Opinion Polarization by Learning from Social Feedback

Sven Banisch & Eckehard Olbrich MPI for Mathematics in the Sciences (Leipzig)

Opinion Dynamics and Collective Decision, Bremen 2017 H2020 – FETPROACT-2016 - 732942

Opinion Dynamics and Cultural Confict odycceus.eu in European Spaces

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A controversy, most on which final decisions are based. Importantly, often, involves a series the weights assigned to of issues that are logicOver the course of a the different issues as across the participants ally and cognitivelydiscourse not all the Over the course of a issues playing a role are discourse not all the well as the understanding and this has important

linked. onarguments a

about their mutual onarguments a

visible from the very issues playing a role are consequences for the onarguments a A proper spatial relatedness differ beginning. New issues visible from the very process of finding For a debate to be fruitful the emergence of representation of such multi-dimensionality multi-dimensionalitymay arise, first-time beginning. New issues compromise. a common understanding complexity should and interconnectedness multi-dimensionality may arise, first-time A controversy, most In a prolonged debate of the structure of the points at stake. and interconnectedness andissues interconnectedness may cease to be multi-dimensionality underlying a controversy of the points at stake. therefore take into and interconnectedness central, and sometimes issues may cease to be often, involves a series the importance of and We propose to model of the points at stake. multi-dimensionality is necessary. and interconnectedness of the points at stake. this by a network the We propose to model of the points at stake. account the this by a network the We propose to model very few controverse central, and sometimes relations between the nodes of which are key We propose to model of issues that are logic nodes of which are key this by a network the issues in a debate and this by a network the issues in a debate and nodes of which are key nodes of which are key points turn out and very few controverse dimensions of it are issues in a debate and issues in a debate and the links weights for the links weights for ally and cognitively the links weights for the links weights for We propose to model relations in between relations in between relations in between become the key issues points turn out and subject to cognitive relations in between those. linked. those. those. those. this by a network the become the key issues negotiations as well. nodes of which are key A proper spatial issues in a debate and representation of such the links weights for complexity should relations in between therefore take into those. account the Polarization: an important but also controversial issue Te Puzzle of Polarization

