ABSTRACT
The Galois lattice (Birkhoff 1967) is recommended for the analysis of two-mode social networks (Freeman and White 1994, Duquenne 1991). Mohr et al. provide a good summary of the two-mode property as it relates to the lattice: “A Galois lattice, however, has the special property of representing two orders of information in the same structure such that every point contains information on both logical orders simultaneously” (Mohr et al. 2004: 10). A Galois lattice can be viewed as the unfolding of the structure of multidimensional, two-mode binary data. In the lattice-based method of this paper, one mode is a set of n artifact types A = {a1, a2, ... an} and the other mode is the set C = {c1, c2, ... cm} of m cliques, each of which comprises a subset of three or more of the artifact types in set A. A membership relation I ≤ AxC links these two sets (see Merrill and Read 2010). When an artifact type ai belongs to a specifi c clique cj, it follows that (ai,cj) ε I and 0 otherwise. The membership relations between artifact types and cliques can therefore be represented by a nxm binary matrix M, where for any element mij in M, mij = 1 if (ai, cj) ε I and 0 otherwise. The mathematics needed to unfold and graphically show the complete two-mode structure of the clique-artifact type binary matrix as a Galois lattice are beyond the scope of this chapter. The mathematical details of the theory, representation, as well as examples of applications of Galois lattices in the analysis of two mode binary data are available elsewhere (Davey and Priestley 2002, Duquenne 1991, 1999:419-428, Wille 1982, 1984).
