Nonlinear Dynamics of Group Dynamics: Models from Mathematical Sociology

Barbara F. MeekerProfessor Emerita

Sociology DepartmentUniversity of Maryland

College Park, MD

SCTPLS Baltimore July 2012

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

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

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

)C+O*S-O*R(-O+O=O 11-t1,11-t2,11-t1,1-t1,t1, ( 1 ))C+O*S-O*R(-O+O=O 21-t2,21-t1,21-t2,1-t2,t2, ( 2 )

R1*R2 < S1*S2 Convergence (2.1)R1*R2 > S1*S2b Divergence (2.2)

This system is sensitive to values of the parameters:

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)

Simulation, Undifferentiated Dyad

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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’

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.

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

Supreme Court

Upholds Obamacare

U Va Board of Regents Fires President

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

Are Discussion Groups Important?

Photos: B. Meeker

Pictures of two discussion groups

Group 15 2H2L M

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Group 8 2H2L F

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Pictures of two more discussion groups

Bales, Robert F. . 1952 "The Equilibrium Problem in Small Groups" in T. Parsons, T., R. F. Bales and E. A. Shils, eds, Working Papers in the Theory of Action, , Glencoe, Ill: The Free Press. Also, Pp. 444-476 in Small Groups, rev ed., 1965,edited by A. P. Hare, E. F. Borgatta and R. F. Bales. N. Y.: Alfred A. Knopf. Chase, Ivan D. 1980. "Social Process and Hierarchy Formation in Small Groups: A Comparative Perspective" American Sociological Review 45:905-924. Fararo, Thomas J. and John Skvoretz. 1988. "Dynamics of the Formation of Stable Dominance Structures" Pp. 327-350 in Status Generalization: New Theory and Research edited by M. Webster and M. Foschi, Palo Alto, CA: Stanford Univ. Press. Fisek, M. Hamit. 1974. "A Model for the Evolution of Status Structures in Task-oriented Discussion Groups." Pp. 53-83 in Expectation States Theory: A Theoretical Research Program. edited by J. Berger, T. L. Conner, and M. H. Fisek. Cambridge, MA: Winthrop. Fisek, M. Hamit, Joseph Berger, and Robert Z. Norman. 1991. "Participation in Homogeneous and Heterogeneous Groups: A Theoretical Integration." American Journal of Sociology. 97:114-142. Goetsch, G. G. and David D. McFarland. 1980. "Models of the Distribution of Acts in Small Discussion Groups." Social Psychology Quarterly. 43:173-83. Horvath, W. J. 1964. "A Mathematical Model of Participation in Small Groups." Behavioral Science. 10:164-166. Kadane, J. B. and Gordon Lewis. 1969. "The Distribution of Participation in Group Discussions: An Empirical and Theoretical Reappraisal" American Sociological Review 34: 710-722. Leik, Robert K. 1967. "The Distribution of Acts in Small Groups." Sociometry. 30:280-299. Leik, Robert K. and Barbara F. Meeker. 1975. Mathematical Sociology. Englewood Cliffs, N. J.: Prentice-Hall.

Leik, R. K. and B. F.Meeker (1995) "Computer Simulation for Exploring Theories: Models of Interpersonal Cooperation and Competition" Sociological Perspectives. 38:463-482.

Lotka, A. J. 1932. "The Growth of Mixed Populations: Two Species Competing for the Common Food Supply".Journal of the Washington Academy of Sciences. 21:461-469. Meeker, Barbara, 2002,“Getting In Each Other’s Way? Some Mathematical Aspects of Interaction In Small Groups” Presented at Group Processes Session, American Sociological Association Annual Meetings 2002

Skvoretz, John V. 1988. "Models of Participation in Status Differentiated Groups." Social Psychology Quarterly 51:43-57. Skvoretz, John and Thomas J. Fararo. 1996. "Status and Participation in Task Groups: A Dynamic Network Model" American Journal of Sociology 101: 1366-1414.

Skvoretz, J., M. Webster, and J. Whitmeyer. 1999. Status Orders in Task Discussion Groups. Advances in Group Processes 16: 199-218.

Smith-Lovin, Lynn and Charles Brody. 1989. "Interruptions in Group discussions: The Effects of Gender and Group Composition" American Sociological Review. 54: 424-435. Smith-Lovin, Lynn, John V. Skvoretz, and Charlotte Hudson. 1986. "Status and Participation in Six-Person Groups: a Test of Skvoretz's Comparative Status Model." Social Forces 64:992-1005. Stephan, F. and E. G. Mishler. 1952. "The Distribution of Participation in Small Groups: An Exponential Approximation." American Sociological Review. 17:598-608.

Strodtbeck, Fred L. and Richard D. Mann 1956 “Sex Role Differentiation in Jury Deliberations” Sociometry, 19,: 3-11

40 Years of mathematical models

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

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

Group 15 2H2L M

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Group 8 2H2L F

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Pictures of two more discussion groups

IIIb: Extension of Model to four actors

TopHi C=.5 .4 .3 .2Share .40 .20 .20 .20:add 1 1 1 1

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

Actual Data

X1= 3.0000 R1=0.0500 S1=0.0700 C1=0.5000X2= 3.0000 R2=0.0500 S2=0.0700 C2=0.4000X3= 3.0000 R3=0.0500 S3=0.0700 C3=0.3000X4= 3.0000 R4=0.0500 s4=0.0700 C4=0.2000 R1Eq S1Eq C1Fix|R2Eq S2Eq C2Fix|R3Eq S3Eq C3Fix|R4Eq S4Eq C4FixModel is Interactive/OthersOutput is Current/No Delay/N= 4/Up,Dn halffair share is 0.40 0.20 0.20 0.20Add to Output #1 #2 #3 #4 1.00 3.00 1.00 1.00

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

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’

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

C = .5,.4,.3,.2 Share = .4,.2,.2,.2Initial: start=3, r=.05, s=.07

R is ‘UP’, S is ‘Fixed’

R and SBoth ‘UP”

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]