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Comparison of Mathematical Models of Opinion Dynamics AURORA BASINSKI-FERRIS Integrated Science Program, Class of 2018, Mcmaster University

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Comparison of Mathematical Models of Opinion Dynamics AURORA BASINSKI-FERRIS Integrated Science Program, Class of 2018, Mcmaster University MATHEMATICAL PAPER Comparison of Mathematical Models of Opinion Dynamics AURORA BASINSKI-FERRIS Integrated Science Program, Class of 2018, McMaster University SUMMARYSUMMARY SociophysicsSociophysics isis anan emergingemerging interdisciplinary interdisciplinary researchresearch fieldfield thatthat usesuses theoriestheories andand methodsmethods developeddeveloped byby physicistsphysicists asas aa modellingmodelling frameworkframework forfor variousvarious socialsocial sciencescience phenomena.phenomena. ThisThis paperpaper presentspresents thethe SznajdSznajd model,model, whichwhich isis thethe basisbasis ofof oneone ofof thethe simplestsimplest yetyet mostmost efficientefficient methodsmethods toto predictpredict thethe collectivecollective humanhuman behaviourbehaviour inin aa binarybinary opinionopinion society.society. ThisThis modelmodel isis aa variationvariation ofof thethe IsingIsing modelmodel –– aa mathematicalmathematical modelmodel usedused inin statisticalstatistical mechanicsmechanics whenwhen studyingstudying thethe ferromagneticferromagnetic phasephase transitionstransitions forfor thethe magnetizatimagnetizationon ofof aa systemsystem basedbased onon thethe interactioninteraction betweenbetween thethe neighbouringneighbouring atomicatomic spinsspins ofof thethe system.system. InIn both the Ising model and the Sznajd model, individuals are only allowed to have a binary both the Ising model and the Sznajd model, individuals are only allowed to have a binary Mathematics opinionopinion –– yesyes (spin(spin up)up) oror nono (spin(spin down).down). TheThe SznajdSznajd modelmodel isis basedbased oonn thethe ideaidea ofof socialsocial validation;validation; thethe modelmodel requiresrequires thatthat twotwo neighboursneighbours shareshare thethe samesame opinionopinion toto bebe ableable toto convinceconvince individuals individuals aroundaround themthem of of theirtheir opinion.opinion. UsingUsing MATLAB,MATLAB, this this paper paper applies applies a a two two-- dimensionaldimensional simplifiedsimplified versionversion ofof thethe SznajdSznajd modelmodel toto ttrackrack thethe opinionopinion movementmovement overover manymany timetime stepssteps inin aa latticelattice ofof individualsindividuals withwith anan initialinitial randomlyrandomly generatedgenerated distributiondistribution ofof opinions.opinions. TheThe resultsresults ofof thisthis simplifiedsimplified modelmodel areare thenthen analyzedanalyzed usingusing psychologicalpsychological principlesprinciples toto examineexamine thethe validityvalidity ofof thethe outoutcomes.comes. Received:Received: 11/1311/13/2016/2016 AcceptedAccepted: 03/07:/2017 03/07 /2017 Published: Published: 03/03/2017 07/3 1/2017 URL:URL: https://journals.mcmaster.ca/iScientist/article/view/1357 https://journals.mcmaster.ca/iScientist/article/view/1152/1005 Keywords:Keywords: sSociophysics,ociophysics, SznajdSznajd Model,Model, oOpinionpinion dDynamics,ynamics, mMathematicalathematical mModelling,odelling, MATLABMATLAB INTRODUCTION Mathematical models of opinion formation examine society as a physical system in which The idea of applying concepts from the natural individuals and their interactions are viewed as the sciences to try to explain social science phenomena microscopic scale of the system governed by rigid is at least 25 centuries old. One of the first rule sets, and the emergent macroscopic trends are recorded comparisons between the properties of used to reflect overall societal opinion. In general, natural sciences and the behaviour of humans these models deal with binary opinion systems in occurred when the Greek philosopher Empedocles which there is a general assumption that the stated an idea to the effect of “Humans are like opinion of an individual is in some way related to liquids: some mix easily like wine and water, and the opinions of their neighbours (Oz, 2008). others like oil and water refuse to mix.” (Stauffer, 2009). Since then, there have been a few Models of opinion dynamics, and sociophysics in ISCIENTIST individuals who drew comparisons between general, rely on the law of large numbers. This law ordered physical systems and the behaviour of is a statistical theorem that states that as the populations. For example, in 1942, Majorana number of randomly generated variables increases, compared quantum physics to social probabilities the average approaches the theoretical mean (Routledge, 2015). A simple system used to in human behaviour. However, with the exception | 2017 | 2017 of a select few individuals, the field of sociophysics understand this is coin flipping. When just