Stable target opinion through power law bias in information exchange
SStable target opinion through power law bias in information exchange
Amitava Datta ∗ Department of Computer Science and Software Engineering, University of Western Australia, Perth, WA 6009, Australia
We study a model of binary decision making when a certain population of agents is initiallyseeded with two different opinions, ‘+’ and ‘ − ’, with fractions p and p respectively, p + p = 1.Individuals can reverse their initial opinion only once based on this information exchange. We studythis model on a completely connected network, where any pair of agents can exchange information,and a two-dimensional square lattice with periodic boundary conditions, where information exchangeis possible only between the nearest neighbors. We propose a model in which each agent maintainstwo counters of opposite opinions and accepts opinions of other agents with a power law bias untila threshold is reached, when they fix their final opinion. Our model is inspired by the study ofnegativity bias and positive-negative asymmetry known in the psychology literature for a long time.Our model can achieve stable intermediate mix of positive and negative opinions in a population. Inparticular, we show that it is possible to achieve close to any fraction p , ≤ p ≤
1, of ‘ − ’ opinionstarting from an initial fraction p of ‘ − ’ opinion by applying a bias through adjusting the powerlaw exponent of p . PACS numbers:
I. INTRODUCTION
The study of emergent behaviour in a population basedon simple interaction rules among individuals has beenan intense area of research in complex systems and socio-physics. Most of these models study opinion formationin a population in the context of two different opinions,majority and minority, or ‘+’ and ‘ − ’. The opinions inthese models evolve either according to some simple localrules, or according to some group dynamics. An exten-sive review of such models can be found in the paper byCastellano et al. [13].The aim of this paper is to investigate negativity bias in opinion formation. There is an extensive literatureon negativity bias in psychology, as reviewed by Rozinand Royzman [1] and Vaish et al. [2]. Negativity bias ismanifested in humans and animals in many different ac-tivities, including, attention and salience, sensation andperception, motivation, mood and decision making [1].Some of these activities are closely related to opinion for-mation and hence, it is interesting to study the effect ofnegativity bias in opinion formation in a population. Wepresent a model of opinion formation that uses negativ-ity bias and has several interesting properties, includinga similarity with the random-field Ising model and alsothe formation of predictable intermediate configurationsof mixed opinions.One of the earliest among opinion formation modelswas the voter model (VM) [6, 7] that can be simulatedon any connected network. Each agent has a state ± − ) state, the probablity ofassuming a (++) or a ( −− ) state in an interaction is each. This simple update rule gives rise to rich dynam-ical behavior and the VM has been studied extensively.The VM always evolves to a homogeneous final state of one of the opinions, the rate of convergence depends onthe initial populations of the two opinions, and has astochastic nature. Hence it is hard to predict the mix ofpopulations at intermediate stages of the evolution.Schweitzer and Behera [10] introduced the nonlinearVM where neighbors with different opinions are weightedwith nonlinear weighting factors. The nonlinear VM hasinteresting configurations where both the opinions coex-ist equally when the starting initial fraction of populationfor each opinion is 0 . contrarian has a different strategy from the otheragents. Contrarians introduce interesting variations inthe evolution of almost all the models we discuss. Ma-suda [15] studied the linear VM by introducing threetypes of contrarians and concluded that contrarians pre-vent the evolution of the linear VM to a homogeneousfinal state of a single opinion and induce a mixed pop-ulation of both the opinions. Masuda [15] derived theequilibrium distributions of the two opinions under dif-ferent assumptions on the contrarians. However it is notclear whether it is possible to get specific mix of popula-tions in the model in [15] and also whether the dynamicsof the model is scale invariant.Group dynamics of binary opinion formation has beenstudied by Galam and his coworkers extensively as dis-cussed in the review paper [3]. Galam’s [4] 2-stateopinion dynamics model is of particular interest for ourpresent work. This model is for a completely connectednetwork where any agent is a neighbor of any other agent.Initially each agent has one of two opinions A and B, andthe density of the two populations is denoted by p c , ex-pressed as a fraction, e.g., p c = indicates a balancedinitial population of agents with A and B opinions. Eachstep of the evolution of the model consists of picking a a r X i v : . [ phy s i c s . s o c - ph ] O c t random group of agents of a predermined size. All agentsin this group adopt the opinion of the local majority.When this update process is repeated, the resulting dy-namics is dependent on p c . The final population is abalance of A and B opinions when p c = and the groupsize is odd. When the group size is even, the final pop-ulation is balanced for a different value of p c , however,any deviation from these p c values makes the final popu-lation to converge to one of the two opinions. The speedof convergence is faster for larger group sizes.Galam [4] also considered the introduction of contrar-ian agents in this model, the contrarians participate inthe group opinion formation exactly in the same way asbefore, however, a contrarian reverses its opinion onceit has left the group. A mixed phase dynamics with aclear separation of majority and minority opinions pre-vails when the density of the contrarians a c is low. Thesepopulations are stable for a fixed value of a c . However,there are thresholds for a c for different group sizes whenno