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Persons Through Groups 2-mode networks

Persons Through Groups 2-mode networks. Overview Breiger: Duality of Persons and Groups Argument Method Sociology Examples Moody: Coauthorship Methods: Finish ego-networks Working w. 2-mode data Constructing a PTG network Constructing a GTP network (Bipartite graphs).

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Persons Through Groups 2-mode networks

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  1. Persons Through Groups 2-mode networks • Overview • Breiger: Duality of Persons and Groups • Argument • Method • Sociology Examples • Moody: Coauthorship • Methods: • Finish ego-networks • Working w. 2-mode data • Constructing a PTG network • Constructing a GTP network • (Bipartite graphs)

  2. Persons Through Groups 2-mode networks Breiger: 1974 - Duality of Persons and Groups Argument: Metaphor: people intersect through their associations, which defines (in part) their individuality. Duality implies that relations among groups implies relations among individuals

  3. Interpersonal Network Intergroup Network 3 C E 5 1 2 B D F 4 A (4.3) 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 1 0 1 0 1 2 0 0 0 1 0 1 0 0 0 2 1 0 (4.4) 0 1 0 0 0 1 0 1 1 1 0 1 0 2 1 0 1 2 0 1 0 1 1 1 0 Persons Through Groups 2-mode networks An Example: Problem: These two representations, though clearly related, are not easily compared.

  4. Persons Through Groups 2-mode networks An Example: To compare them, construct a person-to-group adjacency matrix: Each column is a group, each row a person, and the cell = 1 if the person in that row belongs to that group. You can tell how many groups two people both belong to by comparing the rows: Identify every place that both rows = 1, sum them, and you have the overlap. 1 2 3 4 5 A 0 0 0 0 1 B 1 0 0 0 0 C 1 1 0 0 0 D 0 1 1 1 1 E 0 0 1 0 0 F 0 0 1 1 0 A =

  5. 1 2 3 4 5 A 0 0 0 0 1 B 1 0 0 0 0 C 1 1 0 0 0 D 0 1 1 1 1 E 0 0 1 0 0 F 0 0 1 1 0 1 2 3 4 5 S A 0 0 0 0 1 = 1 F 0 0 1 1 0 = 2 AF 0 0 0 0 0 = 0 A = 1 2 3 4 5 S D 0 1 1 1 1 = 4 F 0 0 1 1 0 = 4 DF 0 0 1 1 0 = 2 Persons Through Groups 2-mode networks An Example: Compare persons A and F Person A is in 1 group, Person F is in two groups, and they are in no groups together. Or persons D and F Person D is in 4 groups, Person F is in two groups, and they are in 2 groups together.

  6. 1 2 3 4 5 A 0 0 0 0 1 B 1 0 0 0 0 C 1 1 0 0 0 D 0 1 1 1 1 E 0 0 1 0 0 F 0 0 1 1 0 1 2 1•2 A 00 0 B 10 0 C 11 1 D 01 0 E 00 0 F 00 0  2 2 1 A = Persons Through Groups 2-mode networks An Example: Similarly for Groups: Group 1 has 2 members, group 2 has 2 members and they overlap by 1 members (C).

  7. Persons Through Groups 2-mode networks In general, you can get the overlap for any pair of groups / persons by summing the multiplied elements of the corresponding rows/columns of the persons-to-groups adjacency matrix. That is: Groups-to-Groups Persons-to-Persons

  8. 1 2 3 4 5 A 0 0 0 0 1 B 1 0 0 0 0 C 1 1 0 0 0 D 0 1 1 1 1 E 0 0 1 0 0 F 0 0 1 1 0 A = A B C D E F 1 0 1 1 0 0 0 2 0 0 1 1 0 0 3 0 0 0 1 1 1 4 0 0 0 1 0 1 5 1 0 0 1 0 0 AT = Persons Through Groups 2-mode networks One can get these easily with a little matrix multiplication. First define AT as the transpose of A (Simply reverse the rows and columns). If A is of size P x G, then AT will be of size G x P.

