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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)
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
Persons Through Groups
2-mode networks
An Example:
Interpersonal Network
C
B
Intergroup Network
E
D
3
F
1
4
A
0
0
0
1
0
0
(4.3)
0 0 1 0
0 1 0 0
1 0 1 0
0 1 0 1
0 0 1 0
0 0 2 1
5
2
0
0
0
2
1
0
0
1
0
0
0
(4.4)
1 0 0
0 1 1
1 0 2
1 2 0
1 1 1
0
1
1
1
0
Problem:
These two
representations,
though clearly
related, are not
easily compared.
Persons Through Groups
2-mode networks
An Example:
To compare them, construct a person-to-group adjacency matrix:
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
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.
Persons Through Groups
2-mode networks
An Example:
Compare persons A and F
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
1
A 0
F 0
AF 0
2
0
0
0
3
0
1
0
4
0
1
0
5
1
0
0
S
= 1
= 2
= 0
Or persons D and F
1
D 0
F 0
DF 0
2
1
0
0
3
1
1
1
4
1
1
1
5
1
0
0
S
= 4
= 4
= 2
Person A is in 1
group, Person F is
in two groups,
and they are in no
groups together.
Person D is in 4
groups, Person F
is in two groups,
and they are in 2
groups together.
Persons Through Groups
2-mode networks
An Example:
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
Similarly for Groups:
1 2 12
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
g
Pij Aik A jk
k 1
Groups-to-Groups
p
Gij Aki Akj
k 1
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.
A Aji
T
ij
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 =
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
Persons Through Groups
2-mode networks
A
B
A= C
D
E
F
= A(AT)
P
G = AT(A)
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
1
2
AT = 3
4
5
A
0
0
0
0
1
A
B
C
D
E
F
A
1
0
0
1
0
0
B
0
1
1
0
0
0
P
C
0
1
2
1
0
0
= P
(6x6)
D
1
0
1
4
1
2
E
0
0
0
1
1
1
F
0
0
0
2
1
2
See: Breiger_ex.sas for an IML example.
C
1
1
0
0
0
D
0
1
1
1
1
E
0
0
1
0
0
F
0
0
1
1
0
(5x6)
(6x5)
A * AT
(6x5)(5x6)
B
1
0
0
0
0
AT * A
= P
(5x6) 6x5) (5x5)
1
2
3
4
5
1
2
1
0
0
0
2
1
2
1
1
1
G
3
0
1
3
2
1
4
0
1
2
2
1
5
0
1
1
1
2
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
NAMES
A.Alonzo
P.Bellair
C.Charles
E.Cooksey
E.Crenshaw
T.Curry
S.Dinitz
D.Downey
W.Form
R.Hamilton
L.Hargens
G.Hinkle
R.Hodson
S.Houseknecht
J.Huber
D.Jacobs
S.Jang
C.Jenkins
R.Jiobu
R.Kaufman
L.Krivo
W.Li
R.Lundman
E.Menaghan
K.Meyer
J.Mirowsky
F.Mott
K.Namboodiri
T.Parcel
R.Peterson
T.Price-Spratlen
L.Richardson
V.Roscigno
C.Ross
K.Slomczynski
V. Taylor
J.Moody
L.Keister
P.Paxton
N.VanDyke
C.Browning
PHLC
1
0
0
1
1
0
0
1
0
0
1
0
0
1
0
0
0
0
0
0
1
1
0
1
0
1
1
1
0
0
1
0
0
1
1
0
1
0
0
0
1
CC
0
1
1
0
1
1
1
0
0
0
0
0
0
0
0
1
1
0
1
0
1
0
1
0
0
0
0
0
0
1
1
0
0
0
0
0
1
0
0
0
1
GWIR CSC
0
0
0
0
0
0
1
0
0
1
1
0
0
0
1
0
1
1
0
1
0
0
0
1
1
1
1
1
1
1
0
1
0
0
0
1
1
0
1
0
1
0
0
1
0
0
1
0
1
1
1
0
0
0
0
1
1
0
0
0
0
0
1
0
1
1
1
0
1
1
1
1
0
0
1
1
0
1
1
1
0
0
Persons Through Groups
Sociology Example
=A
G=(AT)A
PHLC
CC
GWIR
CSC
PHLC
17
5
8
5
G
CC GWIR CSC
5
8
5
14
3
2
3
21 10
2
10 18
Persons Through Groups
PHLC
See OSU_COM_READ.SAS
Sociology Example
GRWI
Crime & Community
Area Overlap Among
OSU Soc Faculty
P = A(AT)
CSC
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
Persons Through Groups
Sociology Coauthorship
Largest Bicomponent, g = 29,462
0.04
0.27
0.50
0.73
0.96
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 2mode network so that the adjacency
matrix has all zeros in the blockdiagonal 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