SOC 8311 Basic Social Statistics

1
Relations Combinations Applying Matrix
AlgebraDavid Knoke University of
Minnesota POLNET Universiteit van Tilburg
June 20, 2007
2
It All Began with Sociograms
Although origins of network analysis lie in
1920s, Jacob Moreno (1934) pioneered social
network analysis for psychodrama therapy.
Moreno used sociomatrices and hand-drawn
sociograms to show boys and girls likes and
dislikes of classmates as directed graphs
(digraphs).
3
Basic Matrix Elements
A sociomatrix (adjacency matrix), designated by a
boldface capital letter such as X, is the most
common matrix form for presenting the social
network information in a graph G . (A one-mode
sociomatrix is a square g X g array of the
graphs N nodes, where both the rows and the
columns are displayed in the identical sequential
order.)
In digraphs, the senders are in the rows and the
receivers are in the columns.
The g2 cell elements, denoted by lowercase xij,
are the values of the L lines in the graph, with
a distinct value for every lti,jgt ordered nodal
pair. Row actor i sends a relation to receiver
in column j. In graphs of nondirected lines, all
pairwise cells have equal values, xij xji
(i.e., the matrix is symmetric). But, a
digraphs matrix may have pairwise values are not
equal. For example, in a binary matrix if i
sends to j but j does not send to i, then xij 1
and xji 0. If a pair reciprocate ties, both
cells have identical values.
A single row or column of a matrix is a called
vector.
4
Heres a nondirected one-mode graph and its
matrix
HARRY ?
SALLY ?
? TOM
DICK ?
? BETTY
B D H S T BETTY 0 1 1 0 0 DICK 1 0 1 1 0
HARRY 0 1 0 0 1 SALLY 0 0 1 0 0 TOM 0 0 0 0 0
5
Affiliation Networks
An affiliation network consists of two-mode data,
different sets connected by relations between but
not within each set. If the two sets are
actors and events, elements within each mode
are indirectly tied, via common links to the
other mode.
Familiar examples of affiliation networks
include persons belonging to voluntary
associations social movement activists
participating in protest events firms creating
strategic alliances nations signing trade and
military treaties.
Formally, a pair of elementary sets connected by
a (0-1 binary or ordinal) relation Set N of g
nodes (actors) N n1, n2, .. ng Set
M of h nodes (events) M m1, m2, mh L
nondirected lines join the gxh ordered pairs of
nodes ltni, mjgt
An affiliation network can be displayed either as
a bipartite graph, or as a gxh affiliation matrix
(A) whose i,j entry indicators whether actor i
participated in event j. Its hxg transpose
matrix (A) shows whether event j attracted actor
i.
6
Duality of Persons Groups
Ronald Breigers (1974) classic article on the
duality of persons and groups discussed (1)
actor-actor connections occurring through their
co-membership or co-attendance at the same
events and (2) event-event connections via the
overlap or interlocks with shared actors.

• These two dual networks can be created by either
  pre- or post-multiplying an affiliation network
  and its transpose to create two one-mode
  matrices
• AA is a gxg symmetrical matrix its main
  diagonal entries show the number events in which
  an actor is affiliated its off-diagonal elements
  are the number of events in which a row column
  pair jointly participated.
• AA is an hxh symmetrical matrix whose main
  diagonal entries show the number actors
  participating in the row event its off-diagonal
  elements are the number of actors affiliated with
  a particular pair events.