  9. Persons Through Groups 2-mode networks 1 2 3 4 5 A 0 0 0 0 1 B 1 0 0 0 0 C 1 1 0 0 0 D 0 1 1 1 1 E 0 0 1 0 0 F 0 0 1 1 0 A B C D E F 1 0 1 1 0 0 0 2 0 0 1 1 0 0 3 0 0 0 1 1 1 4 0 0 0 1 0 1 5 1 0 0 1 0 0 P = A(AT) G = AT(A) A = AT = (5x6) (6x5) A * AT = P (6x5)(5x6) (6x6) AT * A = P (5x6) 6x5) (5x5) P A B C D E F A 1 0 0 1 0 0 B 0 1 1 0 0 0 C 0 1 2 1 0 0 D 1 0 1 4 1 2 E 0 0 0 1 1 1 F 0 0 0 2 1 2 G 1 2 3 4 5 1 2 1 0 0 0 2 1 2 1 1 1 3 0 1 3 2 1 4 0 1 2 2 1 5 0 1 1 1 2 See: Breiger_ex.sas for an IML example.

  10. Persons Through Groups 2-mode networks Theoretically, these two equations define what Breiger means by duality: “With respect to the membership network,…, persons who are actors in one picture (the P matrix) are with equal legitimacy viewed as connections in the dual picture (the G matrix), and conversely for groups.” (p.87) The resulting network: 1) Is always symmetric 2) the diagonal tells you how many groups (persons) a person (group) belongs to (has) In practice, most network software (UCINET, PAJEK) will do all of these operations. It is also simple to do the matrix multiplication in programs like SAS or SPSS

  11. Name Health Fam Devlp Ineq Alessandro Tarozzi 0 0 1 1 Alexander Pfaff-Talikoff 1 0 0 0 Amar Hamoudi 1 1 0 0 Anatoli Yashin 1 0 0 1 Angela M ORand 1 0 0 0 Anna Gassman-Pines 0 1 1 1 Asia Maselko 1 0 0 0 Avshalom Caspi 1 0 1 0 Charlie Cloffelter 0 0 0 1 Christina M. Gibson-Davis 0 1 1 1 Duncan Thomas 1 1 1 1 Elizabeth Frankenberg 1 1 0 1 Elizabeth Oltmans Ananat 0 1 0 1 Frank A. Sloan 1 0 0 0 Jacob L. Vigdor 0 0 1 1 James Moody 0 0 1 1 James S Clark 1 0 0 0 James W. Vaupel 1 0 0 0 Jennan Read 1 0 0 1 Jerry Reiter 1 0 0 0 Kim Blankenship 1 0 0 0 Kathleen Sikkema 1 0 1 0 Keith E Whitfield 1 0 0 0 Kenneth A Dodge 1 1 1 1 Kenneth C Land 1 0 1 1 Linda K George 1 0 0 0 Linda M Burton 1 1 1 1 Lisa A Keister 0 0 0 1 M. Giovanna Merli 1 0 0 0 Manoj Mohanan 1 0 0 0 Marie Lynn Miranda 1 0 1 0 Marjorie B McElroy 0 1 0 0 P. J. Eric Stallard 1 0 0 0 Patrick Bayer 0 0 0 1 Peter Arcidiacono 0 1 0 1 Phil Morgan 0 1 0 0 Philip J. Cook 0 0 0 1 Philip R Costanzo 1 0 1 0 Rachel Kranton 0 0 0 1 Sabrendu Pattanayak 1 0 0 0 Seth Gary Sanders 1 1 1 1 Sherman James 1 0 0 1 Terrie E Moffitt 0 0 1 0 V. Joseph Hotz 0 1 0 1 William \"Sandy\" Darity 0 0 0 1 Zeng Yi 1 1 0 0 Persons Through Groups DuPRI Example =A G=(AT)A Health Fam HDev Ineqy 29 7 9 9 7 14 6 10 9 6 15 10 9 10 10 23

