CONNECTING DATA WITH COMPETING OPINION MODELS Oct. 31, 2014 Keith Burghardt

CONNECTING DATA WITH COMPETING OPINION MODELS Keith Burghardt with Prof. Michelle Girvan and Prof. Bill Rand Oct. 31, 2014 Friday, October 31, 14 OUTLINE Introduction and background Motivation and empirical findings Description of our model and agreement with data Analysis of the model Conclusions and future work Friday, October 31, 14 INTRODUCTION TO OPINION DYNAMICS Basic Idea: opinions or ideas propagate via social interactions between individuals Applications: adoption of languages, products, religions, etc. Why is important to model opinion dynamics? Basic understanding: How do human interactions give rise to complex social phenomena? Practical: how can we create the largest impact, e.g. vote shares, spread of social norms Friday, October 31, 14 INTRO TO OPINION DYNAMICS, CONT’D A statistical mechanical framework for opinion modeling: Atoms : properties of condensed matter → individuals : opinion dynamics How do we understand opinion formation? Empirical data Intuitive models (but are they well motivated?) Friday, October 31, 14 MODELING OPINION DYNAMICS Simplest model: adopt the opinion of a randomly chosen neighbor (Voter Model) Often seen (e.g. Voter Model Universality Class) Influential Clifford and Sudbury, Biometrika (1973) Holley and Liggett, Ann. Prob. (1975) Redner MAPCON (2012) Dornic et al., Phys. Rev. Lett. (2001) Friday, October 31, 14 TYPICAL MODEL ISSUES Inconsistent phenomenological models Motivation often based on results and not the dynamics themselves Unrealistic results (e.g., in equilibrium, everyone has a single opinion) Friday, October 31, 14 GOALS OF OUR MODEL Phenomenological: capture statistical patterns observed in data Realistic macro-dynamics: e.g., slow convergence to consensus/non-consensus in equilibrium Realistic micro-dynamics: Well-motivated mechanistic rules for the spread of ideas Friday, October 31, 14 MOTIVATING DATA Top: universal scaling for a candidate’s share of votes Q: # candidates (opinions) v: # voters for a candidate (opinion holders) N: # votes (nodes) Bottom: vote correlation ~ - log(distance) Stubbornness (resistance to opinion change) seems to increases with time Fortunato and Castellano, Phys. Rev. Lett. (2007) Fernandez-Gracia et al, Phys. Rev. Lett (2014) Friday, October 31, 14 UNIFYING OBSERVATIONS WITH A SINGLE MODEL Our Model: Competitive Contagions with Individual Stubbornness (CCIS Model) Contagion-like spread of opinions (opinions