Both dual matrices may be analyzed as one-mode
networks, measuring such properties as size,
density, reachability, and cohesion.
Interpretations of co-memberships must recognize
that entities are indirectly connected, and that
the specific identities of those indirect paths
cannot be known from the dual matrix (e.g., we
know the number of events a pair attended but not
which events).
7
Affiliations in the GIS
Consider this 1998 affiliation matrix (A) from
the Global Info Sector project, 10 U.S.
computer/software firms participating as partners
in 54 alliances. A small portion of the
bipartite graph appears below.
11111111112222222222333333
3333444444444455555
12345678901234567890123456789012345678901234567890
1234 ------------------------------
------------------------ 1 APPLE
00000000100000100000000100000000000000000000000000
0000 2 COMPAQ 100011100110000101000000000110
000110011100001100110000 3 DELL
10000000000000000000000000000000010001000000001010
0000 4 HP 000100010000010000001101101011
100100000000010000100110 5 IBM
11000000000010001000101000001001111011000000100110
1000 6 INTEL 100011101101000100110010000000
000001000000100101001000 7 MICROSOFT
01111110001101110111010010111111100100110000111011
0000 8 NETSCAPE 000000000000000000000000000000
000000000011101000000001 9 ORACLE
00000001000000000000000001000000000000001100100010
0010 10 SUN 001000000000100010000000010000
000000100000010000000101
Apple ? Compaq ? Dell ? HP
? IBM ? INTEL ?
? SA 1 ? SA 2 ? SA 3 ? SA 4 ? SA 5 ? SA
6
8
How to Multiply Matrices
Two matrices can be multiplied only if theyre
conformable the number of columns in Matrix 1
must equal the number of rows in Matrix 2.
Resulting matrix dimensions rows of Matrix
1 columns of Matrix 2. A row-column cell
value in the resulting matrix is the sum of
products of the elements in the corresponding
Matrix1 row times Matrix 2 column.


Y
X
To illustrate Multiply 5x5 matrix X by 5x2
matrix Y, which is possible only because the
number of columns in X equals the number of rows
in Y. (Why is Y multiplied by X not possible?)
The result is 5x2 matrix Z
0 1 1 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0
0 2 2 0 2 2 2 0 0 2

Z
XY
4 2 4 4 2 2 2 2 0 0
(00 12 12 02 00 4) (02 10
12 00 02 2) (10 02 12 12 00
4) (12 00 12 10 02 4) (00
12 02 02 10 2) (02 10 02
00 12 2) (00 02 12 02 00 2)
(02 00 12 00 02 2) (00 02
02 02 00 0) (02 00 02 00
02 0)

9
To multiply dual network matrices in UCINET, use
Data/Transpose to transpose matrix A into matrix
A, then use Tools/Matrix Algebra to multiply
this pair of matrices in a specific order.
Post-multiplying the two-mode 1998 GIS matrix A
by its transpose, using the UCINET Algebra
command OxOprod(MAT_A,MAT_ATR), creates a 10x10
orgs-by-orgs matrix (OxO).
1 2 3 4 5 6 7 8 9 10
AP CO DE HP IB IN MI NE OR SU
-- -- -- -- -- -- -- -- -- -- 1 APPLE
3 0 0 1 0 1 1 0 0 0 2 COMPAQ 0
19 4 3 7 7 14 1 2 0 3 DELL 0 4
5 2 4 1 2 0 1 0 4 HP 1 3 2 16
4 0 9 0 3 2 5 IBM 0 7 4 4 17
4 6 1 2 3 6 INTEL 1 7 1 0 4 16 9
1 0 0 7 MICROSOFT 1 14 2 9 6 9 31 1
2 1 8 NETSCAPE 0 1 0 0 1 1 1 5 3
1 9 ORACLE 0 2 1 3 2 0 2 3 7 1
10 SUN 0 0 0 2 3 0 1 1 1 8
The main diagonal shows that Microsoft was the
most active in forming partnerships (31) and
Apple the least active (3). The off-diagonal
entries reveal that Microsoft and Compaq
partnered most often (14 times), but 13 pairs
formed no partnerships that year
10
Command ExEprod(MAT_ATR,MAT_A) creates a 54x54
event-by-event matrix (ExE). Its main diagonal
shows the number of organizations forming the
strategic alliance (e.g., 4 partners in event 1)
and the off-diagonal shows the number of actors
common to two alliances (e.g., 3 organizations
participated in both alliance 1 and alliance
34). To save space, I display only the first
five rows of the event-by-event matrix
11111111112222222222333333333344444444
4455555 123456789012345678901234567890123456789
012345678901234 -------------------------------
----------------------- 1 41002220121110021111102
0000120011321131100102212312000 2
12111110001111111111111010112112211111110000211121
1000 3 012111100011111111110100111111111001101100
011110110101 4 0112111100110211011112012021222111
01001100011110210110 5 21113330122201130222011010
1221111112012200102311221000
UCINETs NetDraw Visualization Software can
display ties among both sets in an affiliation
network. Click File/Open/Ucinet dataset/2-Mode
network and create the following diagram.
11
What evidence that a Wintel coalition (Microsoft
Intel) was opposed by the NOIS (Netscape,
Oracle, IBM, Sun) in the 1998 GIS strategic
alliance spider web?
12
Each one-mode matrix can be analyzed using
conventional network methods to reveal properties
of the inter-actor or inter-event indirect ties
Descriptive values for the OxO dual network
are Density Mean alliances 2.31 per dyad
(s.d. 2.94) Network Centralization Degree
24.0 Microsoft 45, Compaq 38, Apple
3 Closeness 49.7 Microsoft 9, IBM
10, Apple 15 Betweenness 8.6 Microsoft
4.1, HP 2.5, Apple Sun 0.2 Clique Analysis
(using only binary ties)
1 Compaq Dell HP IBM Microsoft Oracle 2
Compaq Dell IBM Intel Microsoft 3 Compaq IBM
Intel Microsoft Netscape 4 Compaq IBM Microsoft
Netscape Oracle 5 IBM Microsoft Netscape Oracle
Sun 6 HP IBM Microsoft Oracle Sun 7 Apple HP
Microsoft 8 Apple Intel Microsoft
13
Multidimensional scaling (stress0.14) with
hierarchical clusters (complete link)
14
2-Mode Data Exercise
US National Labor Policy Domain Event Interests
Network UCINET matrix input.txt is a data file
of 117 organizations interests in 36 policy
events in 1988 (Knoke et al. 1996).