opinion dominates and there is no symmetry break-ing to separate the final population into majority andminority opinions. In other words the final population isbalanced between the two opinions even though agentschange their opinion dynamically. It is not clear whetherGalam’s model can achieve arbitrary and stable propor-tions of the two opinions by introducing contrarians.The Majority Rule (MR) model was introduced byKrapivsky and Redner [5] as a simple two-state opiniondynamics model. The MR model has similarities withGalam’s model [4]. A group of agents is chosen at everystep and the agents in the group all assume the opinionof the local majority. The aim in [5] was to study thetime to reach global consensus as a function of N , thetotal number of agents and also the probability of reach-ing a given final state as a function of the initial opiniondensities. The MR model has many interesting proper-ties and one of the characteristics features of the MRmodel that is of interest to us is that even small islandsof one opinion surrounded by the opposite opinion cangrow in size. The growth of a particular opinion variesfrom one initialization to another. There are also inter-mediate metastable states in the MR model that persistfor long times, however again the concentration of opin-ions in these metastable states vary depending on theinitialization.There are some similarities between our proposedmodel and the model in [4]. First, the aim of our modelis to arrive at a final population of a mixed majorityand minortiy population. This is achieved in Galam’smodel when the density of the contrarians is low. Sec-ond, our model behaves similar to Galam’s model whenthe fraction of initial population of agents is balanced,i.e., each. This is manifested in Galam’s model bothin the absence of contrarians and also when the initialpopulation of contrarians is greater than a threshold fordifferent group sizes. However there are distinct dissimilarities between ourmodel and the model in [4], apart from the fact that theupdate rules in Galam’s model are based on groups. Ouraim is to achieve a final population of agents separatedinto majority and minority opinions without the use ofcontrarians. In other words, all agents in our model havea common strategy. Galam’s model without contrarianshas been analysed for d -dimensional lattices by Lanchierand Taylor [8]. They have proven that Galam’s model(or the spatial public debate model in the terminologyof [8]) converges to a stationary distribution where boththe opinions have positive densities. However it is notclear whether any specific mix of the two opinions canbe achieved. Our model on the other hand behaves in asimilar way both on a completely connected network andon a 2D lattice with nearest neighbor connections. Henceour model can be thought of as the formation of globalopinion through simple local interactions. The differncesbetween our model and the MR model are also similar tothe differences mentioned above.We frame our problem in this paper in a general wayas follows. Given an initial population of agents with ‘+’and ‘ − ’ opinions, with fractions p and p respectively, p + p = 1, the goal is to achieve a fraction p of finalpopulation of agents with the ‘ − ’ opinion, 0 ≤ p ≤ p > p . We show that it is possible to achieve closeto a final fraction p of agents with ‘ − ’ opinion by intro-ducing a bias in the exchange of opinion when two agentsmeet. It is interesting that this bias can be expressed asa power law exponent of p , and scale-invariant for boththe completely connected network and the two dimen-sional lattice. Our model has interesting properties thatare similar to other models studied in statistical mechan-ics. For example, a coexistence of opposite opinions hasbeen observed in the nonlinear voter model [10], eventhough this coexistence is not stable and predictable interms of the exact mix of the two opinions. Also prop-erties of our model related to the surface tension of thedomain boundaries of opposite opinions and also first-order phase transition and domain formation are similarto the random-field Ising model [12].The rest of the paper is organized as follows. We dis-cuss our model in section II. We discuss the results froman empirical study of the model without and with thepower-law bias during information exchange in sectionsIII and IV respectively. Finally we conclude in sectionV. II. THE MODEL
A population initially has agents with two opinions incertain fractions p and p , with p + p = 1. Eachagent has the option of choosing one of the opinions astheir final opinion, however, reverting an initial opinionis allowed only once. Agents interact pairwise either on acompletely connected network or on a square lattice withperiodic boundary conditions. We have two free param-eters in our model, β and τ . Each agent maintains twocounters θ + and θ − of the positive and negative opinionsencountered so far. Initially θ + i = 1 and θ − i = 0 if agent i is a ‘+’ agent, and θ + i = 0 and θ − i = 1 if agent i is a‘ − ’ agent. When agents i and j interact, the rules forexchange of opinion from agent i to agent j are given ineqn.(1) and eqn.(2) (the subscripts of the two countersindicate which agent the counter belongs to): if ( θ − i > p β θ + i ) then θ − j = θ − j + 1 (1) if ( θ + i > θ − i ) then θ + j = θ + j + 1 (2)The update of the state of agent j occurs due to one ofthese two equations in a Monte Carlo step. First eqn.(1)is checked and if it is satisfied and an update occurs,eqn.(2) is skipped for that Monte Carlo step. Otherwisethe condition in eqn.(2) is checked and an update oc-curs if the condition in eqn.(2) is satisfied. There is noexchange of information if θ + i = θ − i and the encounteris considered a failure. In other words, we introduce anasymmetry in the updating of the counters of agent j byintroducing the bias factor p β in eqn.