The Ising universality class of kinetic exchange models of opinion dynamics
Sudip Mukherjee, Soumyajyoti Biswas, Arnab Chatterjee, Bikas K. Chakrabarti
aa r X i v : . [ phy s i c s . s o c - ph ] D ec The Ising universality class of kinetic exchange models of opinion dynamics
Sudip Mukherjee a,b, , Soumyajyoti Biswas c, , Arnab Chatterjee d, , Bikas K. Chakrabarti b,e,f, a Department of Physics, Barasat Government College, Barasat, Kolkata 700124, India b Saha Institute of Nuclear Physics, Kolkata 700064, India c Department of Physics, SRM University - AP, Andhra Pradesh - 522502, India d TCS Research, Tata Consultancy Services, New Delhi, India e S. N. Bose National Centre for Basic Sciences, Kolkata 700106, India f Economic Research Unit, Indian Statistical Institute, Kolkata 700108, India
Abstract
We show using scaling arguments and Monte Carlo simulations that a class of binary interactingmodels of opinion evolution belong to the Ising universality class in presence of an annealed noiseterm of finite amplitude. While the zero noise limit is known to show an active-absorbing transition,addition of annealed noise induces a continuous order-disorder transition with Ising universality classin the infinite-range (mean field) limit of the models.
1. Introduction
Study of social dynamics using tools of statistical physics is an active research area, wheresystems and models are being studied qualitatively and quantitatively [1, 2, 3, 4, 5]. The study ofdynamics of opinions, individual and collective choices is one of the most popular topics. One isusually concerned with the question, how interacting individuals choose between different options(vote, opinions, language, culture, etc.), leading to a state of ‘consensus’ or a state of coexistenceof multiple options. With the basic assumption that such options can be quantified and that theindividuals take their decisions in choosing those options through interactions with their ‘neighbors’,physicists have attempted to model this ‘complex’, multi-component system to bring forward theirgeneric and universal features [3, 4, 5].Given a finite interaction term among the individuals, that competes with the noise, usually themodels of social dynamics exhibit collective dynamical phenomena. Several models introduced sofar study the dynamics that leads to different opinion states and the universal nature of transitionsbetween such states. The rich emergent phenomena, resulting out of interaction of a large numberof entities or agents [6] follow various ‘universality classes’ in terms of their characterizations thatare, depending on specific contexts, similar to the models of statistical physics [3, 5].The present study concerns the dynamics of opinions, and how consensus may or may notemerge out of interaction of individual opinions, or, out of interaction of individuals with opinionsevolving out of influence of others. A series of studies on this topic [7, 8, 9, 10, 11] has enriched ourunderstanding of the topic. Opinions are usually modeled as discrete or continuous variables, whichcan undergo spontaneous changes or change due to interaction with others, or external factors. The [email protected] [email protected] [email protected] [email protected] Preprint submitted to Physica A December 22, 2020 Ο ( λ , ζ ) λ N=512 a ζ=0.9ζ=0.8ζ=0.7ζ=0.6ζ=0.5ζ=0.4ζ=0.3ζ=0.2ζ=0.1 Ο ( λ , ζ ) λ N=512 b ζ=0.3ζ=0.2ζ=0.1ζ=0.0 λ ζ order disorder c Figure 1: Plot of order parameter with conviction λ for different values of noise: (a) for ζ =0 . , . , . , . , . , . , . , . , .
9, and (b) for ζ = 0 . , . , . λ , compared with the LCCCmodel ( ζ = 0). The plots are given for N = 512. (c) The phase diagram of the noisy LCCC model, in the λ − ζ plane.The region above the line is the ‘ordered’ phase while that below is the ‘disordered’ phase. study of dynamics of opinions, as well as their steady state properties are interesting. The possibilityof a phase with a spectrum of opinions and another phase where the majority have similar valuesmay demonstrate the existence of different phases – a typical scenario to study phase transitions. Incontinuous opinion models, opinions cluster around a single value (consensus), or two (polarization)or can as well have several values (fragmentation).We focus our attention to a specific class of models [12], having apparent similarity with kineticmodels of wealth exchange [13, 14]. The opinions of individuals that can be continuous or discretevariables, depending on the context, are bound between two extremes ( − ,
2. Kinetic exchange opinion model: Transition driven by external noise
Following the multi-agent statistical model of closed economy [13], Lallouache et al. [12] proposeda minimal multi-agent model for the collective dynamics of opinion formation. Let o i ( t ) ∈ ( − , +1)be the opinion of an individual i at time t . In a system of N individuals, opinions change out ofbinary interaction according to: o i ( t + 1) = λ [ o i ( t ) + ǫo j ( t )] o j ( t + 1) = λ [ o j ( t ) + ǫ ′ o i ( t )] , (1)where ǫ , ǫ ′ are randomly and independently drawn from an uniform distribution in (0 ,
1) and λ is a parameter, interpreted as ‘conviction’, the value of which lies between 0 to 1. In the abovemodel [12] (LCCC model hereafter), everyone has the same value of conviction λ . In the abovedynamics (Eq. 1), agent i meets j , each keeping a fraction λ of their own opinion and are alsoinfluenced by a random fraction of the other agent’s opinion. There are no conservation laws herefor the opinions, but the opinions are bounded, i.e., − ≤ o i ( t ) ≤
1. This bound is ensured bykeeping the magnitude of the opinion values to 1 even if Eq. (1) gives a higher magnitude. The2 b O N β / ν ( λ - λ c )N ν ’ O λ c V N - γ (cid:0) ν ( λ - λ c )N ν ’ V λ a U ( λ - λ c )N ν ’ U λ Figure 2: Finite size scaling behavior for ζ = 0 .