» If people tend to become alike in their beliefs, attitudes, and behavior when they interact, why do not all such diferences eventually disappear? « (Axelrod, 1997) » Since universal ultimate agreement is an ubiquitous outcome of a very broad class of mathematical models, we are naturally led to inquire what on earth one must assume in order to gene- rate the bimodal outcome of community cleavage studies. « (Abelson, 1964) ▶ Tese questions have inspired a lot of modelling work throug- hout decades: the feld is now known as Opinion Dynamics Terminology tćFUFSNx1PMBSJ[BUJPOjJTVTFEEJČFSFOUMZJOEJČFSFOUĕFMET t*O(SPVQ1TZDIPMPHZJUNBJOMZSFGFSTUPBQSPDFTTCZXIJDIEFMJ- CFSBUJOHHSPVQTCFDPNFNPSFFYUSFNF ŕ5BLFOVQJO1PMJUJDBM4DJFODF 4VOTUFJOATxMBXPGHSPVQQPMBSJ[BUJPOj) t*O4PDJPMPHZmainlyBQSPDFTTCZXIJDITUSPOHEJWFSHFODFTPGQPMJ- UJDBMPQJOJPOTDPNFBCPVUBOEBNFBTVSFPGUIFFYUFOEUPXIJDIB DFSUBJOEJTUSJCVUJPOPGPQJOJPOTJTQPMBSJ[FE ŕ%J.BHHJPFUBMEJČFSFOUJBUFBTQFDUT t*O0QJOJPO%ZOBNJDTCPUIOPUJPOTTPNFUJNFTBEPQUFE QIZTJDT NPEFMTWTTPDJPMPHJDBMNPEFMT "MTPPQJOJPOQMVSBMJUZPSDPFYJTUFODF Mechanisms t1PTJUJWFTPDJBMJOĘVFODFPQJOJPOTUFOEUPBMJHOJOJOUFSBDUJPO t7BMVFIPNPQIJMZBHFOUTXJUITJNJMBSPQJOJPOTJOUFSBDUNPSF GSFRVFOUMZ t4UBUVTIPNPQIJMZTPDJPEFNPHSBQIJDBMMZTJNJMBSBHFOUTJOUFS BDUNPSFGSFRVFOUMZ t/FHBUJWFTPDJBMJOĘVFODFBHFOUTXJUIWFSZEJČFSFOUWJFXT BEPQUFWFONPSFEJTUBOUQPTJUJPOTJOUIFPQJOJPOTQBDF t"SHVNFOUQFSTVBTJPOMFBSOJOHOFXTVQQPSUJWFBSHVNFOUT CJBTFEBTTJNJMBUJPOJOGBWPVSPGTVQQPSUJWFWJFXT Mechanisms → Outcomes t1PTJUJWFsocial infuence + status homophily → consensus (static networks, traditional models, binary models) t1PTJUJWFsocial infuence + opinion homophily → plurality with MPDBMBWFSBHJOH "YFMSPE CPVOEFEDPOĕEFODF DPFWPMWJOHOFUXPSLT tPositive + negativ social infuence → polarization (Bourdieu: cultural/aesthetic diferentiation) t1PTJUJWF OFHBUJWinfuence + opinion homophily → consensus tArgument persuasion + opinion homophily → polarization tBiased assimilation + opinion homophily → polarization Polarization Dynamics by Learning Q() Q() from Social Feedback t Social feedback + status homophily → polarization t"HFOUTDBOFYQSFTTUXPEJČFSFOUPQJOJPOT or XJUIJOUIFJS social neighborhood t0OFYQSFTTJOHUIFJSDVSSFOUPQJOJPOBHFOUTSFDFJWFBďSNBUJWF or non-confrming response depending on the current opinion in the neighborhood t"HSFFNFOUMFBETUPBQPTJUJWFFYQFSJFODFXIJDITUSFOHUIFOT TVQQPSUPOUIFFYQSFTTFEPQJOJPO Rooted in Reinforcement Learning Q() Q() t0QJOJPOFYQSFTTJPOBTBDUJPO t"HFOUTMFBSOWBMVFT2 P PGBDUJPOTCBTFEPOSFXBSE XIJDINBZ NPEFMEJČFSFOULJOETTPDJBMGFFECBDL t'PSOPXBHSFFNFOU→S]EJTBHSFFNFOU→ r = -1

3BOEPNBHFOUFYQSFTTFTPiBSHNBY2i P џHSFFEZ o 3BOEPNOFJHICPSSFTQPOETPj

3FXBSESi = oi oj DPEJOH = 1 = -1 4. Micro- and Macrodynamics Q() Q() ▶ Selected individual trajectories (spatial random graph)

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value

0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 time time time Polarization Dynamics Q() Q() ▶ Value diference ∆Q = Q()-Q() can be interpreted as strength of support (interval in continuous models is usually interpreted this way) ) )-V(

emergence of two opposing V = V( V =

∆ opinion groups

time (every Nth step) Micro- and Macrodynamics Q() Q()

▶ Macroscopic polarization measures (DiMaggio et al. 1996)

