In this paper a graph theoretical elaboration of the stochastic cumulative scaling model of Mokken (1970) is given to determine: (a) cumulative scales of vertices on the basis of their relations with other vertices in a simple graph; and (b) cumulative scales of relations in a multigraph.
In Stokman (1977) both graph theoretical elaborations are given and applied to (co-)sponsorship of resolutions in the United Nations General Assembly to determine leadership structures among developing nations. Felling (1974, pp. 270-95) elaborated the applicability of the deterministic cumulative scaling model of Guttman to determine cumulative scales of relations in multigraphs. Graph theoretical elaboration of the stochastic scaling model gives, however, certain new insights that were not considered by Felling.
In Section 1 we introduce a number of graph theoretical concepts that are used in the remainder of the paper. In Section 2 cumulative scales of vertices in a simple graph are treated, and in Section 3, cumulative scales of relations in a multigraph. After determination of the different cumulative dimensions of relations in a multigraph, for each dimension a new graph can be generated and analyzed. In Section 4 it is shown how this can be done. 1. Graph Theoretical Concepts [ 1 ] A graph is an object that contains vertices and edges, each edge being incident with one or two vertices [ 21. Let us consider, as an example, * Paper to be published in the proceedings of the Quantum-SSHA conference on Quantification and Methods in Social Science Research: Possibilities and Problems with the use of Historical and Process-Produced Data held at the