3: (a) Scaling collapse of Binder cumulant with λ c = 0 . ± . N (inset). ν ′ ≡ dν = 2 . ± .
01 is estimated from the scaling collapse.Critical Binder cumulant U ∗ = 0 . ± .
01. (b) Scaling collapse of order parameter O for β = 0 . ± .
01. Inset showsunscaled data for O with λ . (c) Scaling collapse of V with γ = 1 . ± .
01. Inset shows unscaled data for V with λ . ordering in the system is measured by a quantity (order parameter) O = | P i o i | /N . Numericalsimulations show that the multi-agent system (dynamics defined by Eqn. (1)) goes into either ofthe two possible phases: for any λ ≤ λ c , o i = 0 ∀ i , while for λ > λ c , O > O → λ → λ c ≈ /
3. Here λ c is the critical point of the phase transition. The relaxation time, defined asthe time to reach a stationary value of O in time, diverges as τ ∼ | λ − λ c | − z ( z ≈ . ± .
1) when λ → λ c on either side [12]. The order parameter near the critical point behaves as: O ∼ ( λ − λ c ) β with the order parameter exponent β = 0 . ± .
01 [16]. A mean field calculation can be proposedfor the fixed point o ∗ : o ∗ [1 − λ (1 + h ǫ i )] = 0, from which it easily follows that the critical point is λ c = 1 / (1 + h ǫ i ) (where h . . . i refers to average). For uniform random distribution of ǫ , h ǫ i = 1 / λ c = 2 /
3. It was also noted that the underlying topology (1 d , 2 d or infinite range)has little effect on the critical point. A map version of the model [12] has also been proposed: o ( t + 1) = λ (1 + ǫ ( t )) o ( t ), with ǫ ( t ) drawn randomly from a uniform distribution in [0 , o ( t ) is still bounded in [ − , +1] as before. The critical value λ c can be analytically shown to beexp[ − (2 ln 2 − ≈ . o i ( t + 1) = λ [ o i ( t ) + ǫo j ( t )] + ζ i o j ( t + 1) = λ [ o j ( t ) + ǫ ′ o i ( t )] + ζ j , (2)where ζ i , ζ j are drawn randomly and independently from ( − ζ, ζ ) with | ζ | ≤
1. Of course ζ = 0corresponds to the LCCC model. We observe that introducing the noise ζ destroys the active-absorbing nature of the phase transition. Specifically, in the LCCC model, the entire phase λ < λ c was characterized by null value of opinion for all agents, o i = 0 ∀ i and thus O = 0 trivially. Anyfinite ζ remarkably changes this character, and O = 0 close to the critical point (see Fig. 1b).Another interesting observation is the change in the nature of the order parameter curve as thenoise ζ increases. For small values of noise, the rate of change in O is much faster near the criticalpoint i.e., the transition is sharp, while this rate seems to decrease as the noise level increases (seeFig. 1a). This is because, the critical region is smaller, the closer the model is to the absorbingtransition point. With an increased noise parameter, the model is further away from the absorbingtransition point and therefore has a wider critical range and hence show a relatively less sharptransition. But as we shall see in the following discussions, the universality class of the transitionis independent of the noise level.Apart from the order parameter O , we also calculate the following quantities: V = N (cid:2) h O i − h O i (cid:3) analogous to susceptibility per spin, and U = 1 − h O i h O i , the fourth order Binder cumulant. The3 a U ( λ - λ c )N ν ’ U λ b O N β (cid:1) ν ( λ - λ c )N ν ’ O λ c V N - γ (cid:2) ν ( λ - λ c )N ν ’ V λ Figure 3: Finite size scaling behavior for ζ = 0 .
