Uppsal a univ ersitets l ogotyp UPTEC F 21048 Examensarbete 30 hp Juni 2021 Compartmental Models in Social Dynamics Alice Graf Brolund Civilingenj örspr ogrammet i tek nisk fysik Civilingenjörsprogrammet i teknisk fysik Uppsal a univ ersitets l ogotyp Compartmental Models in Social Dynamics Alice Graf Brolund Abstract The dynamics of many aspects of social behaviour, such as spread of fads and fashion, collective action, group decision-making, homophily and disagreement, have been captured by mathematical models. The power of these models is that they can provide novel insight into the emergent dynamics of groups, e.g. 'epidemics' of memes, tipping points for collective action, wisdom of crowds and leadership by small numbers of individuals, segregation and polarisation. A current weakness in the scientific models is their sheer number. 'New' models are continually 'discovered' by physicists, engineers and mathematicians. The models are analysed mathematically, but very seldom provide predictions that can be tested empirically. In this work, we provide a framework of simple models, based on Lotka's original idea of using chemical reactions to describe social interactions. We show how to formulate models for social epidemics, social recovery, cycles, collective action, group decision-making, segregation and polarisation, which we argue encompass the majority of social dynamics models. We present an open-access tool, written in Python, for specifying social interactions, studying them in terms of mass action, and creating spatial simulations of model dynamics. We argue that the models in this article provide a baseline of empirically testable predictions arising from social dynamics, and that before creating new and more complicated versions of the same idea, researchers should explain how their model differs substantially from our baseline models. Tek nisk-naturvetenskapliga fakulteten, Upps ala universitet. Utgivningsort U pps al a/Visby . H andledare: Li nnéa Gyllingberg, Äm nesgranskar e: D avi d Sum pter, Exami nator : T omas Nyberg Teknisk-naturvetenskapliga fakulteten Uppsala universitet, Utgivningsort Uppsala/Visby Handledare: Linnéa Gyllingberg Ämnesgranskare: David Sumpter Examinator: Tomas Nyberg Populärvetenskaplig sammanfattning Matematiska modeller kan hjälpa oss att förstå många typer av sociala fenomen, som ryktesspridning, spridning av memes, gruppbeslut, segregation och radikalisering. Det finns idag otaliga modeller för sociala beteenden hos människor och djur, och fler presenteras kontinuerligt. Det stora antalet modeller försvårar navigering inom forskningsfältet, och många av modellerna är dessutom komplicerade och svåra att verifiera genom experiment. I detta arbete föreslås ett ramverk av grundläggande modeller, som var och en modellerar en aspekt av socialt beteende; det gäller sociala epidemier, cykler, gemensamt handlande, gruppbeslut, segregation och polarisering. Vi menar att dessa modeller utgör majoriteten av de verifierbara aspekter av socialt beteende som studeras, och att de bör behandlas som en utgångspunkt när en ny modell ska introduceras. Vilka av mekanismerna från utgångspunkten finns representerade i modellen? Skiljer den sig ens nämnvärt från utgångspunkten? Genom att ha en god förståelse för grundmodellerna, och genom att förklara på vilket sätt en ny modell skiljer sig från dem, kan forskare undvika att presentera modeller som i praktiken är mer komplicerade varianter av sådana som redan finns. I detta arbete visar vi hur dessa grundläggande modeller kan formuleras och studeras. Modellerna bygger på enkla regler om vad som händer när individer i en befolkning möter varandra. Till exempel, om en person som har vetskap om ett rykte träffar någon som inte har det, kan ryktet spridas vidare. Därför har antaganden om vilka personer som kan träffa varandra stor påverkan på de resultat som modellerna ger. I detta arbete studeras varje modell med två olika metoder: i den ena har alla personer i befolkningen samma sannolikhet att träffa varandra, i den andra representeras befolkningen av ett rutnät, där varje plats motsvarar en individ. I den senare har alltså varje person ett begränsat antal grannar att interagera med. Vilken av dessa två metoder man väljer har stor betydelse för vilka beteenden modellerna förutspår. Som ett komplement till detta arbete presenteras ett verktyg i form av ett Python-program som utför analysen av modellerna. Detta kan användas för att undersöka grundmodellerna som presenteras i detta arbete, men också för att formulera och analysera nya modeller på samma sätt. På det viset kan nya modeller enkelt jämföras mot grundmodellerna. Verktyget är användbart både som introduktion för de som är nya inom social dynamik, men också för de forskare som som vill ta fram nya modeller och föra forskningsfältet vidare. Contents 1 Introduction 4 1.1 Social Behaviour as States . 4 1.2 Well-Mixed Interpretation . 5 1.3 Spatial Interpretation . 7 1.4 Python Tool For Running The Models . 