(1). The other freeparameter τ is used as a threshold of opinion until whichan agent participates in information exchange. Once ei-ther of the counters θ + i or θ − i reaches τ , agent i freezesits state either to ‘+’, or to ‘ − ’, depending on whether θ + i > θ − i or θ + i < θ − i . Hence this freezing of state mayrequire a flipping of the initial state of agent i , this isallowed only once. Once frozen, the state of agent i re-mains the same until all agents have reached the thresh-old which is same for all agents. However agent i stillmaintains its two counters θ + i and θ − i for use in interac-tions with agents who have not yet reached the threshold τ . Though there are no contrarian agents in our model,the β parameter can be viewed as similar to a contrarianstrategy as it introduces a bias in the interactions amongthe agents.The hallmark of negativity bias is to give greater em-phasis to negative perceptions and entities. This empha-sis is manifested in four different ways [1], negative po-tency, steeper negative gradients, negativity dominanceand negative differentiation. We aim to capture nega-tive potency and steeper negative gradient in our model.Negative potency gives a stronger impact for negative en-tities, compared to positive entities. This is captured ineqn (1). Negative gradient emphasises the steeper growthof negative events compared to positive events. This isan outcome of our model as we will explain in later sec-tions. It becomes harder for positive opinion to prevailas negative opinion accummulates more and more in thecounters of the agents in our model. The use of a single exponent in the power law bias for the whole popula-tion tries to capture the inherent negativity bias quan-titatively through a single exponent. Though this is asimplified assumption in eqn.(1), we show later that thebehavior of the system is quite stable when this exponentis allowed to vary randomly within a certain range.The agents in our model maintain more informationcompared to Galam’s model [4] and the MR model [5] inthe sense that an agent has access to both accumulatedpositive and negative opinions of another agent duringan interaction. It may seem that we are assuming a lotmore information for decision making, However one sur-prising aspect of our model is that the convergence to thedesired final state is very fast. In other words, each agentneeds to interact with a much smaller number of otheragents in order to arrive at a ‘correct’ decision, so thatthe overall fraction of desired opinion is achieved. Alsoour model can converge to a desired ‘ − ’ opinion abovethe 0 . − ’ opinion is low. This is not the casefor all the models that we have reviewed above. Galam’smodel [4] can achieve such a target population only byusing contrarian agents. The MR model has intermedi-ate states with mixed populations, however the mix issensitive to instances of the two initial populations evenwhen the fraction of the two initial populations are fixed.We will show in the following sections that our modelhas stable states of mix of opinions that can be tunedfairly accurately using the β and τ parameters. Thesestable states are scale-invariant both for the completelyconnected network and the 2D lattice. However ourmodel is unstable with respect to the τ parameter. Thefinal configurations converge to a single opinion if thethreshold τ is set relatively high. This convergence isfaster on the completely connected network and muchslower on the 2D lattice. However, the parameter τ is ameasure of the number of interactions among the agentsand a lower value indicates that the convergence of ourmodel to a balanced population mix is faster.Forming opinion based on accumulated history maybe a realistic model in the sense that people in real lifeaccept others’ opinions for decision making. Also peo-ple have their own opinion and they usually take others’opinions with some negativity bias. The contagious ef-fects of negative opinions has been studied in the psy-chology literature [1], and it has been noted that nega-tive aspects of human thought process spread faster [11].Moreover people discussing a binary opinion usually talkabout the pros and cons of the two choices. Though it ishard to capture these processes through some numericalestimates, agents in our model abstract such real-life in-teractions and discussions through the counters and alsothrough the bias for accepting opinions. (a) t=0 (b) t=2657346 (d) t=0 (e) t= 2500178, final ‘-'= 0.9 (a) (b)(c) (d)(d) FIG. 1:
Two simulation results on a 256 ×
256 lattice, ‘ − ’agents are shown in black and ‘+’ agents in white,simulation time ( t ) is measured in Monte Carlo (MC) steps. (a) Initial fractions of populations: ‘+’ and ‘ − ’ both 0 . (b) The lattice after all agents reach the threshold τ = 10, at t = 2657346. The final population of ‘ − ’ agentswas a fraction of 0 .
51 of the total; (c)
Initial fractions ofpopulations: ‘ − ’opinion 0 . . (d) Thelattice after all agents reach the threshold τ = 10, at t = 2500178. The final population of ‘ − ’ agents was afraction of 0 . III. DYNAMICS WITHOUT BIAS
We first discuss the dynamics of our model withoutapplying any bias, in other words when β = 0 in eqn.(1).There is no bias for ‘+’ or ‘ − ’ opinion in this case, andagent j accepts the higher of the two counters θ + i and θ − i of agent i for updating its own counter. We study someinteresting dynamics of our model on a 256 ×
256 latticein Fig. 1. We have verified that these results are scale-invariant by simulating them on lattices of size 512 × × ×
256 lattice with wrap-around connec-tions. Panels (a) and (b) show the results when the ini-tial populations of ‘+’ and ‘ − ’ agents are a fraction of0 . τ = 10. We have simulatedthis configuration 100 times and the final fraction of ‘ − ’agents is between 0 . − .
52 in all the simulations. Panel(a) shows the initial population of the agents ( t = 0),black (resp. white) denotes ‘ − ’ (resp. ‘+’) agents. Inpanel (b), all agents reach the threshold by t = 2657346.Since the final fractions of ‘+’ and ‘ − ’ agents are al-most the same, the basic difference between panels (a)and (b) is the rearrangements of the agents into clusters.Panels (c) and (d) in Fig.1 are for a simulation whenthe initial population of ‘ − ’ agents is 0 . − ’ agents in panel (d) is 0 . ± .
