Notation for Social Network Data
Based on Wasserman and Faust (1994) Chapter 3
Graph theoretic notation looks formidable and requires some persistence to
decipher. Don’t worry if its complexity initially seems overwhelming – the notation
system is designed to allow precise communication among specialists. We can
get by with a few basic symbols and some illustrations.
Denote a graph G for a single relation, consisting of two elementary sets (N,L)
– nodes connected by lines, which may be either directed or nondirected:
A set N of g nodes (actors):
N = {n1, n2, … ng}
A set of L lines either joining ordered pairs of nodes <ni, nj> – where ni is
the sender/chooser and nj is the receiver/chosen – or joining nonordered
pairs of nodes (ni, nj): L = {l1, l2, … lg}
Not every possible line connecting the nodes may actually exist; graphs almost
always have some or many null relations. A directed graph (digraph) contains a
maximum of (g)(g-1) = g2 – g lines and a nondirected graph has a maximum of
(g2 – g)/2 nonordered pairs (assuming that self-choices are inapplicable).
Denote multiple relations by a set R comprised of two or more sets of lines for
a single set of nodes. For example, one set might represent the “works with”
relation and a second set might represent the “friends with” relation. The example
graphs below apply only to a single relation.
Graph diagrams (sociograms) of a digraph depict labeled points for the nodes
and arrowhead arcs that indicate the line directions, which could represent
choice, exchange, liking, etc. (nondirected graphs have no arrowheads):
HARRY
SALLY
●
●
● TOM
DICK
●
SOC8412 Social Network Analysis Fall 2009
● BETTY
SOC8412 Social Network Analysis Fall 2009
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A sociomatrix X is a gXg (“g by g”) rectangular array of g sending nodes in the
rows, g receiving nodes in the columns, and (g)(g-1) = g2 – g matrix elements (xij)
consisting of a numerical value for the <i,j> ordered pair’s relation. Values may
be binary (0-1 dichotomy), discrete (± integers), or continuous variables that
represent the strength, intensity, or frequency of the dyadic relations. The nodes
appear in the same sequence in both rows and columns.
The binary adjacency matrix of the 5-actor diagram above (where the 0s on the
main diagonal indicate that self-choices aren’t allowed) is:
Betty
Dick
Harry
Sally
Tom
Betty Dick
0
1
1
0
0
1
0
0
0
0
Harry Sally
1
0
1
1
0
0
1
0
0
0
Tom
0
0
1
0
0
For each of R multiple relations, separate graphs and matrices would be
constructed. Later chapters discuss how to analyze multiple matrices
simultaneously.
Analysts can manipulate adjacency matrices using matrix algebra procedures
(such as symmetrizing, transposing, normalizing, adding, or multiplying) to reveal
additional network properties.
W&F (pp. 85-89) discuss special notation for relations connecting two different
sets of nodes; for example, two-mode relations such as persons attending
political events, or conglomerate firms operating factories in various industries.
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