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)

Breiger: 1974 - Duality of Persons and Groups 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

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

To compare them, construct a person-to-group adjacency matrix: 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 =

An Example: Compare persons A and F A = Or persons D and F 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. 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 = 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 Or persons D and F Person D is in 4 groups, Person F is in two groups, and they are in 2 groups together. 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

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

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: Persons-to-Persons Groups-to-Groups

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. 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 =

P = A(AT) G = AT(A) (5x6) (6x5) A * AT = P AT * A = P (6x5)(5x6) (6x6) 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.

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

=A G=(AT)A Persons Through Groups G PHLC CC GWIR CSC PHLC 17 5 8 5 NAMES PHLC CC GWIR CSC A.Alonzo 1 0 0 0 P.Bellair 0 1 0 0 C.Charles 0 1 0 0 E.Cooksey 1 0 1 0 E.Crenshaw 1 1 0 1 T.Curry 0 1 1 0 S.Dinitz 0 1 0 0 D.Downey 1 0 1 0 W.Form 0 0 1 1 R.Hamilton 0 0 0 1 L.Hargens 1 0 0 0 G.Hinkle 0 0 0 1 R.Hodson 0 0 1 1 S.Houseknecht 1 0 1 1 J.Huber 0 0 1 1 D.Jacobs 0 1 0 1 S.Jang 0 1 0 0 C.Jenkins 0 0 0 1 R.Jiobu 0 1 1 0 R.Kaufman 0 0 1 0 L.Krivo 1 1 1 0 W.Li 1 0 0 1 R.Lundman 0 1 0 0 E.Menaghan 1 0 1 0 K.Meyer 0 0 1 1 J.Mirowsky 1 0 1 0 F.Mott 1 0 0 0 K.Namboodiri 1 0 0 1 T.Parcel 0 0 1 0 R.Peterson 0 1 0 0 T.Price-Spratlen 1 1 0 0 L.Richardson 0 0 1 0 V.Roscigno 0 0 1 1 C.Ross 1 0 1 0 K.Slomczynski 1 0 1 1 V. Taylor 0 0 1 1 J.Moody 1 1 0 0 L.Keister 0 0 1 1 P.Paxton 0 0 0 1 N.VanDyke 0 0 1 1 C.Browning 1 1 0 0 Persons Through Groups Sociology Example =A G=(AT)A G PHLC CC GWIR CSC PHLC 17 5 8 5 CC 5 14 3 2 GWIR 8 3 21 10 CSC 5 2 10 18

Area Overlap Among OSU Soc Faculty P = A(AT) Persons Through Groups Sociology Example PHLC P = A(AT) Crime & Community GRWI CSC See OSU_COM_READ.SAS

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)

Persons Through Groups Sociology Coauthorship Sociology Coauthorship Networks

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

Persons Through Groups Sociology Coauthorship 3-degrees of Dan Lichter

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

Persons Through Groups Sociology Coauthorship

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

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

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:

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:

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:

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

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

Persons Through Groups Bipartite “Two-Mode” graphs 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 0 0 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 It is always possible to arrange a 2-mode network so that the adjacency matrix has all zeros in the block-diagonal cells.

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

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

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

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

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