A graph theoretic approach to the investigation of system‐environment relationships†

Work reported here was supported by Grant MH‐22743 from the National Institute of Mental Health.

For any empirical structure consisting of a system S and its environment E, there is an associated digraph D whose points and arcs (directed lines) correspond to the elements and relationships of the structure. The arcs of D are thus of four types: (1) internal arcs, which join two points of S; (b) external arcs, which join two points of E; (c) out‐liaisons of S, which join a point of S to one of E; and (d) in‐liaisons of S, which join a point of E to one of S. The boundary of S is defined as the subgraph of D induced by the liaisons of S and corresponds to those elements and relationships of the structure directly involved in transactions between the system and its environment. The basic structural properties of boundaries are then identified, and it is shown how the points of S and E can be stratified according to their distances to (or from) the boundary of S. Next, several results are derived concerning system‐environment relationships in structures whose digraphs are symmetric, transitive, or signed. The concept of convexity is then introduced to deal with a certain kind of segregation of a system relative to its environment. And, finally, it is shown how the adjacency matrix of D can be employed to facilitate the analysis of such structures and the calculation of various indexes of system‐environment relationships.

Notes

Work reported here was supported by Grant MH‐22743 from the National Institute of Mental Health.