1. Run UCINET Data/Import/DL to create and save this
   large file as a 2-mode orgs-by-events affiliation
   matrix.
2. Data/Extract to reduce the large file by keeping
   no more than 25 orgs and 10 events (read
   codebooks and examine matrix to decide which
   ones).
3. Display with UCINET NetDraws File/Open/Ucinet
   dataset/2-Mode network (try various display
   options). Interpret structural relations of orgs
   events.
4. Data/Transpose to transpose the reduced network
   into a 2-mode events-by-orgs affiliation matrix
   and save.
5. Tools/Matrix Algebra to create both 1-mode OxO
   and ExE matrices. By visual inspection, which
   orgs/pairs were the most/least active? Which
   policy events/pairs attracted the
   biggest/smallest audiences?
6. Analyze various network properties of both 1-mode
   matrices (e.g., density, centrality,
   multidimensional scaling) and interpret your
   findings.

15
Relational Algebra
Relational algebra, also known as role algebra,
uncovers the structure of social roles by
investigating indirect connections across
multiple relations. Role structures describe how
social roles are associated in the networks,
independent of individual actors occupying those
roles.
Scott Boorman and Harrison White (1976) proposed
using role algebra methods as an extension of
blockmodel analysis. But, relational algebra can
be learned and applied independently of
blockmodeling.

• RELATIONAL ALGEBRA a formal structure
  consisting of two elements - sets of relations
  and operations to manipulate those relations
• Dichotomous primitive relations (generator
  relations) among actors, represented by capital
  letters e.g., relations F and E for Friend
  Enemy
• A composition operation (?) that combines two or
  more primitive relations.
• Compound relation FE (Enemy of my Friend) results
  from composition where tie i(F ? E)j occurs if
  there exists some third actor k such that iFk and
  kEj

16
High Tech Managers Roles
Consider these graphs and matrices representing
the advice-giver (A) and friendship (F) images
for a three-position blockmodel of David
Krackhardts 21 high-tech managers (Wasserman
Faust 1994439-442). The circular arrows show
that high densities of ties occur among the
individual managers within three jointly-occupied
blocks
ADVICE (A) FRIENDSHIP
(F)

