Persons Through Groups

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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
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
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Persons Through Groups 2-mode networks
An Example
Problem These two representations, though
clearly related, are not easily compared.
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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
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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.
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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).
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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
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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.
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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.
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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
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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
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Persons Through Groups Sociology Example
Area Overlap Among OSU Soc Faculty
PHLC
P A(AT)
Crime Community
GRWI
CSC
See OSU_COM_READ.SAS
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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)
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Bearman and Everett The Structure of Social
Protest
Uses Protest events to create a Groups by
Issues matrix. They then create a
Group-to-Group network and an Issue-to-issue
network and analyze changes in these networks
over time. They find that 1) Central groups
define salient repertoires 2) Identity movements
(NSMs) increase in centrality 3) Labor is an
important element in all time periods
(hidden)
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Bearman and Everett The Structure of Social
Protest
7
5
6
(61-63)
(66-68)
(71-73)
(76-78)
(hidden)
See paper for group compositions
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Bearman and Everett The Structure of Social
Protest
(hidden)
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Persons Through Groups Sociology Coauthorship
Sociology Coauthorship Networks
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Persons Through Groups Sociology Coauthorship
(2-mode)
(1-mode projection)
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Persons Through Groups Sociology Coauthorship
3-degrees of Dan Lichter
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Persons Through Groups Sociology Coauthorship
The likelihood of coauthorship varies by type of
work
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Persons Through Groups Sociology Coauthorship
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Persons Through Groups Sociology Coauthorship
Largest Bicomponent, g 29,462
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Persons Through Groups Sociology Coauthorship
Largest Bicomponent, n 29,462
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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 Womens data
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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
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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
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Persons Through Groups Bipartite Two-Mode graphs
Instead, you could analyze the network as a joint
network, with two types of nodes
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Persons Through Groups Bipartite Two-Mode graphs
Instead, you could analyze the network as a joint
network, with two types of nodes
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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.
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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
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Persons Through Groups Bipartite Two-Mode graphs
Galois Lattices
Imagine you had the following data on actors and
events
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Persons Through Groups Bipartite Two-Mode graphs
Galois Lattices
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Persons Through Groups Bipartite Two-Mode graphs
Galois Lattices
The Davis data in Lattice form
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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