By few simple extensions it is possible to model radical groups, charismatic leaders and processes of radicalization in the bounded confidence framework. In the resulting model we get a lot of surprising (non-)monotonicities. In certain regions of the parameter space more radicals or more 'charismaticity' may lead to less radicalisation

Hegselmann, Rainer

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Bounded Confidence, Radical Groups, andCharismatic LeadersRainer HegselmannBayreuth UniversityGermanyEmail: rainer.hegselmann@uni-bayreuth.deAbstract—By few simple extensions it is possible to model rad-ical groups, charismatic leaders and processes of radicalizationin the bounded confidence framework. In the resulting model weget a lot of surprising (non-)monotonicities. In certain regions ofthe parameter space more radicals or more ‘charismaticity’ maylead to less radicalisation.I. INTRODUCTIONThere are some stylized facts about charismatic leaders,radical groups and processes of radicalisation. Among them wepresumably find the following three: First, a radical group hascompared to ‘normal’ agents a comparatively stable in-groupconsensus on an extreme opinion. Second, a charismatic leadercounts for ‘normal’ agents that are under his/her influencemuch more than other ‘normal’ agents. Third, in a process ofradicalisation people tend to get less and less open-minded.The three facts inspire an applied and modified version ofthe well known bounded confidence model as introduced byHegselmann and Krause in [1]. The modified version is stillan extremely simple conceptual model. Under some assump-tions the whole parameter space can be analysed. The modelshows some surprising results and mechanism that inspire newpossible explanations, new perspectives for empirical studies,and new ideas for prevention policies.II. BASICS OF THE BOUNDED CONFIDENCE MODEL(BC-MODEL)The basic assumptions of the BC-model are:• There is a set of n individuals; i, j ∈ I .• Time is discrete; t = 0, 1, 2, . . . .• Each individual starts with a certain opinion, given by areal number; xi(t0) ∈ [0, 1].• The profile of all opinions at time t is X(t) =x1(t), . . . , xi(t), xj(t), . . . , xn(t).• Each individual i takes into account only ‘reasonable’others. Reasonable are those individuals j whose opinionsare not too far away, i.e. for which |xi(t) − xj(t)| ≤ ,where  is the confidence level that determines the sizeof the confidence interval.• The set of all others that i takes into account at time tis:I(i,X(t))= {j∣∣|xi(t)− xj(t)| ≤ } (1)• The individuals update their opinions. The next period’sopinion of individual i is the average opinion of all thosewhich i takes seriously:xi(t+ 1) =1#(I(i,Xt))) ∑j∈I(i,X(t))xj(t) (2)III. MODIFICATIONS OF THE BC-MODELFor the modified BC-model we now assume that thereare two groups of agents: The first group, the normals, haveopinions from the interval [0, 1], they all have a positivenormals > 0, and they update according to equation (2). Thesecond group, the radicals, have all the opinion R, with Ragain from the unit interval, but more or less close to the upperbound, e.g. R = 0.9. The radicals’ confidence level radicalsis constantly and homogeneously 0. Consequently, they updateaccording toxradicalsi (t+ 1) = xradicalsi (t) = R (3)Figure 1 shows single runs with the same uniform startdistribution for 50 normal agents with an normals = 0.2.In the left figure there are no radicals. Light grey verticallines between neighboring opinions indicate that their distanceis not greater than normals. In the figure in the centre agroup of 5 radicals is added. Their opinion is R = 0.9. Theblack horizontal line is their trajectory. Dark grey vertical linesindicate the chain of direct or indirect (i.e. via a chain ofothers) influence of radicals on normals. In period 4 that chainbreaks. The dark grey area indicates that part of the opinionspace in which all normals, given the size of their confidenceinterval, are under the direct influence of the radicals. (Theright figure will be explained below.)Charismatic leaders can be covered by reinterpretation: Weconsider a radical group with m members as one person thatcounts m-times for all normal agents that have the charismaticleader within their confidence interval. Thereby, the radicals’group size m turns into a kind of degree of charismaticity.Assuming that R is the same in all cases, the conceptualmodel covers it all: a radical group, a charismatic leader, orany combination of both.To include that in a process of radicalisation normals getless and less open-minded, requires a simple, but substantialmodification of the original BC-model: We apply the BCMiguel, Amblard, Barceló & Madella (eds.) Advances in Computational Social Science and Social SimulationBarcelona: Autònoma University of Barcelona, 2014, DDD repository <http://ddd.uab.cat/record/125597>Fig. 1. 