are “caught” like a cold) The longer an individual has an opinion the less likely they are to change it (stubbornness) Individuals occasionally re-evaluate their opinions and “recover” to the unopinionated state Well motivated micro-dynamics: Contagion-like spread, stubbornness, and recovery are all in qualitative agreement with empirical data Friday, October 31, 14 MODEL ASSUMPTIONS Fixed, undirected, network with N nodes, and Q opinion states A node i is in state {0, 1, 2, ...., Q} (0 is the un-opinionated state) At time t = 0, n0 nodes are in state 0, n1 with opinion 1, ... , nQ with opinion Q. Friday, October 31, 14 MODELING DYNAMICS ON NETWORKS Wikipedia Dynamics modeled on a variety of networks Representative of real social networks (c) Lattice Network Friday, October 31, 14 (d) “Small-World” Network ARE ASSUMPTIONS REASONABLE? Fixed number of opinions: e.g. jury verdict (“guilty” vs. “not guilty”) Assume a few (not necessarily all) individuals exhibit an innate opinion preference Candidates not well known in the beginning Some individuals seeded with preferences, e.g. candidate himself Friday, October 31, 14 MODEL OVERVIEW Convert un-opinionated neighbor with probability ! Adoption probability decreases with time an individual i has had an infection ("i) After "i > 1/#, an opinion “freezes” Opinionated (even frozen) individuals recover at a rate $ Friday, October 31, 14 β Padopt 0 µ-1 τ τi β β (1-τi μ) τj δ No Opinion Opinion 1 Opinion 2 β (1-τj μ) δ EXPLAINING VOTE DISTRIBUTIONS Goal: Fit several voter distributions with consistent parameters 3 fitting parameters Persuasiveness (! = 0.5) Initial fraction with a candidate preference (6%) Network degree distribution (p(k) ~ k-2.9) All other parameters fixed Friday, October 31, 14 Vote probability distributions, shifted by decades (inset: original data collapse). 100 Ê Ê 0 Ò· Ê Á · Á Ò¢ Á ı @ · Û ‡ Ê Á Ú · Û Ú Û Ù ı Ú Ù ı Û ÙÙıÙıÙıÙı Ò· 0 ıı Ò£ Ù ı Ú 0 Ù Á Ù ‡ Ù Ù Ú  Û @ ‡ · ‡ 0 ı ı Ò ‡ Á Ù ‡ Ò Ò0 Á ¢ £ £ Ê @ Ù ‡ ¢ Ú ı ‡ · Ò Û Ú · £ 0 £ Ò0 ¢ Ù @ @ ¢ £ ı  Ú Ù ¢ Á Ù ‡ ¢ ¢ £ Á ¢ Û £ ‡ Û £ Ê ı ‡ 0 Ú £ ı ı · @ @ ¢ Ù Ù ‡ Á Ú 0 0 £ @ @ ¢ Û  Ò ¢ ‡ Ù @ £ ı ‡ · · Ú Á £ @ Á ¢ Ò Û ı ‡  Û Ú @ Ù Ù · Á £ · Ò0 ¢ Á @ @ @ Ù ¢ ı ¢ ı Ò ‡ Ù Ê £ ¢ ı Û @ ‡ ı Ê Ú ‡ Ú 0 0 0 · £ £ ·ÊÊ 0 Û Û Á Ò‡ ı ÙıÙÙ Ù Á Á Ò‡ £ @ Ò¢ ÒÁ Ú Û £ · Ò@ £ Ù Ù ¢ Ò£ Ù £ ı · Ú @ Û ¢ ı ‡ ı £ ı Û ¢ Ù ı @ @ ‡ Ú Á Ú Ù 0 0 0 0 ‡ ı ı £ Ú Ê Ê ı · Ù ı Ò Ò Ò‡ ‡ Ê ¢ Ù Û Û @ @ Ù ı · · · · Ú Û ı Ú 0 0 0 Ê Á Á Ù ¢ Ê Ê Ú Ê Û Ò Ò · · ı Ê ‡  ¢ @ @ @ @ Á £ ı ı Ê 0 0 0 0 0 Û ı Ú · · Û