looses its stabilityand systems is polarized, i.e. opinions in groups X and Y split into opposite directions. From λ max changing itssign we get a relationship between the K xy and K yx : K yx = (cid:18) − cαKcα (cid:19) × K xy . (9) If K yx = K xy the system is in the polarised phase andits steady state is x t →∞ = − y t →∞ which can be foundby solving numerically for x t →∞ the following relation: x t →∞ = ( K − K xy ) c tanh( αx t →∞ ) . (10)Eq. (10) can be written in a normalised form u = ( K − K xy ) αc tanh( u ) when u = αx t →∞ . Since the solution u of the last equation increases from 0 to ( K − K xy ) αc whenthe product ( K − K xy ) αc increases from 1 to ∞ thus for( K − K xy ) αc (cid:29) x t →∞ ≈ ( K − K xy ) c, (11) time o p i n i o n (a) XYX-MFY-MF 0 25 50 75 100 time o p i n i o n (b) time o p i n i o n (c) FIG. 5. Example trajectories in the asymmetric coupling parameters scenario for α = 1 , , . which explains the difference in the behaviour we observein Fig. 3.In Fig. 2 we show the two aforementioned phases - con-sensus and polarisation. Plots show the mean opinion ofeach layer as a function of time. The agent based sim-ulations are not deterministic and therefore we show 20independent realisations and compare against the meanfield prediction. It is apparent that below the criticalvalue of cα the whole system converges at zero - bothlayers reach a neutral consensus (Fig. 2a). As the con-trol parameter is increased the situation changes and apolarisation phase occurs (Fig. 2b). The two layers nowstand in opposition to one another and no consensus ispossible.Using the mean field theory we estimate the criticalvalue of cα and present the test of our predictions inFig. 3. As mentioned before it depends whether we mod-ulated α or c and we show that in Fig. 3a and Fig. 3b,respectively. When c = const. the system reaches aplateau, however, when c is increased the final opinionvalue of the system also increases indefinitely. In bothscenarios we see a phase transition (a supercritical pitch-fork bifurcation [60]) at a certain critical value and areasonably decent fit from the mean field approximation.We also present a heatmap (Fig. 4) of the coupling pa-rameters phase space with cα ≈ .
06. The colour thereshows the absolute value of the mean opinion of the sys-tem. Again we see a transition from consensus to polar-isation with a good match from the mean field approachand specifically the Eq. (9).We find this setting to be representative of a typicalecho chamber situation in context of two rivalling groupssuch as political parties. If the animosity from one to theother or mutually is strong enough then no consensus ispossible - while the groups may not be as radical as ininitially they will always persist in their view opposite tothe other. This essentially shows that prejudice has thepotential to lock society into a predetermined antagonis-tic state.
B. Opinion oscillations
Here we present the asymmetric case where one group”likes” the other but the feeling is not mutual. Accord-ing to the mean field theory we ought to see two possiblebehaviours of the system - dampened or sustained oscil-lations depending on the values of the control parameter.As before it does depend whether we change c or α . InFig. 5 and 6 we show time and phase trajectories respec-tively. In both cases it is apparent that the two afore-mentioned behaviours are present. Namely the systemhas two possible attractors - a point or an orbit. Whilethere is a slight shift as to when the transition occurswhen comparing agent based simulations and the meanfield approximation we find the analytical approach to bequalitatively successful.This effect is due to the product K xy K yx being nega-tive and then the eigenvalues are complex, and the sys-tem exhibits a supercritical Hopf bifurcation [60]. When Kcα < ,
0) i.e. there is a consensus amongst thegroups. When
Kcα > x ) isbounded).What is also interesting in this case is how the sus-tained oscillations change as we modulate α or c . Asbefore we choose to modulate c via the parameter m . InFig. 7 we show both frequencies and amplitudes as func-tions of cα with either α or m modulation. A supercriti-cal Hopf bifurcation takes place at the point Kcα = 1 andthe frequency of the emerging periodic orbit at the crit-ical point should be equal to f crit = ca (cid:112) K xy K yx / π ≈ . x )), the mean field predictions are show-ing a good qualitative match to agent based simulations.The frequency f is slightly different in the overcriticalregion as compared to the critical value f crit that is inagreement with the theory of Hopf bifurcation [61]. Forlarge values of cα the amplitude of oscillations saturates opinion X o p i n i o n Y opinion X o p i n i o n Y opinion X o p i n i o n Y opinion X o p i n i o n Y MF FIG. 6. Y(X) trajectories in the asymmetric coupling scenario for α = 1 , , ,