36 36 36 36 36 28 28 28 28 52 28 55 84 52 19 55 84 52 19 84 52 19 84 52 19 55 84 19 23 29 23 29 5523 29 5523 29 23 85 29 9531 85 9531 85 9531 85 9531 85 9531 10 34 10 34 10 34 10 34 10 373425 4 377325299 4 377325299 4 377325299 4 377325299 4 73299 70 9 66 5170 9 66 5170 9 66 5170 9 66 5170 80 9 66 51 9880 93 9880 93 9880 93 9880 93 98 93 18 18 18 18 6 18 660 15 660 15 660 15 660 39 15 60 39 15 39 39 39 81 33 81 90 33 81 90 33 81 90 33 81 90 33 90 91 91 91 41 91 41 26 91 41 26 41 26 41 26 26 21 38 21 38 21 38 21 38 21 38 71 71 71 71 69 72 71 69 94 72 69 94 72 69 94 72 69 94 7230 94 3065 3065 32 13 3065 32 3065 32 13 65 32 13 24 32 13 24 24 13 24 24 12 77 20 12 5 77 20 12 5 77 100 20 12 5 77 100 61 20 12 5 77 100 61 20 5 58 100 61 58 100 61 58 61 58 47 5896 47 96 47 96 47 96 47 96 75 75 17 75 17 75 17 75 17 49 17 63 49 63 49 63 49 63 49 62 63 40 62 40 62 40 62 87 40 62 82 57 87 40 22 82 57 44 87 89 22 82 57 44 87 89 22 82 57 44 87 89 22 82 57 44 89 22 44 89 3 76 3 74 76 3 74 76 3 74 76 3 74 76 74 78 53 78 53 78 53 83 78 53 83 78 53 83 83 83 79 11 79 11 79 11 79 11 79 11 45 45 45 45 45 92 92 92 92 42 92 48 42 48 42 48 42 48 42 48 88 56 88 5456 16 88 5456 16 88 5456 88 5456 16 54 16 50 50 16 50 50 50 67 46 67 46 67 8 46 67 8 46 67 46 35 8 7 35 8 27 1 7 64 35 27 1 7 64 35 27 7 35 8 27 1 7 64 27 1 64 1 64 68 86 68 86 86 68 86 68 86 14 14 14 68 14 59 14 59 59 97 59 97 59 97 97 97 43 43 43 43 43 1

0.8

0.6

0.4

Macroscopic Measures Support Strength 1 Dispersion 0.2 Bimodality Average Opinion Dissimilarity 0 0 10 20 30 40 50 60 70 80 90 100 time (every Nth step) Mathematical Characterizations Q() Q() tćFNPEFMJTTFUVQBTBDPNNVOJDBUJPO DPPSEJOBUJPO HBNF UIBUBHFOUTQMBZPOBOFUXPSL t"HFOUT»solve«UIJTHBNFCZUFNQPSBMEJGGFSFODFSFJOGPSDFNFOU MFBSOJOHXIJDIJTLOPXOUPDPOWFSHFUPUIFFYQFDUFEHBNFQBZ- PGGT VOEFSDFSUBJODPOEJUJPOT

ŕ8FDBOFYQMPJUUIJTGBDUBOEVTFHBNFUIFPSFUJDBSHVNFOUTUP BOBMZUJDBMMZDIBSBDUFSJ[FTUBCMFTZTUFNDPOGJHVSBUJPOT Opinion Games I Q() Q() t%FDJTJPOQSPCMFNGSPNUIFJOEJWJEVBMQFSTQFDUJWF TJOHMFBHFOUXJUILOFJHICPSTJOBĕYFEFOWJSPONFOU t0QJOJPOJOOFJHICPSIPPE t&YQFDUFESFXBSE→ oN J ]→ -oN J

* * t'JYFEQPJOUT ε > 0 2 i oN J | 2 i -oN J * * tɠ2i 2 i 2 i oN J →3BEJDBMJ[BUJPOJOVOJGPSNTPDJBMFOWJSPONFOUT! Opinion Games II t4USVDUVSBMIPMFT A B t4JOHMFTIPUHBNFCJNBUSJY t/PUJDF$PPSEJOBUJPOHBNF GPSUXPQMBZFST

t'PSLABOELB /BTIFRVJMJCSJVNHJWFOCZ PA]PB = 1) → »(BUFLFFQJOH«BUTUSVDUVSBMIPMFT → *NQPSUBOUSPMFQMBZFECZDPNNVOJUZTUSVDUVSF Convergence of Q-Learning A B tDynamical evolution of two agents A and B each linked

to a fxed community of opposed sign (kA = 10, kB = 2) ▶ Quick convergence to the payofs of the associated opinion game