ı Û Ê Ù ı Û ¢ Ò Á Û Ù  Ù Û ‡ £ £ £ ı ÒÚ Ê ¢ ¢ Û ı Ù @ @ @ @ @ @ @ ı · · Ê · Ù -2 ¢ 0 0 ı Ù Á Á Ù ı ÒÚ £ ı Ù ¢ Ú ‡ £ · · Ù Ù  Ê Ê Û Û Û ı Ò Ò ‡ Û ı Û ı £ @ ı Û ¢ ıı Á Á ‡ ÚÚ · ÛÛ Ù Û ı @ Ò Ò Ò· Ê Ú Û Û @ @ Ú ‡ ‡ Ú Ú Ù 00 0 0 0 0 0 Ù 0 Ú Û Ù Ú ı  £ · · · 10 Ú ı Û Ú Û Ò Ò Ò Ê ¢ ¢ ¢ ¢ Ù ı Ù Ù £ Ú Á Á Á Á ÁÁ Ù Ù ı Ê Û Û Ù ¢  Û Ú ‡ ı Ú Ù £ £ ıı Ú Û ‡ ı ı ¢ ¢ Ú ı · · · · Ú ‡ ÒÒ Ò ÒÙ Ê Ê Ê Ê Û £ £ £ £ ı Ù Ù Û Ú Ù ¢ Ú  ‡ Û Ê Û Ú Ú Ú Ù Ù ı  Û Û Ú Ú ‡ ‡ ‡ ‡ ı Ù Ú Ù £ Û Ù Ù ÛÚÚ Ùı ÊÙ Ê ÛÚ Û Ù ı ı Ê Ú Ú Ú Ú ÚÚ Ú Ù Ù ı ııÙÛÙ ÛÛÛÛ 0 Û Ú Û Ù ÛÛÚÛ Ú 00 Ù ¢ 0 Û ¢ Ù Ù Ú 0 ¢ ¢ ¢ ¢ ¢ ¢ Ù ¢ Ú 0 0 ¢¢0 Û 0¢0¢¢00 Ù ÙÛıÛÛÛ Û ÚÚ ı ÚÛÛ Ù 000 ¢¢¢¢ ¢ ¢ Û ¢ 0 0 -4Ù 00 ÚÚ0000 0 ¢ 0 ¢ @ ¢ 0¢ ¢0¢ 0 ÚÚ @£@@ Ù10 Û £@££@ ££ 0¢ ¢ @ Ú Ú £ £ £ £ @ £ @ @ @ £ £ ¢ 0 Û @ £ Ù 000 0 Ù ÚÚ Ù Poland, 2001 £@@ Ú Ú 00 ¢@ £££ ££@£@£ @ ¢¢¢¢0 @@ £ ı @ @@@ @¢¢@¢@ £ · ·· £ · £££@@ Ù @ @ @ ££@ @ ‡‡·‡·‡·‡·‡‡‡‡‡‡·‡‡·‡· Ú Poland, 2011 £ @ @ £@@ ‡·‡·‡ £ Û ‡·‡·‡· · ‡ ‡‡ £· ££ -6 · £·£ £ ···· ·‡ ‡ ÒÒ Ò ¢ Finland, 1995 ·     Ò ‡ 10· · ·· £·‡‡‡ Ò Ò Ò Ò Ò Ò Ê  Ò Ò Ò ·‡‡ 0 ·  Ò  Ò · · · ‡‡ Ê £ Finland, 2003 Ò Ò Ò Ò  Ò ·· · · Ò Ò‡ Ò‡ Ò ÁÁÁÊ ÁÊ @  Á Á Á Ê Ò  Á ‡‡ ‡ Ò ‡‡ Ò Á Ê ÒÒÒÒ  Á Á Ò Ò Ê Á ÁÁ Ê Á Ò  ÊÊ Á Ê Ê ÁÁ Á Ù Ê Á Á Ê Á Á Ê Ê Ê ÁÁ ÊÊÊ Ò ÙÒ Ò -8 ÁÊ Ê Á ÊÊ Ê  Á ÙÁ ÁÊÁÊÊ Á Ù ÊÊ Á ÊÊ Ê ÊÊ Ù Ò 10 10-10 -2 10  Ò 10-1 100 101 ‡ ·  Ò Ê Á Italy, 1958 Italy, 1979 Switzerland, 2011 102 v QêN Closed markers represent empirical data. Open markers represent model results. EXPLAINING OBSERVATIONS: LONG-RANGE CORRELATIONS Correlation of US Presidential Elections Assume if ! (recovery) → 0, model behaves diffusively C(r) ~ - log(r)/log(t) (2 dimensions) Slow consensus! Simulations corroborate this behavior Friday, October 31, 14 2000 Presidential Election AGREEMENT WITH SIMULATION 0.8 Á 2D grid with rewired connections (with no recovery or stubbornness) CHrL 0.6 0.4 0.2 2D: Correlation falls as ~ log(r), 0.0 Á Á Á Á Á Á Á Á Á Á Á Á Á Á 0.1 Rewire Probability Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á 101 Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á ÁÁÁ Á 0 Á Á Á Á ÁÁ ÁÁ ÁÁ ÁÁÁ Á ÁÁÁÁ Á ÁÁÁÁÁ Á ÁÁ Á ÁÁÁÁ Á Á Á Á Á Á ÁÁ Á Á ÁÁ Á Á ÁÁ ÁÁÁ ÁÁÁÁ Á Á ÁÁ Á Á Á Á Á Á ÁÁ Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á ÁÁÁ Á ÁÁÁÁ ÁÁ ÁÁÁ Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á ÁÁÁÁÁ Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á ÁÁ ÁÁ ÁÁ ÁÁ ÁÁ ÁÁ ÁÁ ÁÁ ÁÁ ÁÁ ÁÁÁ ÁÁ ÁÁ ÁÁ ÁÁ ÁÁ ÁÁ ÁÁ ÁÁ ÁÁ ÁÁ ÁÁ ÁÁ Á ÁÁÁ Á Á Á ÁÁ ÁÁ Á Á Á ÁÁ ÁÁ ÁÁ ÁÁ ÁÁ Á ÁÁÁÁ ÁÁ ÁÁ ÁÁ ÁÁ ÁÁÁ ÁÁÁ Á ÁÁ ÁÁ Á ÁÁ Á Á Á Á ÁÁ ÁÁ ÁÁ ÁÁ ÁÁ ÁÁ ÁÁÁ ÁÁ ÁÁ 