10 with dashed lines representing 20 independentrealisations of the agent based simulation and solid lines showing the mean field solution. We observe in detail that the systemhas two possible attractors - a point and an orbit. For
Kcα > as the function of the parameter α and is a linear func-tion of the parameter c . This behaviour is similar toplots at Fig. 3 and it related to scaling observed for theasymptotic steady state solution x t →∞ , see Eq. (11).While this scenario might be slightly less obvious tointerpret we do believe there are certain parallels to bedrawn here. It may seem as though one groups is a trendsetter while the other are followers . In such a case thereis a very similar sort of a feedback dynamic that we ob-serve in our model. One group - the followers - is posi-tively oriented towards the other - the trend setters - asthey look up to them and would like to be, act, thinklike them etc. On the other hand, the setters share anegative attitude towards the followers in this context.While they might appreciate the following they wouldvery much want to move away from it in terms of theopinion in question. This leads to this chasing and os-cillating behaviour. However, should the attitudes mag-nitudes within the groups be not strong enough the dy-namic simply dies down as neither the followers are notas interested in following nor setters in trend setting. C. External bias
We can also study a cumulative effect of a bi-layer envi-ronment via an addition of external bias to a mono-layersystem. This bias can represent cumulative effect of an-other group (Y) or just the medium in which the systemoperates. This external bias - B - is either supporting Xor working in opposition to X. We assume it is of a linearkind.In such case we can say that:˙ x i = − x i + K N (cid:88) j A ij ( t ) tanh( αx j ) + B, (12)and by averaging x i we get˙ (cid:104) x (cid:105) = −(cid:104) x (cid:105) + Kc tanh( α (cid:104) x (cid:105) ) + B. (13)The dynamical system described by the (13) exhibits acusp catastrophe [62, 63]. If Kcα <
Kcα > B is smaller than some critical value B c c a m p li t u d e (a) c f r e q u e n c y (b) c a m p li t u d e (c) c f r e q u e n c y (d) XYX-MFY-MF
FIG. 7. The oscillations frequencies and amplitudes dependence on the parameter modulation in the asymmetric coupling case.Top row is for α modulation and the bottom one for m . Dashed lines represent an average over 20 independent realisations ofagent based simulations with a 95% confidence interval present as the error bands. Solid lines show the mean field solution.Differences in asymptotic behaviours of amplitudes in (a) and (c) panels are similar to differences observed for values of thesteady state solutions x t →∞ at the left and right panel of Fig. 3. then equation (13) possesses two stable and one unstablefixed point. It means the group can possess a polarisationtowards or against the external bias B . When B is largerthan some critical B c then the equation (13) possessesonly one solution and the polarisation directed againstthe external bias is not possible. It means that at somecritical B c a discontinuous transition takes place (see Fig.8). Values of B c can be found from the stability analysisof (12) or (13).In this scenario we can no longer linearise at (cid:104) x (cid:105) = 0and must introduce an unknown x c instead. In the caseof (13) we get the Lyapunov exponent [64] at x c as λ = − Kcα sech ( αx c ) (14) In the case of (12) the Jacobian becomes: J | x i = x c = − Kcα sech ( αx c ) . . .Kcα sech ( αx c ) − . . . ... , (15)with the largest eigenvalue λ max = − N − N Kcα sech ( αx c ) , (16)When N → ∞ then the solutions (14) and (16) coincide.Combining the condition for the steady state of (13) andthe condition for changing the sign of the eigenvalue λ max (16) we get a solution for the critical value of the external FIG. 8. Bifurcation and hysteresis loop in the system of ex-ternal bias: for α > / ( Kc ) the system is bistable and oncea critical value of B c is reached there is a switch from a po-larisation against the field to towards it (an vice-versa for − B c ). Also in such a case, we cannot reach a neutral solution( x = 0) for any B >
0. For α < / ( Kc ) we have only onestable solution and such effects do not take place. bias B c : B c = x c − Kc tanh( αx c ) x c = α cosh − (cid:18)(cid:113) N − N Kcα (cid:19) N →∞ ∼ α cosh − (cid:0) √ Kcα (cid:1) . (17)In Fig. 9 we show that a phase transition occurs fromthe system’s state to that of the bias. E.g. if the systemconverges on a positive (average) opinion and we set thebias to a negative and sufficiently large value the systemwill suddenly jump to the opposite side. In Fig. 9a wepresent an example of that. We wait until the systemreaches its steady state and then activate the bias withan opposite sign. If the value is below the critical onethe system merely shifts slightly towards zero, however,if | B | > B c a sudden jump occurs. In Fig. 9b we showthis in the B − α phase space. Yet again the mean fieldapproach - Eq. (17) - allows us to predict this behaviour,however, unlike previously the discrepancy in the actualvalue of the critical point is quite significant. This is mostlikely due to finite nature of the simulated system as isusual for the mean field theory.We consider this case study as illustrative of how forexample a propaganda may or may not be successful. Weuse propaganda here as a neutral term without concern-ing ourselves whether it is good or bad. One can easilyimagine situations that are either. Such a scenario boilsdown to the strength of the campaign in question, how- time < x > (a) B=0.3B=0.15bias activation
