Relational Algebras
Based on Wasserman and Faust (1994) Chapter 11
Relational algebras, also called role algebras, are methods for analyzing the
structure of social roles by emphasizing multiple relations rather than actors. Role
structures describe how relations are associated in the network, independent of the
actors occupying those roles. Harrison White and his students pioneered this
approach in the 1970s as an extension to blockmodeling. Relational algebras
involve perhaps the most complex mathematics in network analysis, requiring
intensive study to attain full comprehension.
RELATIONAL ALBEBRA – a formal structure consisting of sets of relations and
operations to manipulate those relations.
Relational algebra can trace the indirect connections across multirelational
networks. For a social structural analysis, the two required elements are:
(1) Dichotomous primitive relations (generator relations) among actors,
represented by capital letters (e.g., U,T)
(2) The composition operation () that combines two or more primitive
relations. Compound relation TU results from the composition operation,
where tie i(T  U)j occurs if there exists some actor k such that iTk and kUj
Consider these graphs and matrices representing the advice-giver (A) and
friendship (F) images for a three-position blockmodel of Krackhardt’s high-tech
managers (W&F:439-442). The circling arrows show that high densities of ties
occur among the individual actors within some jointly-occupied blocks:
ADVICE (A):
a
b
A
a
b
c
FRIENDSHIP (F):
●
a
●
a
0
0
0
c
c
●
●
b
b
1
1
0
●
c
1
1
0
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F
a
b
c
a
1
0
1
b
0
0
1
c
1
0
1
FORMING COMPOUNDS
A compound relation can be formed by the Boolean multiplication of two or
sociomatrices. Boolean multiplication resembles to ordinary matrix multiplication,
except that any cell entry greater than “0” is replaced by a “1”. A nonzero entry
means that a compound relationship exist between that pair of actors/blocks.
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.
In the example above, where the two UCINET binary matrices are named
Advice & Friendship, the Boolean matrix product AF uses the command:
“AF=bprod(Advice,Friendship)” Here’s the result:
AF
a
b
c
a
1
1
0
b
1
1
0
c
1
1
0
Each 1 entry in AF identifies an advice-giving block connected via one
intermediary position to a friendship block . For example, in the advice
network, block a gives advice to block c (aAc), while in the friendship network
block c cites block b as its friend (cFb). Hence, the AF compound reveals a
connection from block a to block b: (aAc)(cFb) = (aAFb).
Note that nonzero diagonal blocks in the image matrices are always used
when composing a compound. Thus, block b’s compound tie to block c
involves the latter’s within-block friendships: (bAc)(cFc) = (bAFc).
Because square images are conformable for matrix multiplication in reverse order,
several distinct compound relations can be formed, including some self-multiplied
compounds.
The FA product shows how one block can use their friends to pass advice to
another 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
0
0
b
1
0
1
c
1
0
1
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The FF product identifies that classic compound relationship “the friend of a
friend”. Notice that, because block b cites no direct friends, it also can’t reach
any indirect friends. However, the other blocks have friends who are
connected to other friends: (aFc)(cFb) = (aFFb).
FF
a
b
c
a
1
0
1
b
1
0
1
c
1
0
1
Finally, the AA composition yields the “advisors of advisors” (evidently the
experts’ mavens). But this compound matrix is identical to the original advicegiving ties! The relationship AA=A reveals that this advising network is
transitive; for example, the compound (aAb)(bAc)=(aAAc) which is already a
direct tie (aAc). Hence, we don’t need AA because A encompasses all the
compound relations in its direct advising ties.
AA
a
b
c
a
0
0
0
b
1
1
0
c
1
1
0
ROLE TABLES for RELATIONS
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. Role algebraists inductively produce a “dictionary” containing
the unique words (matrices/images), with the fewest letters, required for a complete
description of a multiple-network system’s 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 word’s
sociomatrix. Words with identical matrices or images are equivalent, and the set of
all words with identical images comprise an equivalence class.
For the Krackhardt high-tech images, the shortest unique words in the
dictionary are A, F, AF, FA, and FF. See W&F440 for examples of longer
words in those equivalence classes.
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
W&F show Fig. 11.2), each equivalence class in the table is labeled by the graph’s
word. The cell entries in the table contain the smallest word that results from
multiplying the row matrix by the column matrix.
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 Matrix multiplication is associative: the order of performing successive
multiplications does not affect the result: ABC=(AB)C=A(BC)
 Matrix multiplication is not commutative: the result of multiplying two
matrices may differ by the sequence: 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, 1992, 2000; Pattison 1993). If all compositions of the primitive relations are
also members of the set, then a semigroup is closed under associative matrix
multiplication (437: the role table contains “all possible images that can result from
the operation of composition on the primitive relations”).
The role table for advice and friendship (W&F Fig. 11.5) shows that
composing any pair of the five unique words yields four of these words. For
example, multiplying (AF)(FA) = (AFFA). But, the first three terms 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 initial (AF)(FA) product reduces to just (A), as displayed
in row 3 column 4.

A
F
AF
FA
FF
A
F
AF
FA
FF
A
FA
A
FA
FA
AF
FF
AF
FF
FF
AF
FF
AF
FF
FF
A
FA
A
FA
FA
FF
FF
AF
FF
FF
Note that rows 1 & 3 are identical, as are rows 2, 4, & 5, as well as two
column pairs, 1 & 4 and 2 & 3. These identities imply opportunities for further
simplification of the social structure represented in Krackhardt high-tech
manager role table.
SIMPLIFYING a ROLE TABLE
Role table simplification involves reducing the number of network images or words
but 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.” (W&F:443). Unfortunately, a unique or “best” reduction may not be possible for
some networks.
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Image simplification strategies include: (1) substantive approaches that combine
images with the identical meaning or similar operation; (2) sociometric approaches
equate images with similar ties that 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 W&F:444-445 for an
application of the latter technique to the A&F role table.
A homomorphic reduction of an original role table involves a mapping that
preserves the composition operation. More than one homomorphic image may
exist.
One homomorphic image for the A&F role table permutes and partitions the
original 5x5 multiplication table into two groups of rows (and corresponding
columns) 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).
An alternative homomorphic image groups the columns {1,4} and {2,3,5}, with
word equivalences {A,FA} and {F,AF,FF}. Note that 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 the composition” (448).
COMPARING ROLE STRUCTURES
If the same or comparable relations are measured for two or more network
systems, their role tables can be compared to describe and measure their formal
similarities and/or differences. Boorman and White (1976) proposed the joint
homomorphic reduction of two role structures as a method for summarizing
common features (see W&F:451-460). JHR 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. Breiger and
Pattison (1978) interpreted the joint homomorphic reduction of three networks
among political elites in a German town and a U.S. city as an instance of
Granovetter’s strong-weak tie hypothesis.
REFERENCES
Breiger, Ronald L. and Philippa E. Pattison. 1978. “The Joint Role Structure of Two Communities’ Elites.” Sociological
Methods & Research 7:213-226.
Boyd, John P. 1990. Social Semigroups: A Unified Theory of Scaling and Blockmodeling as Applied to Social Networks.
Fairfax, VA: George Mason University Press.
Boyd, John P. 1992. “Relational Homomorphisms.” Social Networks 14:163-186.
Boyd, John P. 2000. “Social Networks and Semigroups.” Politica y Sociedad 33:105-112.
Pattison, Phillipa. 1993. Algebraic Models for Social Networks. New York: Cambridge University Press.
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