Ockham’s Razor in Opinion Dynamics Modeling and Analysis of Weighted-Median Influence Process Wenjun Mei Oct. 18th, 2018, IfA Coffee Talk Institut f¨ ur Automatik, ETH Zurich

Collaborators (a) F. D¨ orfler, ETH (b) F. Bullo, UCSB (c) G. Chen, CAS (d) N. Friedkin, UCSB 1

Outline • Introduction and motivation • The weighted-median opinion dynamics • Theoretical analysis • Numerical study • Conclusion and discussion 2

Introduction and Motivation

General introduction of opinion dynamics Opinion dynamics (influence process) • a group of individuals discussing certain issue • local interaction → individual opinion change → macroscopic phenomena • role of network structure in shaping dynamical behavior • social engineering: intervention & manipulation (a) Movie: “12 Angry Men” (b) 2016 US election 3

General introduction of opinion dynamics “Axioms” of opinion dynamics • individual opinions denoted by real umbers • opinion update due to interactions with social neighbors Classic DeGroot model [J. R. P. French 1956] • weighted-average opinion update xi (t + 1) = n j=1 wij xi (t) • wij : weight assigned to j by i = influence of j on i • W = (wij )n×n : row-stochastic, defines the influence matrix G(W ) J. R. P. French, “A formal theory of social power”, Psychological Review, 63(3):181-194, 1956, doi: 10.1037/h0046123 4

Important Related Literature [1] L. Moreau. “Stability of Multiagent Systems with Time-dependent Communication Links.” IEEE Transactions on Automatic Control, 50(2):169182, 2005. doi:10.1109/TAC.2004.841888. [2] N. E. Friedkin, E. C. Johnsen. “Social influence networks and opinion change” Advances in Group Processes, 16:1-29, 1999, Emerald Group Publishing Limited, ISBN: 0762304529. [3] R. Hegselmann, U. Krause. “Opinion dynamics and bounded confidence models, analysis, and simulations” Journal of Artificial Societies and Social Simulation, 5(3), 2002. [4] N. E. Friedkin, A. V. Proskurnikov, R. Tempo, S. E. Parsegov. “Network Science on Belief System Dynamics Under Logic Constraints” Science, 354(6310):321-326, 2016, doi: 10.1126/science.aag2624 [5] P. Jia, A. MirTabatabaei, N. E. Friedkin, F. Bullo. “Opinion Dynamics and The Evolution of Social Power in Influence Networks” Siam Review, 57(3):367-397, 2015, doi: 10.1137/130913250 [6] C. Altafini. “Consensus problems on networks with antagonistic interactions” IEEE Transactions on Automatic Control, 58(4): 935-946, 2013, doi: 10.1109/TAC.2012.2224251 [7] D. Bindel, J. Kleinberg, S. Oren. “How bad is forming your own opinion?” Games and Economic Behavior, 92:248-265, 2015, doi: 10.1016/j.geb.2014.06.004 [8] S. A. Marvel, J. Kleinberg, R. D. Kleinberg, S. H. Strogatz. “Continuous-time model of structural balance” PNAS, 108:1771-1776, 2011, doi: 10.1073/pnas.1013213108 [9] D. Acemoglu, A. Ozdaglar. “Opinion Dynamics and Learning in Social Networks.” Dynamic Games and Applications, 1(1):3-49, 2011. doi: 10.1007/s13235-010-0004-1. 5

Numerous extensions proposed by control community • time-varying graph/switching topology • gossip-like dynamics • negative weights • quantized opinion dynamics • multiple issues with logical constraints • evolution of social power along issue sequence • state-dependent interpersonal influences • stochastic bounded confidence • unilateral confidence bounds • opinion dynamics with noise • opinion dynamics with leaders • ...... 6

Convergence Rate of the Gossip-like Quantized Opinion Dynamics with non-Euclidean Opinion Spaces and Unilateral Confidence Bounds on Time-varying Topology with Communication Delay and Antagonistic Interactions? 7

x(t + 1) = W x(t) 8

DeGroot Model and its Weakness DeGroot opinion dynamics: x(t + 1) = Wx(t) Advantages • xmin (0) ≤ xi (t) ≤ xmax (0): consistent with empirical data • x(∞) = vleft (W ) x(0)1n : social power = eigenvector centrality 9