6: (a) Scaling collapse of Binder cumulant with λ c = 0 . ± . N (inset). ν ′ ≡ dν = 2 . ± .
01 is estimated from the scaling collapse.Critical Binder cumulant U ∗ = 0 . ± . O for β = 0 . ± .
01. Insetshows unscaled data for O with λ . (c) Scaling collapse of V with γ = 1 . ± .
01. Inset shows unscaled data for V with λ . order parameter O behaves as O ∼ | λ − λ c | β and the susceptibility V as V ∼ | λ − λ c | − γ near thecritical point, i.e., for small values of | λ − λ c | . We apply finite-size scaling (FSS) theory to calculatethe critical exponents. For large system sizes, we expect an asymptotic FSS behavior of the form O = L − β/ν F O ( x ) [1 + . . . ] (3) V = L γ/ν F V ( x ) [1 + . . . ] , (4)where β, γ are scaling exponents for the order parameter and the susceptibility respectively, F beingthe scaling functions with x = ( λ − λ c ) L /ν as the scaling variable, L being the linear dimensionof the system. The dots in [1 + . . . ] are the corrections to the scaling terms. The crossing of theBinder cumulant is used to estimate λ c since Binder cumulant U is independent of system size atthe critical point λ c , i.e., U L ( λ c ) = U ∗ . Scaling collapse of U with ( λ − λ c ) L /ν provides an estimateof critical exponent ν . We study the infinite dimension (mean field) version of the model, whereany agent can interact with any other agent. In that case the linear dimension L and system size N are related as L d = N , where d will be the upper critical dimension. Hence, L /ν will be replacedby N /ν ′ , where ν ′ = νd .We estimate the critical point λ c for values of the noise parameter ζ in (0 : 1), and thus get aphase diagram in the λ − ζ plane (Fig. 1c). The solid line is the critical line of λ c (or ζ c if we hadfixed the noise parameter and observed the transition by driving λ ). It seems that the critical point λ c increases very slowly by increasing ζ until ζ ≃ .
4, after which the rate of change is markedlyfaster.The value of the critical Binder cumulant ( U ∗ ) for all the noise amplitudes we studied ( ζ =0 . , . , .
9) show a value close to 0 .
30 (see Fig. 5), which is near the value ( ≈ .
27) predicted forthe critical Binder cumulant of the Ising model (using field theoretic ǫ expansion [20]; see also thenumerical estimate ≈ .
30 [21]). Numerical analysis here of the model gives dν ≃ β ≃ / γ ≃ ζ . It then implies from thehyper scaling relation that the specific heat exponent α = 2; dν = 0. Then the Rushbrooke scalingrelation α + 2 β + γ = 2 gets satisfied. These observations suggest that the steady state statistics ofthe model belongs to the Ising universality class for any amount of annealed noise
3. Kinetic exchange opinion model: Noise in the interaction
In the infinite range interacting kinetic exchange opinion model introduced by Biswas, Chatterjeeand Sen [18], the dynamical equation for the evolution of the individual opinion o i ( t ) of the i -th4 a U ( λ - λ c )N ν ’ U λ b O N β / ν ( λ - λ c )N ν ’ O λ c V N - γ / ν ( λ - λ c )N ν ’ V λ Figure 4: Finite size scaling behavior for ζ = 0 .
9: (a) Scaling collapse of Binder cumulant with λ c = 0 . ± . N (inset). ν ′ ≡ dν = 1 . ± .
01 is estimated from the scaling collapse.Critical Binder cumulant U ∗ = 0 . ± .
01. (b) Scaling collapse of order parameter O for β = 0 . ± .
