Algebraic and Topological Properties of Connecting Networks

Abstract

A connecting network is an arrangement of switches and transmission links allowing a certain set of terminals to be connected together in various combinations, usually by disjoint chains (paths): e.g., a central office, toll center, or military communications system. Some of the basic combinatory properties of connecting networks are studied in the present paper. Three of these properties are defined informally as follows: A network is rearrangeable if, given any set of calls in progress and any pair of idle terminals, the calls can be reassigned new routes (if necessary) so as to make it possible to connect the idle pair. A state of a network is a blocking state if some pair of idle terminals cannot be connected. A network is nonblocking in the wide sense if by suitably choosing routes for new calls it is possible to avoid all the blocking states and still satisfy all demands for connection as they arise, without rearranging existing calls. Finally, a network is nonblocking in the strict sense if it has no blocking states. A distance between states can be defined as the number of calls one would have to add or remove to change one state into the other. This distance defines a topology on the set of states. Also, the states can be partially ordered by inclusion in a natural way. This partial ordering and its dual define two more topologies for the set of states. The three topologies so obtained are used to characterize (i.e., give necessary and sufficient conditions for the truth of) the three properties of rearrangeability, nonblocking in the wide sense, and nonblocking in the strict sense. Each of these three properties represents a degree of abundance of nonblocking states; the mathematical concept used to express these degrees is the topological notion of denseness.