2 10 r Real network has small geodesic distance (“small world”) Rewire: small world, with long-range correlations A Lattice and “Small World”-type Graph Friday, October 31, 14 ANALYTIC TREATMENT: MEAN FIELD APPROXIMATION Time Density Of Individuals With Opinion A A (t, !) Length Of Time Since Adopting A (!) Friday, October 31, 14 (t) (t) MEAN FIELD ANALYSIS Mean field PDE: [@t + @⌧ ]⇢A (t, ⌧ ) = PA (t) ⌘ Z 1 ⇢A (t, ⌧ ) ⇢A (t, ⌧ )d⌧ P B6=A k✓(1 Cyan: losses due to “recovery” 1.0 DPeq Blue: losses due to competing opinions Right: Difference in Q = 2 opinion densities versus persuasiveness. ⌧ µ]⇢A (t, ⌧ )PB (t) Difference in Opinion Densities Versus Persuasiveness 0 Close agreement with simulation: ⌧ µ)[1 · · · · · · · · · · · · · · · · · · Û Û Û · · Û · Û Û · · · Û Û Û · Û Û Û · Û · Û Û · · Ì Û Û Û ÌÁ · ÌÁ ÌÁ ÌÁ ÌÁ · ÌÁ · ÌÁ Û ÌÁ · Û Û · Û · Û ÌÁ Û ÌÁ · ÌÁ ÌÁ Û Û ÌÁ · ÌÁ Û · ÌÁ · Û ÌÁ Û ÌÁ Û ÌÁ · Û · Û ÌÁ ÌÁ · Û ÌÁ · ÌÁ Û · Û ÌÁ ÌÁ ÌÁ ÌÁ · Û Û · Ì Û Û · Ì Û · Ì Ì · Á Ì Û Á Û Ì Ì Á · Ì · Û Ì Û · Ì Á Û Ì Á Û · Ì Á · Û Ì · Á Û Á Ì Á ·ÛÛÛÛÛÌÌÌÌÌÁ Á · Á Á · Á · · Ì Á · Û Á Ì · ÛÛÛÌÌÌÁ Á Á · · Á 0.8 Ì Û · Á · ÛÛÛ ÌÌÁ · Á · Á DPH0L · Û ÌÌÁ · · ÛÛ ÌÌ 0.6 · · ÛÌÁ Á 0.02 Á · Û · · Û · ÛÛ ÌÌÌ Á Á · · Ì 0.05 Ì · 0.4 Û Á · Û · Á · ÛÛ ÌÌÌ ÁÁ · · · · Û 0.1 · ÛÛÛÛÛ ÌÌÌ Á · · · · · · Á · Á · Û · · 0.2·· ÛÛÛÛÛÛÛÛÛÛÌÌÌÌÌÌÌÁÁÁ · 0.2 ÁÁ ÛÛÛÛÌÌÌÌÌÌÌÌÌÌÌ Á ÛÛÛÛ ÛÛÛ Á Á Á Á ÌÁ ÌÁ Á ÌÁ ÌÁ ÌÁ ÌÁ Á Á ÌÁ ÌÁ Á ÌÁ ÌÁ ÌÁ ÌÁ ÌÁ Á Á Á Á Á Á Á Á Á Á Á 0.0 0.0 0.2 0.4 0.6 0.8 1.0 b N = 105, K-regular random graph with K = 10 Friday, October 31, 14 A CLOSER LOOK AT THE MEAN FIELD EQUATION [@t + @⌧ ]⇢A (t, ⌧ ) = X ⇢A (t, ⌧ ) k✓(1 ⌧ µ)[1 ⌧ µ]⇢A (t, ⌧ )PB (t) B6=A A Cyan/Blue: Loss/gain due to conversion to/from Blue: contrary opinions Cyan: the unopinionated state (t, !) Red: Total loss Green: Total gain ! -1 " + + ⇢A (t, 0 ) = (0 ) k[ X PB (t)] B6=A [1 PA (t) + X B6=A Friday, October 31, 14 PB (t)+ Z µ 0 1 ⇢B (t, ⌧ 0 )[1 ⌧ 0 µ]d⌧ 0 ] MOTIVATION FOR A SLOW CONSENSUS MODEL Relatively little consensus in day-today ideas, even after many years: Example of Consensus 1.0 GM vs. Toyota? rA HtL 0.8 0.6 Obama vs. Romney? Tcons 0.4 0.2 0.0 0 2000 4000 6000 8000 Long-range vote-correlations often = model with consensus t Realistic models should have either slow consensus or non-consensus. Friday, October 31, 14 SLOW CONSENSUS SIMULATIONS 105 4 10 Tcons 10 Ê Á d Á Ê Û Ú 3 Consensus time minimized at non-trivial parameter values Ï Ì Ï Ì ~m2 -1 10 10-2 10-3 Ú Û Ï Ì 4.0 ~m Ê Á Ï Ì Ï Ì ~m4.3 ÌÏ ÌÏ Ï ÌÏ ÌÏÏ ÌÌ Ï ÌÏ Ì Ï Ì Ú Û