a ?
a ?
c ?
c ?
b ?
b ?
A a b c a 0 1 1 b 0 1 1 c 0 0 0
F a b c a 1 0 1 b 0 0 0 c 1 1 1
What role structure is revealed by the four
compound relations?
17
Forming Compound Relations
A compound relation is formed by the Boolean
multiplication of two or sociomatrices. It
resembles ordinary matrix multiplication, except
that any cell entry greater than 0 is replaced
by a 1. A nonzero entry means that a compound
relation exist between a pair of blocks/actors.
Use Tools/Matrix Algebra in UCINET to open a
window in which to write the matrix
multiplication commands. (Consult the Help
Manual entries under Algebra Package and
Algebra, Binary Operation for proper syntax.)
For Krackhardts three manager blocks, the
Boolean matrix product command for AF is
AFbprod(Advice,Friendship)
AF a b c a 1 1 1 b 1 1 1 c 0 0 0
Each 1 entry in AF identifies an advice-giving
block connected via an intermediary position to a
friendship block (i.e., friends of advisors)
In the original advice network A, block a gives
advice to block c (aAc), while in the friendship
network F, block c cites block b as its friend
(cFb). Hence, the compound relation AF reveals
indirect connection from block a to block b
(aAc)(cFb) (aAFb). The nonzero diagonal
blocks in the original image matrices are always
used when composing a relational algebraic
compound. Thus, block bs compound tie to block
c involves the latters within-block friendships
(bAc)(cFc) (bAFc).
18
FA, advisors of friends, shows how one block
might use their friends to contact an advisory
block. For example, block c cites block a as
friend and block a advises block b. Thus, the
compound matrix reveals (cFa)(aAb) (cFAb).
FA a b c a 0 1 1 b 0 0 0 c 0 1 1
The FF product identifies that classic compound
relation, a friend of a friend. Because block b
cites no direct friends, it also cant reach any
indirect friends. But, the other blocks have
friends who are connected to other friends
(aFc)(cFb) (aFFb).
FF a b c a 0 1 1 b 0 0 0 c 0 1 1
Finally, the AA composition yields the advisors
of advisors (evidently the experts mavens).
But this compound matrix is identical to the
original advice-giving image! The composition
AAA reveals that this advising network is
transitive for example, the compound
(aAb)(bAc)(aAAc) which is already a direct tie
(aAc). Hence, we dont need AA because A
includes all those compound relations within its
direct advising ties.
AA a b c a 0 1 1 b 0 1 1 c 0 0 0
19
Words Equivalence
Composition can involve sequences longer than two
compounded primitive relations e.g., AFFAAFA.
A string of letters is a word, whose length is
the number of primitive relations in it. Role
algebraists inductively generate a dictionary
of the unique words (matrices/images), with the
fewest letters, required for a complete
description of a multiple-network systems social
role structure. When a researcher generates a
longer new word, she compares its sociomatrix to
see whether any simpler word already in the
dictionary also has that longer words
sociomatrix. Words with identical matrices or
images are equivalent, and the set of all words
with identical images comprise an equivalence
class.

For Krackhardts high-tech manager data, the five
shortest unique words in its dictionary are A, F,
AF, FA, and FF (but not AA). See WF (1994440)
for some examples of longer words in those
equivalence classes.
20
Role Table

• In a multiplication table, or role table, each
  row and column entry corresponds to a unique
  primitive or compound relation. Instead of
  displaying network images (as WF show in Fig.
  11.2), each equivalence class in the table is
  labeled by the graphs word. The cell entries in
  the table contain the smallest word resulting
  from multiplying the row matrix by the column
  matrix.
• Matrix multiplication is associative the order
  of performing successive multiplications does not
  affect the result ABC(AB)CA(BC)
• Matrix multiplication is not commutative the
  result of multiplying two matrices may differ by
  the sequential order AB?BA
• In mathematical theory, a semigroup is defined as
  a set elements with an associative binary
  operator on it. Thus, a social network semigroup
  is the set of images/matrices formed by a set of
  relations and the composition operation (Boyd
  1990 Pattison 1993). If all compositions of the
  primitive relations are also members of the set,
  then a semigroup is closed under associative
  matrix multiplication. A role table contains
  all possible images that can result from the
  operation of composition on the primitive
  relations (WF 1994437).

21
The Dictionary
The role table for the Krackhardt managers
advice and friendship networks shows that
composing any pairs of the five unique words in
the dictionary yields four of these words (see
WF Fig. 11.5). For example, multiplying
(AF)(FA) (AFFA). But, the first three terms on
the rightside can be factored (AFF)(AF)(F) and
we find in the table that (AF)(F)(AF). Hence,
by substitution (AFFA)(AFF)(A)(AF)(A).
Finally, the role table shows that (AF)(A)(A),
so the multiplication (AF)(FA)(AFFA) reduces to
just (A), as displayed in row 3 column 4.