01. Inset showsunscaled data for O with λ . (c) Scaling collapse of V with γ = 1 . ± .
05. Inset shows unscaled data for V with λ . agent at time t , in presence of an external influence (denoted by a field term h ), can be extended(following [18]) to: o i ( t + 1) = o i ( t ) + µ ij o j ( t ) + h · sgn ( o i ( t )) , (5)where the agent-agent interaction term µ ij = − p and 1 with probability (1 − p ),and the external influence term | h | , a stochastic variable, is equal to 1 with probability q and equalto 0 with probability (1 − q ). It acts like a local reinforcement, triggered by external influences(e.g., consumption of news from favored but unreliable sources). The bound in the opinion valuesi.e., | o i | ≤ o i and o j could come closer together or move furher apart, depending on µ ij , which was not the case for Eq.(1). The similarity, therefore, is in the binary interactive nature of the models and also the factthat the noisy variant shows Ising universality class, as we shall discuss.If f , f − and f are the fraction of agents having opinion vales 1, − f f (1 − p ) + f f − (2 p −
1) + f f − (1 − p ) q + p (1 − q )( f − − f ) = 0 , (6)where f − f − = O , the order parameter. Further exact evaluation of the probabilities along thisline turns out to be difficult. But one can write down a general polynomial expansion by collectingthe q dependent (linear) terms and the q independent terms, from an equation of the form Eq. (6).We can have a + bO + cO + dO · · · = e · q (1 + O ( O ) + . . . ) , (7)where a, b, c, d, e are constants.First, a = 0, since we must have O = 0 when q = 0 and p = p c = 1 /
4. We must also have c = 0, because O ( − h ) = − O ( h ) (note that e can depend on h , and thereby taking care of the signon the right hand side). Eq. (7) could be compared with Eq. (3) of Ref. [18] in absence of q .The linear term is present there, but not the cubic term. We argue that the cubic term is the onlypossibility beyond the linear term and not the quadratic term, due to the reasons mentioned above.The cubic term, and possibly other higher order terms, will appear in the equation when we takeinto account variations of the order parameter in two or more successive scatterings (we only takeone step scattering in Ref. [18], but for the system to be in the steady state, we can take the othersteps in general). Stopping at the cubic term puts the requirement that b/d ∝ p − p c . Also, from Eq.5 U * ζ theoretical estimate Figure 5: The variation of the critical Binder cumulant value ( U ∗ ) is shown for different noise parameters ( ζ ). Thisvalue, known to indicate a universality class, is showing no systematic variation with ζ . It is close to the theoreticallyestimated [20] value ≈ .
27 expected for the Ising model universality class in the mean field limit. This also suggeststhat the kinetic exchange opinion model with finite noise amplitude belongs to the Ising universality class. (3) of Ref. [18], it can be seen explicitly that the coefficient of the linear in O term is proportionalto p − p c (take f = 1 /
3, then p − f (1 − p ) becomes proportional to p − p c ). Combining these twoobservations suggest that b ∼ ( p − p c ), and in absence of the opinion influencing field ( e = 0). Eq.(7) then gives O ∼ b/d ∼ ( p − p c ) β , where β = 1/2 (as already obtained in Ref. [18])Finally, for p ≈ p c keeping upto the linear terms in O and q , which we take to be very small, wehave bO ≈ e · q. (8)Then the susceptibility would be χ = ∂O∂q (cid:12)(cid:12)(cid:12)(cid:12) q → . (9)Implying, χ ∼ b ∼ ( p − p c ) − γ , (10)where γ = 1, agreeing well with the numerical estimate in Ref. [18].Given that the order parameter exponent β , susceptibility exponent γ values obtained here andthe numerically estimated value of dν = 2 .
4. Discussions
We have studied two variants of the infinite range interacting kinetic exchange opinion model,namely that by Lallouache et al. [12] (with an added annealed noise to get continuous order-disorder transition, avoiding the active-absorbing one) and by Biswas, Chatterjee and Sen [18],using Monte Carlo simulations (analyzing finite size scaling, Binder cumulant behavior, etc.) andscaling arguments (see Landau type expansion of the steady state opinion probabilities in section3). We show that at least in the infinite range limit of interactions between the agents in these two6inds of kinetic exchange models (of opinion formation dynamics), the steady state statistics clearlybelong to the mean-field Ising universality class ( β = 1 / , γ = 1 and dν = 2, satisfying Rushbrookescaling 2 β + γ = dν , giving ν = 1 / d = 4). Additionally, the valueof the Binder cumulant at the critical point ( U ∗ ) shows a value close to 0 .
30 consistently for thesemodels, which is close to the field theoretically estimated value ( ≈ .
27) for the Ising model in themean field [20] and comparable with similar numerical estimates [21]. The numerical results forlower dimensional systems (see e.g., [19]) also indicate the same. There could be interesting futuredirections in extending the model for more realistic networks and/or models with diluted bonds(representing mutually non-interacting agents), to check if the universality class changes.It may be mentioned that this observation of Ising Universality class in the steady state statisticsof some kinetic models help comprehending the missing link connecting the observed Ising univer-sality class of liquid gas transition in the extended kinetic model of ideal gas, bypassing the use oflattice-gas model.
Acknowledgments
We are honored to have this opportunity to contribute in this special issue of Physica A inmemory of Prof Dietrich Stauffer, who pioneered in several interdisciplinary researches in statisticalphysics. The authors thank Parongama Sen for her valuable comments on the manuscript.
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