2 1

1 Q B

) Q i i A (o Q

i Q 0 B

A Q Q (1) B B Q A 0 Q (1) A (1) B B A (1) A 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000 time (every 10th step) time (every 10th step) Cohesive Sets and Stable Play Q() Q() t$IBSBDUFSJ[FBDUJPOPQJOJPODPOĕHVSBUJPOTPWFSBOFUXPSLUIBUBSF TUBCMFJOUIFHBNFUIFPSFUJDTFOTF ŕJF/BTIOPQMBZFSJOEJWJEVBMMZJTCFUUFSPČCZTXJUDIJOH t"OZOPOVOJGPSNPQJOJPODPOĕHVSBUJPOQBSUJUJPOTUIFOPEFTFUJOUP

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and SJTTUBCMFJČDPI S1 ɖBOEDPI S ɖ ŕ5IBUJT FWFSZBHFOUIPMETUIFNBKPSJUZPQJOJPOJOJUTOFJHICPSIPPE On Two-Island Graphs t(FOFSBMJ[BUJPOPGQSFWJPVTFYBNQMF ŕ$POTJEFSUXPDPNNVOJUJFT.BOE-PGBHFOUTTUSPOHMZDPOOFDUFE XJUI Q XJUIJOBOEXFBLMZ Q BDSPTTDPNNVOJUJFT ŕ4JNQMFHSBQINPEFMUIBUBMMPXTDPNCJOBUPSJBMBOBMZTJTPGDPIFTJWFOFTT t7JFXFBDISPXPGBEKBDFODZNBUSJYBT#FSOPVMMJTFRVFODF ▶ 'PSJՈ.QSPCBCJMJUZUIBUNJOMJOLTBOEMPVUMJOLT

▶1SPCBCJMJUZUIBUJJTMFTTUIBODPIFTJWF On Two-Island Graphs

▶ qM and qL probability that one agent is less than 1/2-cohesive ▶ There are M respectively L agents in the communities and probability that none of them is less than 1/2-cohesive is

M = 10 | M = 500 | ▶ Comparison of the theoretical L = 10 M = 100 | M = 50 | M = 10 |

= 500 results with an average over

M = 50 |

L ½-cohesive] L L

L = 50

= 100 = 50 100 network realizations

= 100

and S ▶ Notice that for large system M Network Realizations limit there is a sharp transition Pr[S Theory at p = 1/2 p Stability of Polarization on Two-Island Graphs ▶ Polarized initialization of the two-community network such that agents in M maximally support 1 and agents in L support -1 ▶ 100 model realizations à 20000 × N steps Simulations: M = L = 10 ▶ Combinatorial analysis of M = L = 50 M = L = 100 cohesiveness provides reasona- M = L = 500 Cohesiveness: ble approximation M = L = 10 M = L = 50 ▶ Loss of stability for smaller p M = L = 100

Pr[ stable polarization ] M = L = 500 in large systems due to finite- size fluctuations (learning rate) p On Socio-Structural Conditions for Polarization ▶ Polarization measure (Flache/Macy) and consensus probability as a function of molularity in a stochastic block model (C = 10) ▶ (Almost) linear relation between modularity and polarization