DeGroot Model and its Weakness DeGroot opinion dynamics: x(t + 1) = Wx(t) Weakness • macroscopic • always predicts consensus under mild conditions • reality: stable (multi-modal) opinion distributions (a) 2004 (b) 2008 (c) 2012 (d) 2016 Opinion distribution among European people on the issue:“Country’s cultural life is undermined by immigrants.” 0: strongly agree; 10: strongly disagree. Data from European Social Survey (http://nesstar.ess.nsd.uib.no/webview/) 9

DeGroot Model and its Weakness DeGroot opinion dynamics: x(t + 1) = Wx(t) Weakness • macroscopic: unrealistic prediction of consensus • microscopic: • weighted average ⇒ distant opinions are more attractive • physical intuition behind the always-consensus prediction 9

Extension of DeGroot Model Motivation: to remedy macroscopic shortcoming of DeGroot model Existence of absolutely stubborn agents • xi (t) = xi (0) • too strong assumption for judgemental issues with no opinion manipulators Friedkin-Johnsen model • convex combination of averaging and stubbornness xi (t + 1) = (1 − γi ) j wij xj (t) + γi xi (0) • over-parametrized: n-dimension state x, n parameters: γi , i = 1, . . . , n • predicts multi-modal opinion distribution by tuning parameters [NEF:15] [NEF:15] N. E. Friedkin. “The problem of social control and coordination of complex systems in sociology: A look at the community cleavage problem.” IEEE Control Systems, 35(3):40-51, 2015. 10

Extensions of DeGroot Model Bounded-confidence model • Equations: for any i xi (t + 1) = j: |xj (t)−xi (t)|

Statement of Objectives Ockham’s Razor: simplicity of form, rich in behavior “Suppose there exist two explanations for an occurrence. In this case the one that requires the least speculation is usually better.” A novel opinion dynamics model • As simple as the DeGroot model (parameter-free) • Based on reasonable microscopic mechanisms • Has rich dynamical behavior • Captures some real phenomena that the other models fail to 12

Weighted-Median Opinion Dynamics

Weighted-Median Opinion Dynamics xi (t + 1) = Med x(i)(t), w(i) ( ) • ordered discrete set of opinions • Ockham’s razor: simplicity of form • Weighted median: given (x1, . . . , xm ) and weights (w1, . . . , wm ), Med(x, w) = x∗ : i: xi x∗ wi ≤ 1/2 • Assumption: uniqueness of weighted median. j∈θ wij = 1/2, for any θ ⊂ {1, . . . , n} 13

Weighted-Median Opinion Dynamics xi (t + 1) = Med x(i)(t), w(i) ( ) • ordered discrete set of opinions • Ockham’s razor: simplicity of form • meaningful intuition • median as a typical value • overlook the outliers (robustness) • connection with median voter model in political science 13

Weighted-Median Opinion Dynamics xi (t + 1) = Med x(i)(t), w(i) ( ) • ordered discrete set of opinions • Ockham’s razor: simplicity of form • meaningful intuition 13

Game-theoretical Interpretation DeGroot model as a best-response dynamics [DB-JK-SO:15] • individual cognitive dissonance ui (xi ; x−i ) = j wij (xi − xj )2 • well supported by psychology literature • weighted averaging = best-response dynamics x+ i = j wij xj ⇔ x+ i = argmin z j wij (xj − z)2 • quadratic cost function: taken for granted but questionable [DB-JK-SO:15]: D. Bindel, J. Kleinberg, S. Oren. “How bad is forming your own opinion?” Games and Economic Behavior, 92:248-265, 2015. 14

Game-theoretic Interpretation Generalized best-response opinion dynamics x+ i ∈ argminz j wij |xj − z|α ($) α > 1(α < 1 resp.): Distant(near-by resp.) opinions are more attractive α = 1: Opinion attractiveness does not depend on opinion distance. Equation ($) with α = 1 is equivalent to the weighted-median model ( ). 15

Theoretical Analysis: Convergence, Consensus, and Disagreement

Convergence Analysis of the WM Opinion Dynamics WM opinion dynamics: xi (t + 1) = Med x(i)(t), w(i) ( ) Assume: asynchronous random updates Initial opinions: i.i.d. from some continuous distribution on X ⊂ Rn Theorem: almost-sure finite-time convergence For any initial condition x0 ∈ Xn, the solution to the weighted-median opinion dynamics ( ) almost surely converges to a fixed point in finite time. 16

