STRUCTURE AND STABILITY IN WEIGHTED DIGRAPH MODELS *

Many of the problems our society faces are concerned with large, complex, and difficult-to-understand systems. This is true of problems involving energy supply and demand, food supply, transportation, pollution, urban services, health care, and so on. The application of mathematical techniques, so frequently successful in dealing with small, well-defined, and mechanistic physical or engineering problems, is not always very helpful in dealing with these more complex systems. However, mathematics has a role to play, if we are more modzst in what we expect to learn about a system whose behavior is being studied mathematically. In this paper, we shall describe some techniques which are part of a general approach to the study of complex systems known as structural modeling. In structural modeling, we concern ourselves mostly with qualitative predictions about systems. We are mostly interested in learning about “geometric” properties of these systems, properties which describe their type of growth, their stability or instability, their sensitivity, and so on, in qualitative terms: they grow without bound, they oscillate wildly, etc. More specifically, we are interested in relating these qualitative properties of a system to certain structural properties of the system. Techniques of structural modeling have been applied to such diverse problems as energy use, air pollution, and transportation systems,’z*22-27*29 health care delivery, water policy, and environmental policy,7-’o naval manpower,” the analysis of coastal resources and the transportation of coal in inland and the study of ecosystem^.'^"^ Recent surveys of structural modeling have been carried out by Cearlock? and Lendaris and Wakeland.I3 We shall concentrate on the technique of pulse process analysis developed by R ~ b e r t s , ~ ’ ~ ~ and Roberts and and generalized by McLean16 and McLean and We shall solve some of the problems posed in Robertsz4 and extend some of the results of Roberts and Brown.29 We shall discuss applications of pulse process analysis to problems of health care delivery, energy use in food production, and transportation.

[1]  J. Kane A primer for a new cross-impact language— KSIM , 1972 .

[2]  Fred S. Roberts,et al.  Building and Analyzing an Energy Demand Signed Digraph , 1973 .

[3]  F. Roberts Structural characterizations of stability of signed digraphs under pulse processes , 1974 .

[4]  Ilan Vertinsky,et al.  Health care delivery: a policy simulator , 1972 .

[5]  T. B. Boffey,et al.  Applied Graph Theory , 1973 .

[6]  R Levins,et al.  DISCUSSION PAPER: THE QUALITATIVE ANALYSIS OF PARTIALLY SPECIFIED SYSTEMS , 1974, Annals of the New York Academy of Sciences.

[7]  D. Pimentel,et al.  Food Production and the Energy Crisis , 1973, Science.

[8]  Ilan Vertinsky,et al.  KSIM: A methodology for interactive resource policy simulation , 1973 .

[9]  Thomas A. Brown,et al.  Signed Digraphs and the Energy Crisis , 1975 .

[10]  F. Roberts Discrete Mathematical Models with Applications to Social, Biological, and Environmental Problems. , 1976 .

[11]  Wai-Kai Chen On Directed Graph Solutions of Linear Algebraic Equations , 1967 .

[12]  Fred S. Roberts,et al.  Signed Digraphs and the Growing Demand for Energy , 1971 .

[13]  F. Harary A Graph Theoretic Method for the Complete Reduction of a Matrix with a View Toward Finding its Eigenvalues , 1959 .

[14]  Mick McLean,et al.  The importance of model structure , 1976 .

[15]  Fred S. Roberts,et al.  Weighted Digraph Models for the Assessment of Energy Use and Air Pollution in Transportation Systems , 1975 .

[16]  J. M. McLean,et al.  Feedback processes in dynamic models , 1978 .

[17]  W. R. Lynn,et al.  Workshop on research methodologies for studies of energy, food, man, and environment. Phase II , 1974 .