10 200.10.20.30.40.5 B (b) MF 0.40.60.81.0
FIG. 9. Phase transition under the influence of an externalbias. The top panel - (a) - shows two examples of an averageopinion of the system as a function of time. One trajectoryis for a value of external bias above the critical threshold andthe other below. A solid vertical line signifying the momentwe enable the external bias is present. The bottom panel -(b) - shows the B − α phase space, where colour is (cid:104)| opinion |(cid:105) ,with a visible phase transition to an opposite opinion and themean field approximation - Eq. (17). ever, it is not simply the stronger you push the moresupporters you get. The dynamic of change is non-linearand the transition very sudden. This implies that it maybe rather difficult to react to the propaganda machinein time to stop the society from drastically shifting itsstance. D. Weakly coupled groups
Here we discuss the behaviour of the model when thetwo layers are positively but weakly coupled via the cou-pling parameter δ .We introduce this weak-coupling parameter 0 < δ < K xy and K yx are positive andfor simplicity let us assume K xy = K yx = δK . The meanfield equations for the expected values can be written as: (cid:40) ˙ (cid:104) x (cid:105) = −(cid:104) x (cid:105) + Kc tanh( α (cid:104) x (cid:105) ) + δKc tanh( α (cid:104) y (cid:105) )˙ (cid:104) y (cid:105) = −(cid:104) y (cid:105) + Kc tanh( α (cid:104) y (cid:105) ) + δKc tanh( α (cid:104) x (cid:105) ) . (18)With positive coupling the two groups ought to merge forsome critical value δ c . However, before that happens acoexistence of two oppositely polarised groups is possible.In such a case x c = − y c in the steady state and sincesech ( x ) is an even function we need not worry aboutthe sign. Again by writing out the Jacobian and lookingat the larges eigenvalue and the steady state solution itis easy to obtain that: (cid:40) δ c = Kcα cosh ( αx c ) −
10 = − x c + Kc (1 − δ c ) tanh( αx c ) , (19)which must be solved numerically.We find that there exists a critical value δ c for which aphase transition occurs from a polarisation state to a non-neutral consensus state (or a so called radicalisation).In Fig. 10 we present this behaviour. The two layersstart in opposition and we wait until the system reachesthe steady state and then we enable a positive but weak0 < δ < δ increasesthe groups final average opinions slowly and smoothlyapproach each other until the critical value of δ c , cor-responding to a bifurcation point, and the two groupsmerge into one with a radicalised opinion. Whether thefinal opinion is positive or negative is down to infinitesi-mal fluctuations and thus the apparent bifurcation visi-ble in Fig. 10a. When we ignore the sign and look at thedistance between the groups’ averages we clearly see atransition from polarisation to radicalisation (Fig. 10b).We also see a decent match of the mean field approachhere, however, only for relatively small values of α and δ - see the Fig. 10c. Here we show a heatmap of the δ − α phase space where colour denotes the distance betweenaverages and similarly to the case of an external bias wesee qualitative but not quantitative agreement betweenthe simulation and the mean field solution - Eq. (19).Weakly coupled layers case study stands somewhat incontrast to the opposition scenario described before. Saywe can somehow influence the attitudes of the layers suchthat we soften the animosities towards more amicable andmaybe even eventually slightly cordial side of things thenwould that be enough? Or do we need to completely flippeoples attitudes to make consensus possible. Our modelsuggests that it can be enough, indeed. This implies thatwhile prejudice can cause society to split there is alsoroom for hope as not so drastic changes in attitudes cancause the layers to merge in opinion albeit not a neutralone. V. CONCLUSIONS