Convergence Analysis of the WM Opinion Dynamics WM opinion dynamics: xi (t + 1) = Med x(i)(t), w(i) ( ) Assume: asynchronous random updates Initial opinions: i.i.d. from some continuous distribution on X ⊂ Rn Theorem: almost-sure finite-time convergence For any initial condition x0 ∈ Xn, the solution to the weighted-median opinion dynamics ( ) almost surely converges to a fixed point in finite time. Proof method: “monkey typewriter argument” • For ∀x0 , ∃ update sequence {i1, . . . , iT } s.t. x(T) reaches a fixed point. • Sooner or later, some x0 and its corresponding update sequence will occur. 16

Sketch of Proof Important notion: cohesive set [SM:00] • subset of nodes, M is cohesive if j∈M wij > 1/2, ∀ i ∈ M • more generalized definition used in linear threshold model [DA-AO-EY:11] • Nodes in cohesive sets do not change their opinions. • Cohesive set grows to the maximal cohesive set (unique expansion). [SM:00] S. Morris. “Contagion.” The Review of Economic Studies, 67(1):57-78, 2000. [DA-AO-EY:11] D. Acemoglu, A. Ozdaglar, E. Yildiz. “Diffusion of innovations in social networks.” CDC, 2329-2334, Orlando, USA, December 2011. 17

Consensus or Persistent Disagreement New notion: (i, j) is a decisive out-link if ∃ θ ⊂ i’s out-neighbor set, s.t. • j ∈ θ; • k∈θ wik > 1/2; • k∈θ\{j} wik ¡1/2. Define Gdecisive (W ) as G(W ) with all the indecisive links removed. 18

Consensus or Persistent Disagreement Theorem: consensus or disagreement • {1, . . . , n} is the only maximal cohesive set ⇒ almost-sure consensus; • ∃ non-trivial maximal cohesive set ⇒ non-zero probability of disagreement • globally reachable node in Gdecisive (W ) ⇒ almost-sure disagreement • ∃ globally reachable node ⇒ non-zero probability of consensus (conjecture) Ockham’s razor: richness of behavior. 19

Numerical Study: Comparison of Different Models

Which models to compare with? • DeGroot model with stubborn agents(DS): • X(t + 1) = Wx(t) • randomly pick 5% of the nodes and let them be stubborn • Friedkin-Johnsen model(FJ) • X(t + 1) = (I − diag(λ))Wx(t) + diag(λ)x(0) • λi ∼ Unif[0, 1] • Bounded-confidence model on networks(BC) • The networked version could be: (no theoretical result) xi (t + 1) = j∈Ni : |xj (t)−xi (t)|

Simulation Results 1. WM Model predicts steady multi-modal opinion distribution. • Simulation Set-up • Network: Barab´ asi-Albert scale-free network, n = 5000 • Initial opinions: i.i.d. • Uniform: xi (0) ∼ Unif[0, 1] • Unimodal: xi (0) ∼ Beta(2, 2) • Bimodal: Y ∼ Beta(2, 10), xi (0) = Y or 1 − Y with equal probability • 3-modal: bimodal + unbiased unimodal • Results (also hold for Watts-Strogatz small-world networks) • Uniform initial distribution 21

Simulation Results 1. WM Model predicts steady multi-modal opinion distribution. • Simulation Set-up • Network: Barab´ asi-Albert scale-free network, n = 5000 • Initial opinions: i.i.d. • Uniform: xi (0) ∼ Unif[0, 1] • Unimodal: xi (0) ∼ Beta(2, 2) • Bimodal: Y ∼ Beta(2, 10), xi (0) = Y or 1 − Y with equal probability • 3-modal: bimodal + unbiased unimodal • Results (also hold for Watts-Strogatz small-world networks) • Unimodal initial distribution 21

Simulation Results 1. WM Model predicts steady multi-modal opinion distribution. • Simulation Set-up • Network: Barab´ asi-Albert scale-free network, n = 5000 • Initial opinions: i.i.d. • Uniform: xi (0) ∼ Unif[0, 1] • Unimodal: xi (0) ∼ Beta(2, 2) • Bimodal: Y ∼ Beta(2, 10), xi (0) = Y or 1 − Y with equal probability • 3-modal: bimodal + unbiased unimodal • Results (also hold for Watts-Strogatz small-world networks) • Bimodal initial distribution 21

