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That is, a semipath in a directd graph from node x to node y is a sequence of nodes x = x09 xl9 .. , xk - y such that (xf _ l9 xf) or (x;, Xj_i) is an edge in the graph for each / = 1, 2, . . , k. A semipath from a vertex to itself is termed a semicycle. If there is a semipath from x to y for every pair of nodes x and y in the graph, then the graph is said to be weakly connected. 1, for instance, from node A to node D through node C; indeed, all pairs of vertices in the graph are connected by a semipath.

Tp] be a blockmodel on a set B of blocks (so that T is a multiple network on £), and let R = {Ru R2,... , Rp] be a multiple network on a set X = {1, 2 , . . , n}. Let f be a mapping from the set X of members of the multiple network R onto the set B of blocks in B. For each block b eB, let nb denote the number of elements / in X for which f(i) = b. Then T is DEFINITION. 1 a fat-fit to R if (b, c) e Tk iff (/, /) € Rk, for all /, / e X such that 2 a lean-fit to R if (b, c) e Tk iff (/, /) e Rk, for some z, j eX for which f(i) = b andf(j) = c; & = 1, 2 , .

Vertex and edge indices. Indices representing properties of each vertex or edge in a graph have also been proposed. , Harary, 1959b; Katz, 1953; Langeheine &C Andresen, 1982; Nieminen, 1973). The centrality of a vertex has been conceptualised in a number of ways (Freeman, 1979), for instance, in terms of its "closeness" to other vertices, the extent to which it lies on shortest paths between other vertices, and the number of other vertices to which it is directly connected (Freeman, 1979; also, Bolland, 1988; Donninger, 1986; Gould, 1987).

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