02 for 100 simulations. We note here
Time steps in every 10000 MC step N u m be r o f agen t s (a) Change of opinion on a 256x256 lattice '-' to '+', initial '-' is 0.5'+' to '-'. initial '-' is 0.5'-' to '+', initial '-' is 0.7'+' to '-', initial '-' is 0.7 Time steps in every 5000 MC step N u m be r o f agen t s × (b) Change of opinion on a 65536 CCN '-' to '+', τ =3'-' to '+', τ =3'-' to '+', τ =10'+' to '-', τ =10 (a)(b) FIG. 2:
The conversion of agents from ‘+’ to ‘ − ’ and ‘ − ’ to‘+’. (a) On a 256 ×
256 lattice, corresponding to thesimulations in Fig. 1. The middle two plots are for thesimulation when the initial population is balanced, 0 . − ’ agents. The top and bottom plots are for aninitial population of ‘ − ’ agents 0 . − ’ is much higher (top plot), comparedto the conversion of ‘ − ’ to ‘+’ (bottom plot). (b) On acompletely connected network (CCN). The two middle plotsare for a small threshold τ = 3, and the conversions aresimilar. The conversions vary widely from run to run(bottom and top plots) when the threshold is higher( τ = 10). that the behavior of our model has some similarity withthe nonlinear voter model [10] in this aspect of clusterformation.Fig.2 shows the conversion of agents from ‘ − ’ to ‘+’and vice versa. Panel (a) in Fig.2 shows this conversionfor the simulations in Fig.1 on a 256 ×
256 lattice. Wehave plotted these graphs by counting agents that hadan initial ‘ − ’ opinion, but had θ + > θ − at a particularsimulation step, or vice versa. The error bars are verysmall compared to the values of the data points, hencethey are not shown. When the initial populations arebalanced, the conversions are almost in equal numbersand the conversions stop at around t = 1500000, there-after the agents remain either ‘+ or ‘ − ’ and graduallyreach their thresholds. This implies a rearrangement ofthe agents in distinct clusters similar to the first orderphase transition in a 2D random-field Ising model [9].When the simulation starts with an initial fraction 0 . − ’ agents, the final fraction of ‘ − ’ agents is 0 . ± .
02 for 100 simulations. The conversionof ‘+’ to ‘ − ’ agents in this case is much higher and thefraction of ‘ − ’ agents increases from a fraction of 0 . (a) t=0 (b) t=1985384 (a) (b) FIG. 3: Time evolution of a droplet of ‘ − ’ agents in ourmodel. (a) t=0; (b) t=1985384.over 0 .
9. However, still there are some conversions from‘ − ’ to ‘+’ agents. The final configuaration at t = 2500178shows islands of ‘+’ agents due to strong surface tension.Panel (b) in Fig.2 shows similar conversions of agentsfor a simulation on a completely connected network(CCN) of size 65536. These simulations are very sen-sisitive to the value of τ . Both the simulations have aninitial fraction of ‘ − ’ agents as 0 .
5, increasing the value of τ quickly pushes the final population to either all ‘ − ’ orall ‘+’ agents and this final population differs from sim-ulation to simulation. The final population is an equalmix of ‘ − ’ and ‘+’ agents for τ = 3.The formation of clusters on the lattice is symptomaticin our model, as there is strong surface tension alongthe boundaries between regions of ‘ − ’ and ‘+’ opinions.We also exerimentally verified the surface tension in ourmodel by seeding a lattice of size 256 ×
256 with a dropletof negative opinion and let the system evolve until all theagents reach their thresholds, as shown in Fig. 3. Therewas almost no change in the shape of the droplet ex-cept for minute changes on the boundary. This is differ-ent from the voter model, as the coarsening of a similardroplet under the voter model results due to lack of sur-face tension and the droplet disintegrates into a regionwith an irregular boundary [19].The effects of threshold in our model for a completelyconnected network and for a lattice are quite different.Increasing the threshold for the lattice has a much slowereffect. This we can again attribute to the strong surfacetension in our model. The formation of the clusters orislands of ‘ − ’ agents is fairly rapid irrespective of thethrehsold and the main effect of the threshold is the in-crease in convergence time when all the agents reach theirthresholds within the clusters. On the other hand themodel converges to an all ‘ − ’ or all ‘+’ population withhigher threshold for the completely connected network.We studied the formation of clusters of ‘+’ and ‘ − ’agents on a lattice. The clusters of ‘+’ and ‘ − ’ agentsare of similar size when the starting population of ‘+’and ‘ − ’ agents is 0 . − ’ sites, whereas the sites on the surface of clusters have a mixed number of ‘+’ and ‘ − ’ neighbors. In Fig.4(a) westudy the change in the population of lattice points withdifferent numbers of ‘+’ and ‘ − ’ neighbors on a latticeof size 1024 × − ’neighbors are similar. In Fig.4(a), the number of ‘+’ siteswith a single ‘ − ’ neighbor grows very fast and stabilizesat a high level as more and more lattice sites becomeparts of larger clusters. These ‘+’ sites with a single‘ − ’ neighbor are on the surface or the boundary of theclusters. On the other hand the number of ‘+’ sites withtwo to four ‘ − ’ neighbors decrese rapidly and stabilize atlower levels, as these sites become parts of clusters.We study a ratio φ in Fig.4(b). This is the ratio oflattice sites on the cluster boundaries (sites that haveneighbors of opposite opinion) and the total number ofpossible neighbors in the entire lattice. We have plottedthree graphs with starting populations of ‘ − ’ agents as0 .
