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):
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 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
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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SOC8412 Social Network Analysis Fall 2009