Dynamic Social Network Modeling and Analysis Workshop Summary and Papers (2003) / Chapter Skim
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Session I: Social Network Theory Perspectives
Session I: Social Network Theory Perspectives
Finding Social Groups: A Meta-Analysis of the Southern Women Data
Pages 37-77
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SESSION I Social Network Theo PerspecOves
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and Davis, Gardner and Gardner (1941) began to collect systematic data on ' The author owes a considerable debt to Moms H
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2. Southern Women Data Set In the 1930s, five ethnographers, Allison Davis, Elizabeth Stubbs Davis, BurIeigh B
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that can be examined using tools for one-mode analysis. Because of these properties, this DGG data set has become something of a touchstone for comparing analytic methods in social network analysis.
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Those individuals who participate together most often and at the most intimate affairs are called core members; those who participate with core members upon some occasions but never as a group by themselves alone are called primary members; while individuals on the fringes, who participate only infrequently, constitute the secondary members of a clique.
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When the two analyses are based on the use of different data sets comparisons are, of course, not possible.
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used matrix algebra to show that the original two-mode, woman by event DGG data matrix could be used to generate a pair of matrices that are, 46 DYNAMIC SOCIAL NETWORK MODELING kD ISIS
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Breiger s designation has been picked up in a number of later works includin;, the data set released as part of the UCINET progran1 (Borgatti, Everett and Freeman.
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of the DGG data set produced a partition. Women ~ through 7 and 9 were assigned to one group and 10 through IS alone with ~ were assigned to the other.
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Bonacich began with the two-mode data set reported by DGG. His aim was to find a procedure for simplifying the data in such a way that as much as possible—all the unions (a or by, intersections (a and b)
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SVD can be used to analyze either one-mode or two-mode data. In this case, Bonacich applied it directly to the two-mode, women by events, data set.
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used the concept of hypergraphs to generalize their regular colonng to two-mode data sets. And Freeman and Duquenne (1993)
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produced another one-mode analysis of the DGG data. He began with the definition of group proposed by Homans (1950)
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Three Ant {Rli 197 Rli`.~07 an`1 Rli`~071 ~ ~ ~ ~ _ ~ · ~ ~ ~—~ ~ ~ ~ ~ J As part of a broad examination of techniques for the analysis of two-mode data, Borgatti and Everett (1997) used three procedures for finding groups in the DOG data.
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The tabu search found that the best two-group partition for the women was through 9 and 10 through 18. In their third analysis Borgatti and Everett used the regular two-mode DOG data.
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In order to uncover the group structure in this idealized data set, ~ clustered the data produced by the mode! using Johnson's (1967)
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DGG Data After Roberts' Iterative Proportional Fitting Roberts then subjected the transformed data to a singular value decomposition. The result is a procedure like correspondence analysis or pnncipal components analysis, but one in which all of the marg~nal effects have been removed.
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And eleven procedures went on to specify 58 DYNAMIC SOCIAL NETWORK A1ODEL~G ED THESIS
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In each of these sections, meta-analysis will be used to try to answer the two questions posed in Section ~ above: (] ~ By considering all of the results together, can we come up with an informed description of the group structure revealed in the DGG data?
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Thus, consensus analysis can be used to address both questions of interest in the current context. It can determine the "true" answers and it can determine the "competence" of each procedure.
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And finally, loadings on the first principal axis are determined using singular value decomposition. ~ =, The first axis of each matches and covariance provides an index of competence.
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, ~ (Bonacich's correspondence analysis) , 15 (Borgatti and Everett's taboo algorithm)
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When it came to dealing with group overlap, these analyses certainly did not agree. S.2 Specifying Positions: Core and Periphery Next we tune to the question of the assignment of individuals to core and peripheral positions.
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Matrix of Frequencies of Assignment to a "Closer to the Core" Position by Il Analytic Procedures for the First Group of Nine 64 DYNAMIC SOCIAL NETWORK MODELING AIID ANALYSIS
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Analytic Procedures for the Second Group of Nine These two data sets were first subjected to canonical analysis of asymmetry. As it is used here, the canonical analysis is based on an on/off all or none model.
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The only difference is found in the first group. The canonical analysis shows a tie between woman 5 and 66 DYNAMIC SOCIAL NETWORK HODEL~G~D ISIS
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0.962 52 0.923 52 2 HOMED Homans, Intuidon 1.000 17 1.000 17 6 BCH78 Bonacich, Boolean Algebra -~.111 9 0.~00 10 7 DOR79 Doreian, PJgebra~c Topology 0.329 28 0.929 28 8 BCH91 Bonacich, Correspondence Analysis 0.313 67 0.324 68 12 FWl;93 Freeman and White, Full Lath ce 0.953 43 0.~53 43 13 FVV293 Freeman and White, Su~Latice 0.867 45 O.870 46 14 BEl97 Borgati and Everett, Bi-Clique O.385 39 0.350 40 17 S&F99 Skvore~and Faust,p~ 0.932 59 0.933 60 18 ROB00 Roberts, Correspondence Analysis 0.844 64 0.846 65 20 N~1 Newman, Weighted C~P4tendance 0.~6 58 O.932 59 Figure 14. Gammas Showing the Degree to which 11 Analvses A~ree`1 with the Twn Standards In Assigning individuals to Core and Peripheral Positions DY7 JAMIC SOCIAL NETWORK MODELING AND ANALYSIS 67
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Fues3 BCH78 ~ ~ BeR74 ~ DOR78 BE197 ~~ r FRE8Q Baled nOM5O S&F— Romo BE3er Bat ~ ·P"" SINEWS FR193 ~1 FWI93 Figure 15. Axes 2 and 3 Produced by the Singular Value Decomposition of the Matches in the Assignments of Women 68 DYNAMIC SOCIAL NETWORK HODE~G ED ISIS
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In Figure 15, the basis for determining why this was the "best" partition becomes apparent. That partition was specified exactly by six of the analytic procedures: P&C72 (The corrected version of Phillips and Conviser's inflation theoretic algorithms, BCH91 (Bonacich's correspondence analysis)
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2 MISSING AND 3 MISSING AND 1 MISSING FRFE T 20VERLAP 1 OV~RW P C 6 MISSING 4 MISSING | \ | / 1 WRONG SIDE / O \ 0 2 MISSING O ~ WRONG SIDE / O O O MISSING ~ l I ~ OWR~P 2 WRONG SIDE 3 MISSING ~ 6 WRONG SIDE Figure 16. Axes 2 and 3 of the Matches Labeled by Structural Form OveratI, then, it is clear that there was a consensus about assigning women to groups.
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Procedure N Average Score Statistical hJlodel 1 0.957 Eigen Structure 3 0.954 C: ptimal Partition ~ 0.941 Transitivity 1 0.926 Cliques 1 0.916 Algebraic Duality 6 0.914 Intuition 2 0.887 Figure IS. Average Competences of the Various Classes of Procedures A number of features of Figure 17 are worth noting.
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The assumption has been that because they had a huge amount of ethnographic experience in the community, DGG had an edge- they somehow knew the ' true" group structure. But, particularly in the light of the present results, there is no compelling reason to award DGG any special exalted status vis-a-vis their ability to assign individuals to groups.
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Arable 1975 An algorithm for clustering relational data, with applications to social network analysis and comparison to multidimensional scaling. Journal of Mathematical Psychology 12:328-383.
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for the analysis of cross-cIassified data: Reconciliation and synthesis of some methods of Pearson, Yule, and Fisher, and also some methods of correspondence analysis and association analysis. Journal of the American Statistical Association 91: 408-428.
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algorithm. Invited talk, American Statistical Society Meeting.
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