ボトルネック法による強連結グラフの分割法とその応用

抄録

The macroscopic structures of large scale systems are conveniently expressed with directed graphs. But it is still hard to understand the essential points if the graphs are too much complex. One of the best ways of understanding a complex system is to decompose it into some subsystems and to study the interrelation among them. Though many algorithms (4)∼(6) have been suggested for the decomposition of non-strongly connected graphs, there is no effective way for strongly connected graphs except Steward's method⁷⁾. Steward's method suggested for solving large scale simultaneous equations is not necessarily suitable for clustering strongly connected graphs.
In this paper, the authors suggest a new algorithm of decomposition of strongly connected graphs. Its principle is to make the graph non-strongly connected by eliminating a few bottle neck arcs. The grade of bottle neck of any arc is represented by an index that is the number of the shortest paths going though the arc. The shortest paths are the routes which connect any couples of nodes in a graph with the minimum number of arcs. If the index of an arc is large, the structure of the graph must be changed greatly when the arc is eliminated. Therefore, we can change a strongly connected graph into a non-strongly connected one by elimating some of the bottle neck arcs and also find a hierarchy structure consisting of few numbers of subsystems by using ordinary methods. After that the eliminated arcs are retouched and the decomposition is completed.
This method is applied to two examples, one of which is the procedure of solving a large scale simultaneous equations and another is the identification of the structure of complex urban social problems. The results are quite satisfactory.