0.8

0.6 indicative of Consensus Prob. N = 100 Consensus Prob. N = 500 0.4 meaningful Consensus Prob. N = 1000 Polarization N = 100 communities Polarization N = 500 0.2 Polarization N = 1000

polarization | consensus probability 0 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 modularity A Bigger Picture (Discussion) tćFUSBEHFEZPGDPPSEJOBUJPO »MPDBMDPPSEJOBUJPONBZMFBEUPHMPCBMQPMBSJ[BUJPO« t*OGBDU JUEPFTTPJOBXJEFSBOHFPGTPDJBMTUSVDUVSFT tćFNPEFMJTWFSZBCTUSBDUBOEDPVMECFSFMBUFEUP ŕ&YUSFNJ[BUJPOJOSVSBMQFSJQIFSZ ŕ&NFSHFODFPGEJČFSFOUDVMUVSFTMBOHVBHFTJOJTPMBUFETPDJBM HSPVQTBOEUIFJSFYUJODUJPO FTQMBOHVBHFT BTBGVODUJPOPG JODSFBTJOHDPOUBDUUPPUIFSHSPVQT Te Big Picture (Discussion) t"CVODIPGSFDFOUTUVEJFTJOOFVSPCJPMPHZTVHHFTU FWFODMBJN UIBUxTPDJBMJOĘVFODFNFEJBUFTWFSZCBTJDWBMVFTJHOBMTJO LOPXOSFJOGPSDFNFOUDJSDVJUZj $BNQCFMM.FJLMFKPIO txćFSFXBSEJOHQSPQFSUJFTPGTPDJBMCFIBWJPVSNBZIBWFFWPMWFE UPGBDJMJUBUFHSPVQDPIFTJPOBOEDPPQFSBUJPOj 3VČ tćFNPEFMTVHHFTUTUIBUQPMBSJ[BUJPO BTPQQPTFEUPDPIFTJPO BDSPTTHSPVQTNBZCFBTJEFFČFDUPGUIFTFSFXBSEJOHQSPQFSUJFT ŕ*OEJWJEVBMTMFBSOUPDPPSEJOBUFXJUIUIFJSJOHSPVQBUUIFFY QFOTFPGBMJLFMZBMJFOBUJPOUPPVUHSPVQT ŕ4PDJBMTUSVDUVSFJTUIFFMFNFOUBSZEFUFSNJOBOUJOUIBU Conclusion V() V() t.PEFMGPSQPMBSJ[BUJPOEZOBNJDTCBTFEPOB NJOJNBMTFUPGBTTVNQUJPOT "HSFFNFOUMFBETUPBQPTJUJWF EJTBHSFFNFOUUPBOFHBUJWFGFFECBDL *OUFSBDUJPOTBSFTUSVDUVSFECZBOFUXPSL t%ZOBNJDTTJNJMBSUPQFSTVBTJPONPEFMT JOUFSQSFUBUJPOEJČFSFOU ŕ*OEJWJEVBMTMFBSOJOBWFSZCBTJDSFJOGPSDFNFOUDJSDVJUZUPDPPSEJOBUF XJUIUIFJSJOHSPVQBUUIFFYQFOTFPGBMJLFMZBMJFOBUJPOUPPVUHSPVQT ŕćFGBDUUIBUEFMJCFSBUJWFBOEBČFDUJWFNPEFTPGDPNNVOJDBUJPO NBZGPTUFSQPMBSJ[BUJPOQPJOUTBUGVOEBNFOUBMDIBMMFOHFTJODPMM FDUJWFEFDJTJPOQSPDFTTFT Online Demonstrations ▶ www.universecity.de/demos/OpinionValuesSmall.html t1PMBSJ[BUJPO.PEFMXJUIBHFOUT ▶ XXXVOJWFSTFDJUZEFEFNPT0QJOJPO7BMVFT#JHIUNM t1PMBSJ[BUJPO.PEFMXJUIBHFOUT t-JOLTBSFOPUWJTVBMJ[FE QFSGPSNBODFSFBTPOT CFDBVTFUIFSFBSFNBOZ ▶ XXXVOJWFSTFDJUZEFEFNPT4DIFMMJOH4NBMMIUNM t4DIFMMJOH.PEFMPGTFHSFHBUJPOXJUIBHFOUT

▶ In BMMNPEFMT JOUFSBDUJPOTEFĕOFECZBSBOEPNTQBUJBMHSBQI