  12. Area Overlap Among DuPRI Faculty Human Dev P = A(AT) Health (Inequality) Family

  13. Persons Through Groups Sociology Example Or consider ties formed by sharing membership on a student committee (MA, exams, etc). (all committee memberships, line thickness proportional to number of joint appearances)

  14. Persons Through Groups Sociology Coauthorship Sociology Coauthorship Networks

  15. Persons Through Groups Sociology Coauthorship (2-mode) (1-mode projection)

  16. Persons Through Groups Sociology Coauthorship 3-degrees of Phil Morgan

  17. Persons Through Groups Sociology Coauthorship 3-degrees of Phil Morgan

  18. Persons Through Groups Sociology Coauthorship The likelihood of coauthorship varies by type of work

  19. Persons Through Groups Sociology Coauthorship

  20. 0.04 0.27 0.50 0.73 0.96 Persons Through Groups Sociology Coauthorship Largest Bicomponent, g = 29,462

  21. Persons Through Groups Sociology Coauthorship Largest Bicomponent, n = 29,462

  22. Persons Through Groups Director Interlocks Val Burris – Interlocks & Political Cohesion

  23. Persons Through Groups Director Interlocks Val Burris – Interlocks & Political Cohesion

  24. Persons Through Groups Director Interlocks Val Burris – Interlocks & Political Cohesion

  25. Persons Through Groups Director Interlocks Val Burris – Interlocks & Political Cohesion Effect size of indirect ties, by Dependent Variable Party Contribution Presidential Match Presidential Correlation

  26. Persons Through Groups Bipartite “Two-Mode” graphs It is possible to construct a network that links people and their groups directly in a single network. In this case, the nodes are of 2 types: person and groups. Consider the classic example of the Southern Women’s data:

  27. Persons Through Groups Bipartite “Two-Mode” graphs The classic treatment of this network would create a person to person or a group to group network:

  28. Persons Through Groups Bipartite “Two-Mode” graphs The classic treatment of this network would create a person to person or a group to group network:

  29. Persons Through Groups Bipartite “Two-Mode” graphs Instead, you could analyze the network as a joint network, with two types of nodes:

  30. Persons Through Groups Bipartite “Two-Mode” graphs Instead, you could analyze the network as a joint network, with two types of nodes:

  31. 1 2 3 4 5 6 7 8 ---------------------------- Actor 1 1. 0 0 0 1 1 0 0 0 Actor 2 2. 0 0 0 1 1 1 0 1 Actor 3 3. 0 00 1 0 0 1 1 Event 1 4. 1 1 1 0 0 0 0 0 Event 2 5. 1 1 0 0 0 0 0 0 Event 3 6. 0 1 0 0 0 0 0 0 Event 4 7. 0 0 1 0 0 0 0 0 Event 5 8. 0 1 1 0 0 0 0 0 Persons Through Groups Bipartite “Two-Mode” graphs It is always possible to arrange a 2-mode network so that the adjacency matrix has all zeros in the block-diagonal cells.

  32. Persons Through Groups Bipartite “Two-Mode” graphs Galois Lattices A new way to think about bipartite networks is as a collection of ordered sets, and then use some of the tools from discrete mathematics to map the collection of sets. For example, consider the set of all possible combinations of {1,2,3}. This can be represented in a network as: This is known as a Galois Lattice

  33. Persons Through Groups Bipartite “Two-Mode” graphs Galois Lattices Imagine you had the following data on actors and events:

  34. Persons Through Groups Bipartite “Two-Mode” graphs Galois Lattices

  35. Persons Through Groups Bipartite “Two-Mode” graphs Galois Lattices The Davis data in Lattice form:

  36. Methods: Review Ego-Networks. 1) Go over network drawing programs 2) Go over ego-network creation programs 3) Go over ego-network measures programs 4) Go over persons-through-groups creation programs

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