In this paper we consider a temporal bi-layer echochamber and polarisation model on complex networks in- spired by the mono-layer model introduced by Baumannet al. We recognise that there is both a precedent and ap-parent value in studying scenarios where two clearly cutgroups - or layers in a network - are interacting with oneanother. Understanding how layered complex networksevolve in various environments in context of opinion dy-namics can help us better prepare for tackling issues thatpotentially threaten our democracies such as misinforma-tion campaigns or echo chambers.We formulate the dynamics equations for the bi-layersystem (3) and then provide a mean field analysis thatuncovers interesting possible scenarios. The nature ofsystem’s behaviour is different depending on the cou-pling between the layers. We categorise those couplingas opposite polarisation where the groups do not likeeach other, opinion oscillations where one group likes theother, however, the feeling is not mutual, external biaswhere we consider the other group as an external biasacting upon a mono-layer system and finally a weak pos-itive coupling where there is an attraction between thegroups, however, not as strong as withing them.In the opposite polarisation scenario we observe that acoexistence of two groups with different (opposite) opin-ions is possible. The system undergoes a phase transitionfrom a neutral consensus - where the two layers’ opinionsmerge at zero - to a polarised state - where the two groupscoexist each of them having their own opinion, oppositeto the other groups. The details of this pitchfork bifurca-tion and the asymptotic behaviour of the system dependon whether we modulate the non-linearity parameter α or the combined social influence parameter c , or the cou-pling parameters K xy , K yx , however, in both cases themean field approximation gives us a satisfying fit to agentbased simulations.When the coupling parameters are set asymmetrically,in the sense that one is positive and one negative, we de-tect a transition from dampened to sustained oscillationsof the layers’ opinions - a supercritical Hopf bifurcation.In a way one might say that one group is ”chasing” theother with their opinions, while the other is trying toget away. We additionally find that the oscillations arehighly non-linear as the frequency decreases with controlparameter as opposed to increasing as one would expectfrom a linear oscillator. At the same time the amplituderises with the control parameter. We believe the ampli-tude here plays the role of a sort of barrier for the systemto overcome and so the higher the barrier the longer ittakes to be overtaken thus the frequency of the oscilla-tions increase.In the case of a single layer with an external biaspresent we postulate that it might be possible to modeleither a background of some sort or the second layer forthat matter as simply a cumulative effect in the form ofsuch an external bias. We find that there exists a crit-ical value of said bias that when the system is subjectto it a sudden change to an opposite opinion is possibleand the cusp catastrophe is apparent. For small valuesof the control parameter we find a decent match of mean0 X (a) X - Y (b)
20 40 600.20.40.60.81.0 (c)
MF 0.10.20.30.40.5
FIG. 10. Phase transition in a weakly coupled scenario. The left - (a) - and middle - (b) - panels show 20 independentrealisations of an agent based simulation as dots and mean field solution as open circles (with α = 10). In the left one theaverage opinion of layer X is shown (Y omitted for clarity as it would simply be symmetrically opposite) with a clear bifurcationpresent. In the middle one we show the difference (cid:104) x (cid:105) − (cid:104) y (cid:105) . The right panel - (c) - presents the δ − α phase space, where colouris |(cid:104) x (cid:105) − (cid:104) y (cid:105)| , with a visible transition from opposing opinions to a non-neutral consensus. MF is Eq. (19). field approach and agent based simulations, however, forlarger values the two diverge in the prediction as to whenthe transition should occur, most likely due to the finitesize of the simulated system.Finally when the two layers are weakly yet positivelycoupled we see a not very dissimilar behaviour to theone with the external bias. Namely there exists a criticalvalue of the coupling parameter that causes the systemto experience a sudden shift in the opinions. In this casewe observe that there is a transition from an oppositelypolarised state to a polarised consensus (or a radicalisedstate) where all agents (from both layers) share similarand non-zero opinion. Similarly to the previous case thematch between the mean field theory and simulations isqualitatively satisfying, however, for larger values of thecontrol parameter the predictions as to when the transi-tion should happen diverge from the results of numericalexperiments. With each scenario we have drawn parallels to realworld to illustrate what these results could mean for un-derstanding the dynamics of our societies. We under-stand that there are limitations with both the model andthe approach in general as it can be often difficult toconstruct reproducible experiments in sociological con-text, however, we firmly believe that seeing where cer-tain assumptions can lead us is an important and crucialbuilding block of science. ACKNOWLEDGMENTS
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