50 normals, the same start distribution of normals in all pictures, = 0.2, R = 0.9 . Left: no radicals. Centre: 5 radicals, no confidencedynamics. Right: 5 radicals, with confidence dynamics.mechanism, i.e. averaging over elements within one’s confi-dence interval, on both, opinions and the confidence levels.That modification has no effect, if  is the same for all agents.However, in our context radicals have an radicals = 0, whilenormals start with an normals > 0. As a consequence, theconfidence level of normals will shrink if they are directly orindirectly influenced by radicals. Figure 1 right shows such adynamics for the same start distribution as in figure 1 centre.Note that under this confidence dynamics (as we will call it inthe following) one normal less ends up at the radical position.A bit more formally: A confidence dynamics makes theconfidence level normals an individualised and time depen-dent normalsi (t) and the set of agents j that i takes seriouslychanges from (1) toI(i,X(t), i(t))= {j∣∣|xi(t)− xj(t)| ≤ i(t)}. (4)Under a confidence dynamics normals, then, update accordingto two equations:xnormalsi (t+1) =1#(I(i,X(t), i(t))) ∑j∈I(i,X(t),i(t))xj(t),(5)and, additionally, with regard to the confidence level bynormalsi (t+ 1) =1#(I(i,X(t), i(t))) ∑j∈I(i,X(t),i(t)) j(t).(6)The radicals stick to their confidence level 0 and ‘update’accordingly:radicalsi (t+ 1) = radicalsi (t) = 0. (7)(2) together with (3) defines a system without a confidencedynamics. (3) together with (5) to (7) is the correspondingsystems with a confidence dynamics of normals.11Note that in the model the radicals or a charismatic leader, respectively,are simply given. Baurmann, Betz and Cramm present in [2] a model in whichcharismatic opinion leaders can evolve.0 10 20 30 40 5010203040500 10 20 30 40 50102030405001020304050-64-2046Fig. 2. x-axis: the confidence level increases in 50 steps of size 0.01 from0.01 to 0.5. y-axis: the number of radicals increases from 1 to 50. Coloursindicate the number of normals that end up at the radical position which ishere assumed to be R = 0.9 . The total number of normals is always 50.Left: without a confidence dynamics. Right: with a confidence dynamics.IV. FIRST RESULTSThere are many interesting questions about final results, e.g.the number or frequency of normals that finally end up atthe radical position R, the mean and median of the normals’opinions, or – as one of several possible distance measures –something like a root mean square deviation of the normals’opinions with regard to R. It is very natural to think thatfinal results crucially depend upon the number of radicals(or the degree of charismaticity, respectively) compared tothe number of normals, the confidence level at t = 0, andthe radical Position R. All the questions mentioned above,can be answered by systematic simulations that cover thewhole parameter space—admittedly, under some simplifyingassumptions.A first step in this direction is documented in figure 2.The figures show, indicated by color, the number of normalsthat finally end up at the radical position R, which in thisexample is assumed to be R = 0.9 . To the left are the resultswithout, to the right the results with a confidence dynamics.The x-axis gives the value of the normals’ confidence levelat t = 0. It is the same for all normals. In 50 steps of size0.01 the confidence level normals increases from 0.01 to 0.5.The y-axis gives the number of radicals (or the degree ofcharismaticity, respectively). In 50 steps that number increasesfrom 1 to 50. As to the normals we always assume 50agents with opinions all over the opinion space. For each stepcombination x, y we run simulations until the dynamics is(almost) stable. Then we count the number of normals that(almost) ended up at the radical position R. Figure 2 shows,indicated by colors, the number of normals that finally end upat the radical position.There are at least three lessons that we can take away fromfigure 2:1) With or without a confidence dynamics, for a certainrange of confidence levels an increasing number ofradicals leads to less radicalisation in the sense that lessnormals end up at the radical position R. Thus, for acertain range of confidence levels radicalisation is mildlythough clearly monotonically decreasing with respect tothe number of radicals.Miguel, Amblard, Barceló & Madella (eds.) Advances in Computational Social Science and Social SimulationBarcelona: Autònoma University of Barcelona, 2014, DDD repository <http://ddd.uab.cat/record/125597>2) In another range of