Lefties, languages, and (religious) leavers Network structure and social group dynamics

1. Lefties, languages, and (religious) leaversNetwork structure and social group dynamics Daniel M. Abrams Northwestern University Department of Engineering Sciences and Applied Math

2. Modeling social systems • Daily life! • Sometimes called “sociophysics” • Do humans behave like atoms? (cf. Serge Galam) • If so, when, and to what extent? • Similar to “econophysics” • But they look for correlation, not fundamental model • Top-down vs. bottom-up • Particular interest in population changes, migration, voting, fads • Holy grail: quantitative prediction

3. Competitive social systems Groups compete for members: • Lefties vs. righties • Languages (compete for speakers) • Religions (compete for adherents) • Other possibilities: • Operating systems (compete for users) • Political parties (members) • Branded drinks (e.g., pepsi vs. coke) • Food choice (veg. vs. omni- or carnivore) • Reserve currency ($ vs. €, £, etc.)

4. Language death

5. Language death • 5000-7000 languages exist today. • 50-90% expected to disappear “soon”. • No objective method for triage.

6. Problem

7. New speakers added Old speakers removed Change in number of speakers for language X in time interval dt Time interval Current number of speakers of language Y Probability per capita per unit time for XY Probability per capita per unit time for YX Current number of speakers of language X Time interval Model

8. Dividing through by the factor Ndt: Plugging in for y(t) = 1–x(t) and eliminating explicit t’s:

9. Assumptions • Pyx is a function of x and s only. • Pxy(x, s) = Pyx(1-x,1-s) (no language intrinsically “better”) • Pyx increases monotonically in both arguments. • Pyx obeys the following limiting properties: “No one switches to a language with no speakers” “No one switches to a language with no status”

10. Predictions • Exact form of Pyx not important • Model allows only three equilibria, and only two are stable: • monolingual X-speaking population • monolingual Y-speaking population. • Thus, model predicts that bilingual populations are inherently unstable!

11. Equal Status, Different Sizes Example 1 Majority Language • Minority Language

12. Different Status, Same Size Example 2 High Status Language Low Status Language

13. Predictions All X Real data available for some languages: Quechua, Aymara, Scotch Gaelic, Welsh, Irish Gaelic, Alsacien, Galician, Hokkien, Quebec French. Initial Fraction x0 All Y Field work! Relative Status s

14. The down shown are for: • Scottish Gaelic in Sutherland • Quechua in Huanuco, Peru • Welsh in Monmouthshire, Wales • Welsh in all of Wales Blue squares indicate historical data, while red triangles indicate a single modern census.

15. Limitations • No bilingualism • Only two languages • No spatial variation • No social networks • But still surprisingly good agreement! • Suggests preservation technique: Actively control language status (For more info, please see Nature438 (7064), 43: Nov 5, 2005)

16. Includes many aspects of usefulness: moral, spiritual, social, economic, political, security, etc. Religious “leavers” • “Unaffiliated”: fastest growing minority • Includes: atheist, agnostic, secular deist, humanist, among others • Compete with group of religiously affiliated, regardless of denomination Relative utility of group X Fraction in Y X to Y Probability (per unit time) Fraction in X Y to X Probability (per unit time)

17. Some predictions (a) Aaland islands, Finland (b) Schwyz Canton, Switzerland (c) The Netherlands (d) 85 worldwide regions

18. Why does model work? • This “all-to-all” model works surprisingly well • Real society is not all-to-all • Can we generalize to arbitrary networks?

