CONNECTING DATA WITH COMPETING OPINION MODELS

Transcription

1 CONNECTING DATA WITH COMPETING OPINION MODELS Keith Burghardt with Prof. Michelle Girvan and Prof. Bill Rand Oct. 31, 2014

2 OUTLINE Introduction and background Motivation and empirical findings Description of our model and agreement with data Analysis of the model Conclusions and future work

3 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

4 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?)

5 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)

6 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)

7 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

8 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)

9 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

10 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, n 0 nodes are in state 0, n 1 with opinion 1,..., n Q with opinion Q.

11 MODELING DYNAMICS ON NETWORKS Wikipedia Dynamics modeled on a variety of networks Representative of real social networks (c) Lattice Network (d) Small-World Network

12 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

13 MODEL OVERVIEW Convert un-opinionated neighbor with probability! P adopt β 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 $ 0 β (1-τi μ) τj τi β τ β (1-τj μ) µ -1 δ No Opinion Opinion 1 Opinion 2

14 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 ) 10 0 Vote probability distributions, shifted by decades (inset: original data collapse). Ò ÒÒ Ò ÒÒÒÒÒ ÒÒÒ ÒÒ ÒÒÒÒÒ Ò Ù ÙÙÙ Ù Ù ÙÙÙ Ù Ù ÙÙ Ù Ù ÙÙ Ù ÙÙÙ ı ı ı ÚÚÚÚÚÚ ııııı Ú ıııı ııı ÚÚÚÚÚ Û Û Û ıú ııı ÛÛ Û Ú Û ÛÛ ÛÛÛ ÛÛ apple apple apple appleappleapple apple apple apple appleapple ÒÒ Ò Ò ÒÒÒ ÒÒ ÙÙÙÙÙÙÙÙÙÙÙ ÙÙÙÙ ÒÒ Ò ÒÒ Ò apple appleapple Ù Ù Ù Ù Ù Ù ÙÙ ÙÙÙÙ ÙÙ ÙÙÙÙ ı ıı ı ııı ııııııııııııııııııı 10-2 ıııı ÙÙ Ù Ù ı Ù ıı ııı ı ı Ú Ú ÚÚ Ú ÚÚ ÚÚÚÚÚÚÚÚÚÚÚÚÚ Ú ÚÚÚÚ Ú Ú Û ÛÛ Û Û Û ÛÛÛ ÛÛÛÛÛÛÛ ÛÛ ÛÛÛÛÛ Û ÛÛ Û Û Û ÛÛ Û Û ÛÛ ÚÚÚ ÛÛ Ù Ù ÙÛ ÙÛ Ú Ú Ú ÚÚÚÚ Ù ÚÚ Ù @@ 10-6 apple apple apple apple apple apple apple apple apple apple apple apple apple apple apple apple apple apple apple apple apple apple apple apple apple apple apple apple Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò Ò apple apple Ò Ò apple Ò Ò apple apple Ùapple ÙÒ Ò Ò Ò 10-8 Ùapple Ù v QêN Ù ı Poland, 2001 Ú Û Poland, Finland, Finland, 2003 Italy, 1958 apple Ò Italy, 1979 Switzerland, 2011 Closed markers represent empirical data. Open markers represent model results. Ù All other parameters fixed

15 EXPLAINING Correlation of US Presidential Elections OBSERVATIONS: LONG-RANGE CORRELATIONS Assume if! (recovery) 0, model behaves diffusively C(r) ~ - log(r)/log(t) (2 dimensions) 2000 Presidential Election Slow consensus! Simulations corroborate this behavior

16 AGREEMENT WITH SIMULATION 2D grid with rewired connections (with no recovery or stubbornness) 2D: Correlation falls as ~ log(r), Real network has small geodesic distance ( small world ) CHrL Rewire Probability r Rewire: small world, with long-range correlations A Lattice and Small World -type Graph

17 ANALYTIC TREATMENT: MEAN FIELD APPROXIMATION Density Of Individuals With Opinion A A(t,!) Time (t) (t) Length Of Time Since Adopting A (!)