1, 0 . .
4, with varying target populations of ‘ − ’agents (a fraction of 0 . . φ decreases with an increase in the targetfraction, as the clusters of ‘+’ agents decrease in size.However there is a slight increase in φ as the startingpopulation of ‘ − ’ agents is increased. This is due to for-mation of a higher number of ‘ − ’ clusters as a higherinitial population provides a larger number of seeds forthese clusters.As the ‘+’ and ‘ − ’ agents form clusters quite rapidlyin simulations on a lattice, it is natural that most of theagents will be inside the clusters and a relatively smallnumber of agents will be on the surface or the boundaryof the clusters. This behavior of our model is quite sim-ilar to the random-field Ising model [12] in this respect.Moreover there is a power-law relationship between thelattice points inside the clusters and on the surface of theclusters. If we denote the number of lattice sites insidethe clusters (resp. on the surface) as V (resp. S ), thisrelationship can be expressed as S = cV δ (3)We show the log-log plot of this equation in Fig. 5 fordifferent sizes of lattice. A data collapse shows that theexponent δ = − .
29 in case of the simulation with initialfraction of ‘+’ and ‘ − ’ agents 0 . δ depends on the initial population of ‘+’ and ‘ − ’ agents.For example, with a simulation starting with 0 . − ’ and0 . . ± .
02 and δ = − .
59. This is due to the fact that thenumber of clusters of ‘ − ’ agents as well as the sizes of theclusters are smaller in this case and hence the lattice siteson the cluster surfaces are also much smaller in number. Simulation time in steps of 100 × N u m be r o f ' + ' s i t e s w i t h ' - ' ne i ghbo r s × (a)'-' neighbors for '+' sites (1024x1024) one '-' neighbortwo '-' neighborsthree '-' neighborsfour '-' neighbors Target fraction φ (b) φ against target fraction initial '-' 0.1initial '-' 0.2initial '-' 0.4 (a)(b) FIG. 4: (a)
A plot of change of numbers of ‘+’ and ‘ − ’neighbors of lattice sites for a simulation on a lattice ofsize 1024 × − ’ neighbor.The number of ‘+’ sites with two, three and four ‘ − ’neighbors (second from top to bottom) decrease. (b) φ against target fraction for three different initialpopulations of ‘ − ’ agents. Internal lattice points S u r f a c e l a tt i c e po i n t s Surface vs Internal lattice points
FIG. 5: A log-log plot of surface versus internal pointsin clusters for different sizes of lattice. The size of thelattice increases from 128 ×
128 (bottom) to1024 × IV. DYNAMICS WITH BIAS
Our aim in this section is to use a power-law bias toincrease the population of ‘ − ’ agents when the startingpopulation of ‘ − ’ agents is < . − ’ agents is higher than the starting population.As we have noted in the previous section the conversionof agents from ‘+’ to ‘ − ’ and vice versa is rapid in theearly stages of the simulation both for the completelyconnected network and the lattice. The purpose of intro-ducing the power-law bias in eqn.(1) is to influence thisconversion so that a larger proportion of ‘+’ agents con-vert to ‘ − ’ opinion. In eqn.(1), 0 ≤ p ≤ β ≥
0, andwe have discussed the case β = 0 in the previous section.Hence for a fixed p , p β is a monotonically decreasingfunction of β . The condition θ − i > p β θ + i in eqn.(1) en-sures that this condition will be satisfied for lower valuesof θ − i compared to θ + i , as the factor p β < τ discrete integer values of θ − i ,i.e., θ − i, , θ − i, , . . . , θ − i,τ for a fixed threshold τ . Simi-larly we consider the τ discrete integer values of θ + i ,namely, θ + i, , θ + i, , . . . , θ + i,τ . The effect of the factor p β on θ + i is a mapping θ + i → θ − i to partition the values θ + i, , θ + i, , . . . , θ + i,τ into k partitions P k , ≤ k ≤ τ . Themembers of partition P m are mapped within two consec-utive integer values in θ − i . For example, if we assume τ = 10 , p = 0 .
9, and β = 2 . p β = 0 .
76. There are 10values each for θ − i and θ + i , the integers 1 , , . . . ,
10. Hence p β θ + i can be partitioned into eight partitions that arewithin consecutive integer values of θ − i , [0 , [1] , [1 , [2] ,[2 , [3] , [3 , [4 , , [4 , [6] , [5 , [7] , [6 , [8 , , [7 , [10] . Forexample, [6 , [8 , indicates that p β θ + i is between 6 and7 for θ + i = 8 , . × .
08 and 0 . × . β that result in the same partitions, as thecondition in eqn.(1) will evaluate identically for the samepartition. In this example β = 2 . p β θ + i as β = 2 .
