Environment and Planning B, 1981, volume 8, pages 367-391 Graph theory and Q-analysis C F Earl, J H Johnson Centre for Configurational Studies, Faculty of Technology, The Open University, Milton Keynes MK7 6AA, England Received 9 November 1981; in revised form 29 November 1981 Abstract. Structures of graph theory are compared with those of Q-analysis and there are many similarities. The graph and simplicial complex defined by a relation are equivalent in terms of the information they represent, so that the choice between graph theory and g-analysis depends on which gives the most natural and complete description of a system. The higher dimensional graphs are shown to be simplicial families or complexes. Although network theory is very successful in those physical science applications for which it was developed, it is argued that Q-analysis gives a better description of human network systems as patterns of traffic on a backcloth of simplicial complexes. The ^-nearness graph represents the ^-nearness of pairs of simplices for a given ^-value. It is concluded that known results from graph theory could be applied to the #-nearness graph to assist in the investigation of g-connectivity, to introduce the notion of connection defined by graph cuts, and to assist in computation. The application of the q-nearness graph to #-transmission and shomotopy is investigated. 1 Introduction To date the literature on Q-analysis has virtually ignored graph theory, and graph theorists have taken little interest in Q-analysis. Whatever the reasons for this, it is argued that this mutual indifference inhibits a synthesis of results and insights which could be beneficial. Although it will be demonstrated that many of the mathematical structures of graph theory and Q-analysis are similar or identical, it is not implied that the methodology of Q-analysis and the methodology which applies graph theory to social systems are similar or even compatible. In the methodology of physical science, graph theory has been used highly successfully, for example, in the development of network theory for electrical systems and computer science. Graph theory has been less successful in social science, and one must ask the question as to whether a mathematical structure devised primarily for the study of electrical systems need be an appropriate structure (Gould, 1980) to describe, say, road transportation systems (Johnson, 1981b) or even large electrical power transmission and consumption systems (Gould, 1981). Applications of graph theory by social scientists for more than a decade have resulted in little of substance except results in combinatorial mathematics. In our view graph theory is highly successful for those physical science areas for which it was devised, it has been less successful when applied to complicated human systems, but that mathematicians have usefully abstracted and extended the structures born out of applications in the physical and social sciences. As a methodology for social science, Q-analysis has tried to embody the changes in methods of thinking that resulted in today's hard physical science (Atkin, 1974a; 1974b; 1974c; 1975; 1977; 1981). Although its mathematics is born out of the physical science mathematics of algebraic topology, empirical work has required that new and more appropriate mathematics be devised. In its many applications (see Atkin, 1978) Q-analysis has given deep insights into complicated human systems [for example, see Atkin's (1977) study of a University community, Gaspar and Gould's (1981) study of agricultural and communications systems, or Johnson's (1981b) study