9 1.5 Previous Works . 9 1.6 Investigating Models of Social Dynamics . 10 2 Social Epidemic 11 2.1 Model Description . 11 2.2 Dynamical System . 11 2.3 Finding Fixed Points and Nullclines . 11 2.4 Initial Conditions in the Dynamical System . 14 2.5 Spatial Model . 14 2.6 Initial Conditions in the Spatial Model . 16 2.7 Conclusions . 17 3 Temporary Immunity 18 3.1 Model Description . 18 3.2 Dynamical System . 18 3.3 Finding Fixed Points and Nullclines . 18 3.4 Initial Conditions in the Dynamical System . 22 3.5 Spatial Model . 22 3.6 Initial Condition in the Spatial Model . 25 3.7 Conclusions . 26 4 Collective Action 26 4.1 Model Description . 26 4.2 Dynamical System . 27 4.3 Finding Fixed Points and Nullclines . 27 4.4 Initial Conditions in the Dynamical System . 30 4.5 Spatial Model . 31 4.6 Initial Condition in the Spatial Model . 33 4.7 Conclusions . 33 5 Social Recovery 33 5.1 Model Description . 33 5.2 Dynamical System . 34 5.3 Finding Fixed Points and Nullclines . 34 5.4 Initial Conditions in the Dynamical System . 37 5.5 Spatial Model . 37 5.6 Initial Conditions in the Spatial Model . 38 5.7 Conclusions . 39 1 6 Cycles 40 6.1 Model Description . 40 6.2 Dynamical System . 40 6.3 Finding Fixed Points and Nullclines . 41 6.4 Initial Conditions in the Dynamical System . 45 6.5 Spatial Model . 46 6.6 Initial Conditions in the Spatial Model . 49 6.7 Conclusions . 52 7 Cults 52 7.1 Model Description . 52 7.2 Dynamical System . 53 7.3 Finding Fixed Points and Nullclines . 53 7.4 Initial Condition in the Dynamical System . 57 7.5 Spatial Model . 57 7.6 Initial Condition in the Spatial Model . 59 7.7 Conclusions . 59 8 Decision Making - Version One 60 8.1 Model Description . 60 8.2 Dynamical System . 60 8.3 Finding fixed points and nullclines . 60 8.4 Initial Condition in the Dynamical System . 63 8.5 Spatial Model . 63 8.6 Initial Condition in the Spatial Model . 64 8.7 Conclusions . 66 9 Decision Making - Version Two 66 9.1 Model description . 66 9.2 Dynamical System . 67 9.3 Finding Fixed Points and Nullclines . 67 9.4 Initial Condition in the Dynamical System . 70 9.5 Spatial Model . 71 9.6 Initial Condition in the Spatial Model . 74 9.7 Conclusions . 74 10 Segregation - Version One 74 10.1 Model Description . 74 10.2 Dynamical System . 75 10.3 Finding Fixed Points and Nullclines . 75 10.4 Initial Condition in the Dynamical System . 78 10.5 Spatial Model . 78 10.6 Conclusions . 79 2 11 Segregation - Version Two 80 11.1 Model Description . 80 11.2 Dynamical System . 80 11.3 Finding Fixed Points and Nullclines . 80 11.4 Initial Condition in the Dynamical System . 83 11.5 Spatial Model . 83 11.6 Initial Condition in the Spatial Model . 84 11.7 Conclusions . 85 12 Polarisation 86 12.1 Model Description . 86 12.2 Dynamical System . 86 12.3 Finding Fixed Points and Nullclines . 87 12.4 Initial Condition in the Dynamical System . 90 12.5 Spatial Model . 91 12.6 Initial Condition in the Spatial Model . 93 12.7 Conclusions . 94 13 Python Tool for Model Investigation 94 14 Conclusions 95 References 96 3 1 Introduction 1.1 Social Behaviour as States We present a series of models describing how certain social, collective, mech- anisms emerge and play out in a population. The base of the models are the interactions that take place between individual members of the population. To describe the set of interactions that may take place, we make use of compart- mental models. The idea behind compartmental models is that a populations can be divided into mutually exclusive compartments or groups, and we define interactions between members of the different groups. Individuals can transition into other groups, triggered by interactions with members of other groups or simply by chance. Imagine, as a concrete example, that a population is split into three groups, defined by who has and who has not adopted a certain fashion. Those who have not taken to it yet, the Susceptible people (group one), may adopt it from someone who has, who we will call an Infected person (group two). This will result in the susceptible person leaving the susceptible compartment, and instead joining the infected one. We denote this exchange β S + I −! 2I; where S denotes a susceptible person, I denotes an infected, and β is the rate (person per unit time) at which infected people meet susceptible people. Those who have adopted the fashion, may then either spontaneously lose interest in it, which naturally does not involve another person, or realise that the fashion is on decline and therefore stop adhering to it. The latter is the result of a meeting with someone who has already abandoned the fashion, i.e. a Recovered person (group three). Then, we have two options for transitioning into the recovered compartment, the spontaneous transition being γ I −! R; where R denotes a Recovered person, and the transition dependent on interac- tion being I + R −!α 2R: We have described the transitions of our model using chemical reaction equa- tions.