6, and hence the behav-ior of our model will remain the same for either of thesetwo choices for β . Consequently the exponent β has arange instead of a unique value for achieving a final pop-ulation of ‘ − ’ agents and there is no change in the finalpopulation when the β value remains within this range.However the changes in the final population are sharpwhenever the β parameter causes a transition from onepartition to another.When the groups of p β are compressed within lowervalues of θ − i , the result is an increase of the θ − j counter asthe condition in eqn.(1) is satisfied even for lower valuesof θ − i . As a result this favors the ‘ − ’ agents to dominatethe dynamics as more and more agents (both with ini-tial ‘+’ or ‘ − ’ opinions) reach their thresholds for the θ − counters. Hence a high enough threshold makes the sys-tem to converge in an all ‘ − ’ opinion scenario. This con-vergence is faster for the completely connected networkcompared to the lattice, as the erosion of the surface ofthe clusters of ‘+’ nodes is a much slower process for thelattice. A. Dynamics on a completely connected network
We first study the dynamics of the system through anexample for the completely connected network when theinitial fraction of ‘ − ’ (resp. ‘+’) agents is 0 . . − ’ agents is 0 . β = 0, the final fraction of agents with ‘ − ’ opinionreduces further. Also, the final fraction depends on thethreshold, it approaches 0 very fast as the threshold isincreased for the completely connected network. On theother hand a low threshold does not allow enough scopefor the conversion of a large number of ‘+’ agents to ‘ − ’agents that is required for achieving a high fraction of‘ − ’ agents starting from a low fraction. This trade-offfor the threshold exists for the model with a bias as well.In this case the bias gives an impetus for reversing ‘+’agents to ‘ − ’ agents. If the threshold is high, eventuallyall ‘+’ agents will be converted and the final populationwill consist of all ‘ − ’ agents.Fig. 6 shows the comparison between the simulationsof our model with and without bias. This simulation hasbeen done on a completely connected network of 65536agents. The starting population of ‘ − ’ agents is a fraction0 . − ’agents is a fraction of 0 . τ = 5 in this case. The simulation has beendone without bias in (a) , and a lower starting populationof ‘ − ’ agents drives the system to a final population ofall ‘+’ agents. We show in Fig.6(a) the conversion ofagents from ‘+’ to ‘ − ’ and vice versa. There is a smallinitial conversion of ‘+’ agents to ‘ − ’, however, soon thelarger population of ‘+’ agents dominate and all of theinitial fraction of 0 . − ’ agents convert to ‘+’. Thegraph shows a cumulative number of the agents that havehigher value in the θ + or θ − counter until a particularstep. Almost all the conversions occur in the initial stagesof the simulation and the agents reach their thresholdsafterwards over many simulation steps (as indicated bythe red curve in (a)). The biased simulation is shownin (b). The bias in this case drives a conversion of ‘+’agents to ‘ − ’, and as a result the final population of ‘ − ’agents rise to a fraction of 0 . ± .
03 for 100 simulations.The threshold for both the simulations is τ = 5.Increasing this threshold for the simulation with biasquickly pushes the final population to consist only of ‘ − ’agents as a higher threshold gives more scope for ‘ − ’agents to convert ‘+’ agents. For example a simulation Simulation steps in 5000 N u m be r o f ' + ' and ' - ' agen t s t ha t c hanged s t a t e × (a)opinion change for the unbiased model '+' to '-' conversion'-' to '+' conversion Simulation steps in 5000 N u m be r o f ' + ' and ' - ' agen t s t ha t c hanged s t a t e × (b)Opinion change for the biased model '+' to '-' conversion'-' to '+' conversion (a)(b) FIG. 6: The conversion of agents from ‘+’ to ‘ − ’ and‘ − ’ to ‘+’ for the biased and unbiased models for asimulation of 256 ×
256 = 65536 agents on a completelyconnected network. τ = 5 and p = 0 .
9. The startingpopulation of ‘ − ’ agents is 0 . θ − i > θ + i , agent i is identified as a ‘ − ’ agent, and viceversa. (a) β = 0 and almost all ‘ − ’ agents are convertedto ‘+’ agents (the upper curve), conversion of ‘+’ to ‘ − ’is very low (lower curve). (b) β = 6 . − ’(upper curve), compared to ‘ − ’ to ‘+’ conversion (lowercurve).with τ = 10 has a final population of all ‘ − ’ agents.As β = 6 . − ’ agents is p = 0 .