one took off in the early 2000s with the popularization coin is flipped, although the probability of either outcome is 0.5, you cannot predict with any of ideas such as opinion dynamics within academia (Stauffer, 2013). certainty whether that single coin toss will show a head or tails. However, after 500 coin tosses, it is 7 reasonable to assume that the long-term behaviour antiparallel pair contributes +J to the energy is approaching the 0.5 probability. Thus, this law (Stauffer, 2013). applied to opinion dynamics means that with a Therefore, the total energy of the system is given large population, individual fluctuations are by: averaged out, and general trends in data are more visible. In turn, this also implies that the � = −� �&�' justification for the human aspect of these models (&,'* stems from mass psychology rather than individual psychology. To minimize energy, each spin is pushed to be aligned with its neighbours (Castellano, Fortunato This paper offers a brief history of the models and Loreto, 2009). Thus, the Ising model typically developed to deal with the opinions of results in a higher probability of two neighbouring populations. However, the focus will lean more electrons having the same spin (Stauffer, 2009). towards the examination of two variations of a Metropolis dynamics of the Ising model takes each fairly recent model of opinion dynamics called the elementary move to be a single spin change. This Sznajd model. These two model variations were change in spin is accepted with a probability of: simplified and represented in MATLAB to analyze -∆/ and compare the behaviour and validity of these � = � 012 models given an initial random population with varying size. where T is temperature, kB is the Boltzmann constant, and ∆E is the change in energy when this MATHEMATICAL BACKGROUND change takes place (Castellano, Fortunato, and Many models have been proposed to try to Loreto, 2009). As this probability of acceptance is examine the opinion dynamics of populations. In dependent on temperature, there is a critical general, these follow the theme of an initially temperature that denotes whether or not disordered system that is brought to a more macroscopic ordering will occur. Above the critical ordered state under certain conditions (Castellano, temperature, there is macroscopic disorder even Fortunato, and Loreto, 2009). The key part of over many time steps. However, despite the opinion dynamic models is the overarching rule disorder, there are still local clusters of just spin up that guides how interactions between various (+1) or just spin down (-1). Below the critical individuals in the population occur. In most temperature, macroscopic ordering occurs, models, the opinion of a single individual is because these local clusters spread to become updated after each time step; this change is affected global (although limited by the bounds of the Mathematics by the opinions of the individual’s neighbours. lattice). Thus, the stable states of either all positive Additionally, all models must define the space in or all negative spins would emerge (Castellano, which their population exists. The models Fortunato, and Loreto, 2009). presented in this paper treat populations as discrete The dynamics that emerge from the Ising model points on a lattice rather than a continuous plane due to physical interactions can be applied to a (Sen and Chakrabarti, 2013). This setup results in human system to develop a model of opinion an assumption that individuals in the model are dynamics (Stauffer, 2009). In this case, neighbours stationary. on a lattice influence each other such that adjacent individuals ‘want’ to have similar opinions because ISING MODEL AND VARIATIONS energy functions as ‘unhappiness’ due to In physics, the 1924 Ising model is a mathematical disagreement. Due to the application of physical | 2017 | 2017 model used for ferromagnetism (Sznajd-Weron, phenomena to social dynamics, temperature has a 2005). The Ising model looks at each site on a social meaning. The social application of square lattice as an electron with a value assigned temperature denotes both the overall of either +1 (spin up) or -1 (spin down). Thus, approximation of events that influence decision each pair of neighbours at <i,j> has an energy of - making of individuals not included in the JSiSj where J is some proportionality constant and microscopic rules and denotes the tolerance of the ISCIENTIST S denotes spin. This means that each parallel pair individuals in the population. As expected, of spins contributes -J to the total energy, and each changing these factors can change the behaviour of 8 the system, just as going above or below critical individuals of Person A take the opinion of Person temperature will change the ferromagnetic system. B and vice versa. This was meant to emulate the When temperature can grow arbitrarily large, act of an argument whereby neighbours are neighbours
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