Ú Û ÚÁ Û Ì Ï ÌÌ Ï Ï ÌÏ Ì Ì ÚÁ Û Ê Á ÚÁ Û Ê Á Ê Á ÚÁ Û ÚÁ Û Ê Ê Ê Ú Û Ê Ú Û ÊÁ ÚÁ Û ÚÁ Û ÊÁ Ê Á Ï Ì Ê Ú Û Ê Ú Û ÏÏ ÌÌ Ï Ï ÌÏ ÊÁ ÊÁ ÚÁ Û Ê ÚÁ Û Ê ÚÁ Û Ê Ú Û ÊÁ Ú Á Û Ê Á Ê Ê ÚÁ Û ÚÁ Û Ê 2 ÚÁ Û Ê Ú Û Ê Ê Á Ê Á Ê Á Ú Û Ú Û Ê Á Ê Á Ê Á Ú Û Ú Û Ê Á Ê Á Ú Û Ê Á 10 ÚÛ Û Ê Ê Ê ÚÁ ÚÁ Û ÚÁ Û ÚÛ Û ÚÛ Ú ÚÛ Ú Û Ú Û 101 -3 10 10-2 10-1 m Friday, October 31, 14 Analytic treatment: assume Voter Model-like dynamics Ú Û Ú Û ~m4.8 Ú Û 100 Theory: Tcons ~ #2, where # is individual “stubbornness” DEPENDENCE ON NETWORK SIZE In the mean field limit, consensus time scales as: Tcons ⇠ Tcons 103 ‡ 102 Ù Ù Ù Ù Tcons ⇠ 1/N Ù 8000 105 N 10 Ê Tcons 0 104 2 2)/(↵ 1) ‡ ‡ ‡ ‡ ‡ Ù Ê Ù Ù ‡ Ê Ê Ù ‡ Ù Ù Ê Ê Ê Ê Ê Ê Ù Ê Ê 1.0 Ù Ê Ê Tcons 0.5 /µ)2 Complete graph: Friday, October 31, 14 ‡ ‡ ‡ ‡ 101 For scale-free networks: Tcons ⇠ N 2(↵ ‡ 0.0 N hk 2 i( a Ê 2.1 Ù 2.5 ‡ 2.9 Ê Ê Ê 101 Ê Ê Ê 100 102 103 N 104 IMPLICATIONS OF OUR MODEL Checking model assumption: does stubbornness increase in time? 10 Friday, October 31, 14 Ï -1 ÏÏ Ï Ï ÏÚÚÚ ÚÚÏÚ ÚÊ ÊÊ Ú ÊÏ ÚÊ ÊÏ ÏÚ Ê ÊÏ ÊÊ ÚÚÏÏ ÊÊÊ Ú Ï Ú ÏÊ Ú ÚÏ Ê ÏÊÏ Ú Ú Ï ÊÊ ÊÊÊÊ ÚÊÏ Ú ÚÏÏÊÊ ÚÏ Ê Ï ÊÊ Ú ÚÚÏÚÏ -2 10 ÚÏÚÏÚ Ú ÏÏ ÊÊ ÏÚÚ ÊÏÚ Ú ÏÏÚ Ê Ê Ï Ï Ú Ú Ê Ê Ï Ú 10-3 Ú ÏÏ Ê ÚÚ ÊÊ ÊÊ 10-4 Ï 10-2 Brand share probability should follow a distribution alike to the right figure Q = 10 Q = 20 Q = 30 Ê PHv,Q,NL Checking model prediction: if stubbornness = brand loyalty: 100 Rescaled Brand-share Probability Distribution 10-1 100 101 v QêN v: Number of items sold N: total number of items Q: Number of brands 102 SUMMARY OF RESULTS We created a minimal model with: Strong agreement to empirical data Realistic macrodynamics Well motivated microdynamics Found analytically tractable results on dynamics and consensus Presented ways to further corroborate our model Friday, October 31, 14 ADDING GREATER REALISM Opinion Difference Versus Random Flip Rate 1.0 Random opinion flipping Potentially in the Ising Model Universality Class Friday, October 31, 14 0.8 DPeq 3(!) critical points 4.00 ¥ 102 8.00 ¥ 102 1.60 ¥ 103 3.20 ¥ 103 6.40 ¥ 103 1.28 ¥ 104 2.56 ¥ 104 5.12 ¥ 104 0.6 0.4 0.2 0.0 0.00 0.02 0.04 0.06 Flip Rate 0.08 0.10 OPEN QUESTIONS What is the relation between model time-steps and the real world? How can we determine the microdynamics, versus aggregate macrodynamic data? How does stubbornness evolve? Friday, October 31, 14 Thank You Questions? Friday, October 31, 14

CONNECTING DATA WITH COMPETING OPINION MODELS Oct. 31, 2014 Keith Burghardt

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