18 MEAN FIELD ANALYSIS Mean field PDE: t ] A (t, ) = A (t, ) P B6=A k (1 µ)[1 µ] A(t, )P B (t) P A (t) Z 1 0 Cyan: losses due to recovery Blue: losses due to competing opinions Close agreement with simulation: A (t, )d Right: Difference in Q = 2 opinion densities versus persuasiveness. DP eq Difference in Opinion Densities Versus Persuasiveness Ì Û Ì Ì Ì ÌÌÌ ÌÌÌÌ Û ÛÛ Û ÛÛÛÛ DPH0L 0.02 Û ÛÛÛ Û ÛÛÛÛÛ Ì 0.05 Ì ÌÌÌ Û b N = 10 5, K-regular random graph with K = 10

19 A CLOSER LOOK AT THE MEAN FIELD EQUATION t ] A (t, ) = A (t, ) X B6=A k (1 µ)[1 µ] A (t, )P B (t) A(t,!) Cyan/Blue: Loss/gain due to conversion to/from Blue: contrary opinions Cyan: the unopinionated state Red: Total loss Green: Total gain " -1! A (t, 0 + )= (0 + ) k[ X B6=A P B (t)] [1 P A (t) + X B6=A P B (t)+ Z µ 1 0 B (t, 0 )[1 0 µ]d 0 ]

20 MOTIVATION FOR A SLOW CONSENSUS MODEL Example of Consensus Relatively little consensus in day-today ideas, even after many years: GM vs. Toyota? r A HtL T cons Obama vs. Romney? Long-range vote-correlations often = model with consensus t Realistic models should have either slow consensus or non-consensus.

21 SLOW CONSENSUS SIMULATIONS T cons d 10-1 ÚÛ ÏÌ ÏÌ 10-3 ~m 4.0 ÏÌ ÏÌ ÚÛ ÏÏÏÏÏÏÏ ÌÌÌÌÌÌ Ì ÏÌ ÏÌ ÏÏÏ ~m 4.3 ÌÌ ÏÌ ÏÏ Ï Ì ÌÌ Ì ÏÌ ÚÚÚÚÚÚÚÚÚÚÚÚ ÛÛÛÛÛÛÛÛÛÛÛ Ú ÚÛ ÚÛ ÚÚÚÚÚÚÚÚÚÚÚÚÚ ÚÛ ÛÛÛÛÛÛÛÛÛÛÛÛ Ú ÚÛ ÚÛ 10-2 ~m m ÏÌ ÚÛ ~m 4.8 Consensus time minimized at non-trivial parameter values Analytic treatment: assume Voter Model-like dynamics Theory: T cons ~ # 2, where # is individual stubbornness

22 DEPENDENCE ON NETWORK SIZE In the mean field limit, consensus time scales as: T cons Ù a Ù Ù Ù Ù Ù Ù Ù Ù Ù Ù Ù Ù T cons Ù T cons For scale-free networks: T cons N N hk 2 i( /µ) 2 Complete graph: 2( 2)/( 1) T cons N T cons 1/N N

23 IMPLICATIONS OF OUR MODEL Checking model assumption: does stubbornness increase in time? Checking model prediction: if stubbornness = brand loyalty: Brand share probability should follow a distribution alike to the right figure PHv,Q,NL Rescaled Brand-share Probability Distribution ÚÚÚÚÚÚÚÚÚÚÚÚÚ ÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏ ÚÚÚÚÚ ÏÏÏÏÏÏ Ú Ú ÚÚ Ï ÏÏÏ Ï Ú ÚÚ ÚÚ Ú ÚÚ ÏÏ Ú Ú Ï Ï Ï Ú Ú Ú Ú Ï Ï Ï ÚÚ Ú Ú Ï v: Number of items sold N: total number of items Q: Number of brands Q = 10 Ï Q = 20 Ú Q = v QêN

24 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

25 ADDING GREATER REALISM Random opinion flipping 3(!) critical points Potentially in the Ising Model Universality Class DP eq Opinion Difference Versus Random Flip Rate Flip Rate

26 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?

27 Thank You Questions?