9, the bias factor p β = 0 . . = 0 . τ = 5, there are three par-titions of p β θ + i for agent i . These are [0 , [1 , , [1 , [3 , and [2 , [5] . Most of the transitions from ‘+’ to ‘ − ’ occurin the initial steps of the simulation as shown in Fig. 6.There is a small number of conversions from ‘ − ’ to ‘+’as well, however, both of these conversions plateau real-tively early in the simulation. Another interesting aspectof the dynamics is that a complete coversion of ‘+’ to ‘ − ’agents is dependent on τ , rather than β , as mentionedearlier. For example, β = 18 gives only one partition of p β θ + i , [0 − [1 − . However, simulations in this case showa final population of ‘ − ’ agents as a fraction 0 . ± . − ’agent whenever a ‘+’ agents meets a ‘ − ’ agent in this initial population β target population population achieved0.1 3.1 0.6 0.59-0.630.1 3.8 0.7 0.68-0.730.2 1.4 0.6 0.59-0.650.2 3.0 0.7 0.69-0.730.4 0.6 0.7 0.67-0.730.4 1.8 0.8 0.79-0.83 TABLE I: Simulation results on a completely connectednetwork of 256 ×
256 = 65536 agents. The result in eachrow is collected from 100 simulations. The threshold is τ = 5 in all cases. The results were similar whensimulations were run 10 times each on completelyconnected networks of size 512 ×
512 = 262144 and1024 × B. Dynamics on a lattice
We now discuss the dynamics of the system on a latticewhen β is non-zero in eqn.(1). We take the same repre-sentative case when the initial population of ‘ − ’ agentsis a fraction 0 . − ’ agents is a fraction 0 . ×
256 lattice, we have used β = 10 . − ’ agents between afraction 0 . − .
93 for 50 simulations. We have chosen τ = 10, as a higher threshold has a slower effect in drivingthe system to an all ‘ − ’ population, compared to the sim-ulations on completely connected networks. A represen-tative simulation is shown in Fig. 7. Panel (a) shows theinitial configuration with ‘ − ’ and ‘+’ agents 0 . . − ’ agents with the initial ‘ − ’ agents asseeds. The number of ‘ − ’ agents has grown significanltlyeven after t = 100000 Monte Carlo steps. Panels (c) and(d) show the simulation at time steps t = 200000 and t = 400000 respectively and the growth of the clusters of‘ − ’ agents is clearly visible. Panel (e) shows the simula-tion at t = 600000 and the large clusters of ‘ − ’ agentshave already emerged. These clusters further consolidatein panel (f) at t = 1000000 and remain almost unchangeduntil the end of the simulation at t = 2469567. This iseasy to see from panels (f),(g) and (h).We show in Fig. 8 the conversion of agents from ‘ − ’to ‘+’ and vice versa. The behavior is similar to the sim-ulation on the lattice as the large-scale conversion of ‘+’to ‘ − ’ agents occur quite early in the simulation. How-ever, there is more conversion of ‘ − ’ agents to ‘+’ agentsinitially on the lattice compared to the completely con-nected network. The simulation takes a long time tocomplete since the agents reach their thresholds τ = 10at the later stages of the simulation. The fractions of fi- initial population β target population population achieved0.1 3.0 0.6 0.59-0.640.1 5.2 0.7 0.68-0.730.2 1.8 0.6 0.59-0.640.2 3.0 0.7 0.65-0.770.4 1.2 0.7 0.68-0.730.4 2.8 0.8 0.76-0.83 TABLE II: Simulation results on a lattice of 256 × τ = 10 in all cases. Theresults were similar when simulations were run 10 timeseach on lattices of size 512 ×
512 and 1024 × − ’ agents for lattices of size 512 × × β = 10 . τ = 10 are used for the simulations. The partitionsof θ + values with τ = 10 and β = 10 . , [1 − ,[1 − [4 − ,[2 − [7 − , [3 − [10] . It is evident that thedynamics of the system is dominated by the partitions[0 , [1 − , [1 − [4 − as the conversions of ‘+’ to ‘ − ’agents are rapid in the early stages of the simulationwhen the θ + and θ − counters of all agents have rela-tively lower values. This is similar to the simulations onthe completely connected network.Table II shows some more results from our simulations.We should note that it is possible to use higher valuesof τ for achieving sharper and more stable populationfractions closer to the target population. We illustratethis in Fig. 9 with 0 . − ’agents and 0 . τ fortwo fixed values of β . For β = 4 .
6, the target fractionis reached at a lower value of τ = 9, however averagetarget fraction was 0 .
82 for 20 simulations on a 256 × β = 2 . . τ = 25, and the target fraction was 0 . β and τ pairs that allow usto achieve the target fraction accurately.Fig. 9 has some similarities with rate-distortion curvesstudied in information theory [20]. The aim of rate-distortion theory is to establish a connection between thechannel capacity (rate) and output performance (distor-tion) of a communication channel, through minimizingchannel distortion captured through a cost function. Arate-distortion curve separates the plane into two regions,allowable and non-allowable. The points in the allowableregion indicate the minimum required rate to achievea particular distortion in the output signal. Points inthe non-allowable region indicate distortions that are un-achievable using the corresponding rates. Two extremepoints on a rate-distortion curve are the minimum raterequired for zero distortion and the maximum distortionwhen the rate is zero. This also indicates a trade-off be-tween the channel capacity and distortion, as distortion (a) t=0 (b) t=100000 (c) t=200000 (d) t=400000 (e) t=600000 (f) t=1000000 (g) t=1930000 (h) t=2460000 (a) t=0 (b) t=100000 (c) t=200000 (d) t=400000(e) t=600000 (f) t=1000000 (g) t=1930000 (h) t=2460000 FIG. 7: Evolution of agents on a 256 ×
256 lattice, with initial population of negative agents is a fraction 0 . − ’ agents are shown in black and ‘+’ agents in white. The data was collected over 50 simulations and thefinal population of negative agents was 0 . − .