Simulation Results 1. WM Model predicts steady multi-modal opinion distribution. • Simulation Set-up • Network: Barab´ asi-Albert scale-free network, n = 5000 • Initial opinions: i.i.d. • Uniform: xi (0) ∼ Unif[0, 1] • Unimodal: xi (0) ∼ Beta(2, 2) • Bimodal: Y ∼ Beta(2, 10), xi (0) = Y or 1 − Y with equal probability • 3-modal: bimodal + unbiased unimodal • Results (also hold for Watts-Strogatz small-world networks) • 3-modal initial distribution 21

Simulation Results Real data: (a) 2004 (b) 2008 (c) 2012 (d) 2016 Opinion distribution among European people on the issue:“Country’s cultural life is undermined by immigrants.” 0: strongly agree; 10: strongly disagree. Data from European Social Survey (http://nesstar.ess.nsd.uib.no/webview/) 22

Simulation Results 2. Extremists tend to reside in peripheral locations. • Simulation set-up • Network • Barab´ asi-Albert scale-free network, n = 1000 • add self loops, randomize and normalize weights • Initial opinions • xi (0) ∼ Unif[−1, 1], i.i.d • four categories: neutral, moderately biased, biased, extreme • Visualized example of WM model • radius nodes ∼ 1/in-degree • grey scale ∼ extremeness 23

Simulation Results 2. Extreme opinions tend to reside in peripheral locations. • Comparisons: K = 500 realizations, log-pdf of centrality distributions • red: extreme, green: biased, blue: moderately biased, black: neutral • in-degree centrality 24

Simulation Results 2. Extreme opinions tend to reside in peripheral locations. • Comparisons: K = 500 realizations, log-pdf of centrality distributions • red: extreme, green: biased, blue: moderately biased, black: neutral • in-degree centrality • closeness centrality 24

Simulation Results 2. Extreme opinions tend to reside in peripheral locations. • Comparisons: K = 500 realizations, log-pdf of centrality distributions • red: extreme, green: biased, blue: moderately biased, black: neutral • in-degree centrality • closeness centrality Remark: This phenomenon is not simply due to the effect of stubbornness. 24

Simulation Results 3. More difficult to reach consensus in large & clustered networks. • Models to compare • DeGroot, DS, FJ: not even comparable • The networked BC is the only comparable model. • Simulation Set-up • Network: Watts-Strogatz small-world network • For each (n, d): 5000 independent realizations, estimate pconsensus • Result: effect of network size 25

Simulation Results 3. More difficult to reach consensus in large & clustered networks. • Models to compare • DeGroot, DS, FJ: not even comparable • The networked BC is the only comparable model. • Simulation Set-up • Network: Watts-Strogatz small-world network • For each (n, d): 5000 independent realizations, estimate pconsensus • Result: effect of clustering 25

Conclusion and Discussion

Comparison on theoretical results • Classic DeGroot model: consensus or almost-sure disagreement 26

Comparison on theoretical results • Classic DeGroot model: consensus or almost-sure disagreement • DeGroot with stubborn agents: consensus or almost-sure disagreement 27

Comparison on theoretical results • Classic DeGroot model: consensus or almost-sure disagreement • DeGroot with stubborn agents: consensus or almost-sure disagreement • Friedkin-Johnsen model: almost-sure disagreement 28

Comparison on theoretical results • Classic DeGroot model: consensus or almost-sure disagreement • DeGroot with stubborn agents: consensus or almost-sure disagreement • Friedkin-Johnsen model: almost-sure disagreement • Bounded-confident model: no theoretical result for arbitrary graphs 29

Comparison on theoretical results • Classic DeGroot: consensus or almost-sure disagreement • DeGroot with stubborn agents: consensus or almost-sure disagreement • Friedkin-Johnsen: almost-sure disagreement • Bounded-confident: no theoretical result for arbitrary graphs • Weighted-Median: richer dynamical behavior yet more parsimonious 30

Comparison on Model Assumptions and Simulation Results 31

Summary 32