Nonlinear Dynamics of Group Dynamics: Models from Mathematical Sociology

1. Nonlinear Dynamics of Group Dynamics: Models from Mathematical Sociology Barbara F. Meeker Professor Emerita Sociology Department University of Maryland College Park, MD SCTPLS Baltimore July 2012

2. Outline • I: Leik and Meeker’s adaptation of the Lotka-Volterra model of species competition • II: Discussion groups and inequality • III: Application of model to data from 4-person discussion groups

3. Part I: Lotka-Volterra Model and Leik-Meeker Adaptation • Each of two species must compete for resources within an ecological niche, and also must reproduce. The larger the population, the more it can reproduce but also the more it uses up food. Using up food reduces the resources available for the species itself and also for the other species. Source: Lotka, 1932

4. Parameters and variables • O (output) by #1 or #2, at time t • R (Reactivity) output limited by Other’s acts • S (Self-limitation) output limited by own acts • C (Constant) stable rate of growth

5. O1,t = O1,t-1 + O1,t-1(-R1*O2,t-1 – S1*O1,t-1 + C1) (1.1) O2,t = O2,t-1 + O2,t-1(-R2*O1.t-1 – S2*O2,t-1 + C2) (1.2) This system is sensitive to values of the parameters: R1*R2 < S1*S2 Convergence (2.1) R1*R2 > S1*S2b Divergence (2.2)

6. Leik and Meeker’s use of Lotka-Volterra • Actors are participants in a group • Output is amount of task behavior (e.g., talking) per time unit • Limits are time (only one person can talk at once) and task saturation (the overflowing in-box) • Constant is stable individual trait (e.g. talkativeness) • In Addition: Assumes • All actors must contribute something • actors seek to maintain a ‘fair’ ratio of ‘output’

7. Leik-Meeker Model References        "Computer Simulation for Exploring Theories: Models of Interpersonal Cooperation and Competition" R. K. Leik and B. F. Meeker Sociological Perspectives. 1995, 38:463-482.         "Exploring Nonlinear Path Models via Computer Simulation" R. K. Leik and B. F. Meeker Social Science Computer Review 1996, 14:253-268.         "Uses of Computer Simulation for Theory Construction: An Evolving Component of Sociological Research Programs" B. F. Meeker and R. K. Leik, Pp 47-70 in Status, Network and Structure: Theory Construction and Theory Development., edited by Jacek Szmatka, John Skvoretz and Joseph Berger; Stanford Univ. Press. 1997          "Some Philosophy of Science Problems in the Development of Complex Computer Simulations" chapter in The Growth of Social Knowledge: Theory, Simulation, and Empirical Research in Group Processes edited by Jacek Szmatka, Michael J. Lovaglia, and Kinga Wysienska, Praeger, Westport, CT 2002.

8. II: Discussion groups and Inequality • Discussion groups are ubiquitous • They are important • They have been studied for decades • They show regular patterns of inequality of participation • Which are related to influence within the group

9. Are Discussion Groups Important? U Va Board of Regents Fires President Supreme Court Upholds Obamacare Two Drug Makers Near Agreement on Merger U.N. Team Reports on Attack on Syrian Village Collusion Among Bankers Alleged Guilty Verdict in Hit-and-Run Case

10. Pictures of two discussion groups Photos: B. Meeker

11. Pictures of two more discussion groups

12. 40 Years of mathematical models

13. IIIa. Data from 4-person discussion groups • Data from John Skvoretz, Murray Webster, and Joseph Whitmeyer (supported by NSF) • videotapes and coded data from 50 4-member groups • univ undergraduates, previously unacquainted, same gender (80% of groups are female) • task: a ‘survival’ problem: rank order 15 items according to usefulness in a fallout shelter. Talk until consensus, up to 45 minutes

14. Variables • Indep: 3 conditions of ‘status differentiation’, operationalized by class standing (1st year vs 4th year) • 4 equal (all ‘low’) n=17 • 2 High, 2 Low n=16 • 1 High, 3 Low n=17 • Dependent: Number of acts per minute

15. Pictures of two more discussion groups

17. IIIb: Extension of Model to four actors

20. Equal .C=5 .4 .3 .2 share .25 .25 .25 .25 Add 1 1 1 1 TopLo C=.5 1.0 .3 .2 r share .4 .2 .2 .2 Add 1 3 1 1 TopHi C=.5 .4 .3 .2 Share .40 .20 .20 .20:add 1 1 1 1

21. Actual Data

22. X1= 3.0000 R1=0.0500 S1=0.0700 C1=0.5000 X2= 3.0000 R2=0.0500 S2=0.0700 C2=0.4000 X3= 3.0000 R3=0.0500 S3=0.0700 C3=0.3000 X4= 3.0000 R4=0.0500 s4=0.0700 C4=0.2000 R1Eq S1Eq C1Fix|R2Eq S2Eq C2Fix|R3Eq S3Eq C3Fix|R4Eq S4Eq C4Fix Model is Interactive/OthersOutput is Current/No Delay/N= 4/Up,Dn half fair share is 0.40 0.20 0.20 0.20 Add to Output #1 #2 #3 #4 1.00 3.00 1.00 1.00

24. In words • For each minute, we predict that on that minute, each actor will talk as many times as he/she did the previous minute, plus or minus that amount times a factor based on how much others have already talked that minute, how much the actor talked the previous minute and a constant representing talkativeness. • The effect of own and others’ previous minute’s talking increases if the actor’s total share for all minutes departs from the standard of fairness

25. What have we learned? • Model reproduces differences between group types in the pattern and amount of differentiation • Very sensitive to group members’ relative talkativeness and expectation for fair share of talking • Need better theories of ‘contributive justice’

28. Average total number of acts for each actor in the first 27 minutes of group discussion by type of group.

29. Up (want to be ahead); if output is lower than standard, R or S decreases, hence Output increases. If output is higher than standard, R or S increases, hence Output decreases. DeltaUp[i] := rel[i,t] - share[i] C = .5,.4,.3,.2 Share = .4,.2,.2,.2 Initial: start=3, r=.05, s=.07 R is ‘UP’, S is ‘Fixed’ R and S Both ‘UP”