93 of the total. This evolution is shown at different time steps for arepresentative simulation.
Simulation steps in 10000 N u m be r o f ‘ + ' and ' - ' agen t s t ha t c hanged s t a t e (a) Opinion change for the unbiased model '+' to '-' conversion'-' to '+' conversion Simulation steps in 10000 N u m be r o f ' + ' and ' - ' agen t s t ha t c hanged s t a t e × (b) Opinion change for the biased model '+' to '-' conversion'-' to '+' conversion (a)(b)(b) FIG. 8: The conversion of agents from ‘+’ to ‘ − ’ andvice versa for a simulation on a 256 ×
256 lattice, withinitial fraction of ‘ − ’ agents 0 . p = 0 . τ = 10. (a) β = 0, almost all ‘ − ’ agents are converted to ‘+’(upper curve), compared to ‘+’ to ‘ − ’ conversions(lower curve); (b) β = 10 .
6, the bias forces theconversion of a large number of ‘+’ agents to ‘ − ’ (uppercurve) compared to ‘ − ’ to ‘+’ conversion (lower curve).reduces by increasing channel capacity and increases byreducing channel capacity.We can draw a parallel of Fig. 9 with a rate-distortioncurve if we consider the interactions of agents in eqns. (1) initial population β target population achieved0.1 3 . ± . . ± . . ± . . ± . TABLE III: Simulation results on a completelyconnected network of 256 ×
256 agents. The result ineach row is collected from 100 simulations. Thethreshold is τ = 5 in all cases. The results were similarwhen simulations were run 10 times each on completelyconnected networks of 512 ×
512 and 1024 × φ as the output of the channel. Increasing τ increases thenumber of interactions between agents and can be seenas an increase in channel capacity. The distortion is thedifference between the fraction φ achieved with a specificvalue of τ and the target fraction of ‘ − ’ agents. We canstudy a trade-off between τ and φ for a fixed β . Forexample, for β = 2 . τ and draw a vertical line, all thefractions φ of final population of ‘ − ’ agents below the redline are achievable with β = 2 .
3. And no fraction φ abovethe red line is achievable with β = 2 .
3. In other words,if we fix β , the red line divides the plane into allowable(below) and non-allowable (above) regions. A trade-offbetween τ and φ is also noticeable, as the non-allowableregion is larger with smaller values of τ and vice versa.Similar trade-off due to rate-distortion curves has beenobserved in diverse domains like human perception [21],capital asset pricing model for stocks [22] and balancebetween growth and entropy in bacterial cultures [23].0 τ t a r ge t f r a c t i on o f ' - ' initial '-' is 0.3 β =4.6 β =2.3 FIG. 9:
Slow and accurate convergence to the targetfraction 0 . β and τ .initial population β target population achieved0.1 3 . ± . . ± . . ± . . ± . TABLE IV: Simulation results on a lattice of 256 × τ = 10 in all cases. Theresults were similar when simulations were run 10 timeseach on lattices of size 512 ×
512 and 1024 × C. Dynamics with a faulty β We have also experimented with the dynamics of thesystem when the bias β is not constant for all the agents,rather β varies within a range of values that we denoteby β ± R . We choose a value for β from the range β − R to β + R uniformly at random at each Monte Carlostep for each agent. In other words, each agent uses adifferent β within this range at each Monte Carlo step.Though the values of β are different for achieving thedesired fractions of final population of ‘ − ’ agents, thesystem is stable for a range of β that is ± . β for the cases when the central value of β > .
0. The deterioration in achieving the final desiredfraction of ‘ − ’ agents starts beyond the ± . V. DISCUSSION
We have presented a model of opinion dynamics basedon the negativity bias extensively studied in the psychol-ogy literature. Our main aim was to investigate the effectof negativity bias in binary opinion formation. One of theinteresting aspects of our model is the formation of sta- ble target population of ‘ − ’ agents. Our model is close toreal-world exchange of opinions based on negativity bias.People with different opinions usually discuss pros andcons of both alternatives and gives more importance tonegative opinions. We have abstracted this real-world sit-uation in terms of the two counters for individual agents.We have shown that application of a power-law bias dur-ing opinion exchange results in consistent target popula-tions and the bias factors are scale-invariant. Moreover,we have also shown that this consistency is maintainedwith bias factors that can vary randomly and uniformlywithin a range. Another interesting aspect of our modelis its rapid convergence, the composition of the final pop-ulation is reached quite early in the simulation, wheneach agent has interacted with only a few other agents.This is again close to the real-world situation in the sensethat usually people even within a large population inter-act with a few other people while making decisions.There are some similarities between the dynamics ofour model and the dynamics of the random-field Isingmodel. For example, the conversions of ‘+’ to ‘ − ’ opin-ion and vice versa are similar to the first order phasetransition in the random-field Ising model. Similarly, for-mation of clusters of ‘+’ and ‘ − ’ opinions and strong sur-face tension on cluster boundaries are very similar to thedomains of similar spin in the random-field Ising model.We will explore these similarities further in future work. ACKNOWLEDGMENTS
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