19. Model 1: All-to-all coupling • All-to-all • Binary nodes

20. Model 1.5: Stochastic network • Arbitrary net • Binary nodes

21. Model 2: Continuous network • Arbitrary net • Continuous (real-valued) nodes

22. Model 3: Spatial network • Arbitrary net • Continuous “nodes” • Large N lim

23. Perturbation • Why bother with spatial limit? • Perturbation theory likes continuous systems • Take 1D space (convenient) with -1 ≤ ξ ≤ 1 • Coupling kernel G = 1 / 2 corresponds to all-to-all system (model 1 with x replaced by ¯R ) • Perturb: along with R(ξ, t0) = R0(1 + εsgn(ξ))

24. Perturbation

25. Perturbation • Result: perturbed system always goes to same fixed points as all-to-all system • Only change in dynamics is time delay! Perturbation off of all-to-all δ = 0 δ = 0.14 δ = 0.60 δ = 0.98 Delay

26. Summary • New framework for social dynamics • Combines: • Complex networks • Ordinary and partial differential equations • Perturbation theory • Perturbation theory allows predictions • Numerics shows that perturbative predictions accurate even far from all-to-all system • Only a few out-group connections suffice to make all-to-all model good approximation! But must fail at some point. When??? (For more info, please see Phys. Rev. Lett. 107, 088701: Aug 19, 2011)

27. Left-handedness • About 10% of the population • Often persecuted (“sinister”) • Same percentage everywhere • Same percentage since prehistoric times • Different in some hunter-gatherer societies • Different in sports

28. Simple model • Assumptions: • Long time scale • Binary choice (lefty or righty, no ambidextry) Lefties added Lefties removed Assumes neither L nor R has intrinsic advantage Fraction lefty Fraction righty L to R probability (per unit time) Fraction lefty R to L probability (per unit time)

29. Transition rates • Assumptions: • Symmetry in L-R exchange: • Lack of ambidextrous group implies r + l = 1 • To better understand PRL, break it into two pieces: • PRLcomp – “competitive” part • PRLcoop – “cooperative” part

30. Competitive world • PRLcomp • Purely physically competitive society • Lefty-handed minority has frequency-dependent advantage • Advantage disappears as fraction lefty increases to 50% • Evidence: high homicide rate societies, boxing • PRLcomp should decrease monotonically with l Think “surprise advantage” Faurie & Raymond PRSB 2005

31. Cooperative world Think “conformist advantage” • PRLcoop • Purely cooperative society (no physical competition) • Lefty-handed minority has frequency-dependent disadvantage • Disadvantage disappears as fraction lefty increases to 50% • Evidence: differential accident rate, golf  PRLcoop should increase monotonically with l Coren & Halpern Psych Bull 1991

32. Transition rates • General PRL • More realistic society has both cooperation & competition • Write transition rate PRL as superposition: PRL(l) = cPRLcoop(l) + (1−c) PRLcomp(l) where 0 ≤ c ≤ 1 is relative importance of cooperation • A couple more minor assumptions for physically reasonable transition rates (continuity, boundedness)  PRLshould be monotonic with at most 3 inflection points

33. Analysis • l* = 0.5is always a fixed point of this system • The system may have 1, 3 or 5 equilibria for any c and sigmoidal PRL • Either subcritical or supercritical pitchfork occurs at a critical level of cooperativitycc

34. Analysis • Possible bifurcation diagrams: Most animals? Humans (Subcritical) (Supercritical)

35. Empirical validation • Problems: • Only a single evolutionary steady state observable for humans • Animal data sparse and debated • Workaround: Use sports as proxy • Need to supplement model to include selection process

36. Sports selection model Hockey right wings Baseball Boxing, men’s and women’s fencing, hockey (non-wings), men’s and women’s table tennis American football quarterbacks Hockey left wings Men’s & women’s golf

37. Sports selection model Top hitters 1871-2009 Best fit (one free parameter)

38. Conclusions • Matches real-world data well • Implies that physically competitive animal species should show little or no group handedness • Left-handedness is evolutionary adaptation (For more info, please see J. Royal Soc. Interface 0211: April 25, 2012)

39. Final conclusions • Math useful for human systems • Lack of data historically a problem • New datasets available all the time • Much still unknown about network structure impact • Fun!

40. Thanks! • Mark Panaggio, Haley Yaple, Richard Wiener, Steven Strogatz, Luis Morato-Pena, National Science Foundation, James S. McDonnell Foundation, many others!