A F AF FA FF A A AF AF A FF F FA FF FF FA FF AF
A AF AF A AF FA FA FF FF FA FF FF FA FF FF FA FF
Rows 1 3 are identical, as are rows 2, 4, 5,
as well as two column pairs, 1 4 and 2 3.
These identities suggest opportunity to simplify
the social role structure of Krackhardts
high-tech managers (next).
22
Simplifying a Role Table
Role table simplification involves reducing the
number of network images or words while
preserving important structural properties. Each
image in the initial set is mapped onto a smaller
number of images in the simplified set. The
reduction of the role table is a partition of the
distinct images, S, into a smaller collection of
classes, Q. (WF 1994443). Unfortunately, a
unique or best reduction may not be possible
for some networks.

• Image simplification strategies include
• Substantive approaches that combine images with
  the identical meaning or similar operation
• Sociometric approaches that equate images with
  similar ties which may differ substantively.
  Sociometric similarity could be assessed using
  correlation to measure association, or finding
  images that are contained within (subset of)
  another image. See WF (1994444-445) for an
  application of the latter technique to the advice
  and friendship role table.

23
Homomorphic Reduction
A homomorphic reduction of an original role table
involves a mapping that preserves the composition
operation. More than one image may exist.

• One homomorphic image for the AF role table
  permutes and partitions the 5x5 table into two
  groups that produce nearly identical results
  1,3 and 2,4,5 which have the word
  equivalences A,AF and F,FA,FF. The reduced
  matrix expresses a first letter law that any two
  elements always result in an element that is in
  the same class as the first element of the
  composition (447).
• Another homomorphic image groups 1,4 and
  2,3,5, with word equivalences A,FA and
  F,AF,FF. This reduction satisfies a last
  letter law that the composition of any two
  elements results in an element that is in the
  same class as the second element of the
  composition (448).

If the same or comparable relations are measured
for two or more network systems, their role
tables can be compared on formal similarities
and/or differences. Boorman and White (1976)
joint homomorphic reduction of two role
structures summarizes common features. It
involves two mappings that preserve the
composition operation, resulting in a new
multiplication role table that is a reduction of
both original tables. As the union of two roles
structures, the role table contains all the word
equations appearing in one or both systems. For
example, Breiger and Pattison (1978) interpreted
the joint homomorphic reduction of three
political elite networks in a German town and a
U.S. city as an instance of Granovetters
strong-weak tie hypothesis.
24
Role Algebra Exercise
A blockmodel of the send policy information and
give support networks in the U.S. labor domain
yields two 4-block images, whose main members
are (A) unions (B) public interest groups
(C) businesses (D) federal government.
Info A B C D A 1 0 0 0 B 1 1 0 1 C 1 0 1 1 D 1 0 0
1
Support A B C D A 1 1 0 1 B 1 1 0 0 C 0 0 0 1 D 0
0 0 1

• Create text files for both image matrices and add
  DL commands that enable you to import and save
  these networks as two UCINET files.
• Use matrix algebra to compute four possible
  Boolean matrix products corresponding to the
  compound relations among Information and Support
  (e.g., II, IS, etc.). What are your substantive
  observations?
• Produce a role table containing the complete
  dictionary of all the unique words that can be
  formulated by compounding Information Support.
  Provide a substantive interpretation of the
  patterns you observe in this national policy
  domain.

25
References
Boyd, John P. 1990. Social Semigroups A Unified
Theory of Scaling and Blockmodeling as Applied to
Social Networks. Fairfax, VA George Mason
University Press. Boorman Scott A. and Harrison
White C. 1976. Social Structure from Multiple
Networks, II Role Structures. American Journal
of Sociology 811384-1446. Breiger, Ronald L.
1974. The Duality of Persons and Groups. Social
Forces 53181-190. Breiger, Ronald L. and
Philippa E. Pattison. 1978. The Joint Role
Structure of Two Communities Elites.
Sociological Methods Research
7213-226. Knoke, David, Franz Urban Pappi,
Jeffrey Broadbent and Yutaka Tsujinaka. 1996.
Comparing Policy Networks Labor Politics in the
U.S., Germany, and Japan. New York Cambridge
University Press. Krackhardt, David. 1999. Ties
That Torture Simmelian Tie Analysis in
Organizations. Research in the Sociology of
Organizations 16183-210. Moreno, Jacob L. 1934.
Who Shall Survive? Washington, DC Nervous and
Mental Disease Publishing Co. Pattison, Philippa
E. 1993. Algebraic Models for Social Networks.
New York Cambridge University Press. Wasserman,
Stanley and Katherine Faust. 1994. Social Network
Analysis Methods and Applications. New York
Cambridge University Press.