confidence levels, again with orwithout a confidence dynamics, it holds: With only afew radicals no normal ends up with the radical positionR. Above a certain number of radicals suddenly allnormals end up with the radical position. But if wefurther increase the number of radicals, then, suddenly,we get again less radicalisation. Thus, in that range ofconfidence levels radicalisation is not monotonic withrespect to the number of radicals.3) For any number of radicals, with a confidence dynamicsthe sudden transition to a state in which all normals endup radical occurs, compared with the dynamics withouta confidence dynamics, only for much bigger confidencelevels. In general: The confidence dynamics lets theconfidence level of at least some normals shrink. Butthat leads to comparatively less radicalisation.Obviously, in our model works a complicated and some-times counteracting interplay of increasing number of radicalsand/or an increasing size of the confidence level. As a con-sequence we get some counterintuitive (non-)monotonicities.All the effects can be explained by an analysis of single runs.V. NEXT STEPSThe simulations documented in figure 2 show, indicated bycolor, the number of normals that finally end up at the radicalopinion, which was assumed to be R = 0.9. To get a completeoverview we will run simulations for R = 1.0, 0.99, . . . , 0.5.That will be done for both, without and with a confidencedynamics. The results will be visualised by two animationsof 51 pictures of the type used in figure 2. There is muchmore that can be analysed and visualised in the same style:for instance, the mean or the median of the normals’ opinions,the final cluster structure and their distances to the radicalposition.However, there are two major caveats with regard to thepreliminary results: First, it has to be checked, whether or notthe results crucially depend upon the ratio or the absolutenumbers of normals and radicals. And, second, a kind ofconfession: The results in figure 2 are not based on repeatedruns with random start distributions for each x, y-combination.They are based on just one run for each combination. In allruns the same very special, but in a certain sense ‘typical’start distribution of n normals is used: An opinion profileis an ordered profile iff for all i ≤ (n − 1) it holds thatxi(0) ≤ xi+1(0). In all runs of figure 2 we use an orderedstart profile in which the ith normal opinion is i/(n + 1). Insuch an ordered and equidistant start profile the ith opinionis exactly there where it will be at the average over infinitelyrepeated uniform random distributions. On the one side, thatdistribution is therefore very ‘typical’. On the other side, theregular structure of the profile may make us blind for importanteffects that are caused by the typical density fluctuations ofsingle random distributions. Whether or not the use of our startdistribution is a problem or a solution of many problems hasstill to be checked.ACKNOWLEDGMENTThe model was developed 2013 during my time as a seniorfellow at the Alfried Krupp Wissenschaftskolleg in Greifswald.The model is the direct result of intensive discussions withGregor Betz, Michael Baurmann and Rainer Cramm, whichat that time were working there on their own model ofradicalization processes (cf. [2]). I profited a lot from ourdiscussions and the stimulating intellectual atmosphere in theAlfried Krupp Wissenschaftskolleg in Greifswald. In 2014 I’mvisiting international fellow of the department of sociologyat Surrey University. Corinna Elsenbroich, Jen Badham, andPeter Johnson (all of them members of CRESS, the Centrefor Research in Social Simulation, based in the sociologydepartment) helped me a lot to analyze the model, to fix bugs,and to develop ideas for future modifications of the model.Many thanks to all of them.REFERENCES[1] R. Hegselmann and U. Krause, “Truth and cognitive division of labour:First steps towards a computer aided social epistemology,” Journalof Artificial Societies and Social Simulation, vol. 9, 2006. [Online].Available: http://jasss.soc.surrey.ac.uk/9/3/10.html[2] M. Baurmann, G. Betz, and R. Cramm, “Meinungsdynamiken in funda-mentalistischen gruppen. erkla¨rungshypothesen auf der basis von simula-tionsmodellen,” Analyse & Kritik, no. 1, pp. 61 – 102, 2014.Miguel, Amblard, Barceló & Madella (eds.) Advances in Computational Social Science and Social SimulationBarcelona: Autònoma University of Barcelona, 2014, DDD repository <http://ddd.uab.cat/record/125597>

Bounded confidence, radical groups, and charismatic leaders

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Bounded confidence, radical groups, and charismatic leaders

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