On Some Stability Parameters Of Networks

The study of stability (also called vulnerability), in particular the stability of networks, has received considerable progress in applicability from simple networks to computer networks, communication systems, and distribution systems. How to improve the stability of a communication network, composed of processing nodes and communication links, is of prime importance to network designers. As the network begins losing links or nodes, eventually there is a loss in its effectiveness. Obviously, a good communication network must have at least two properties: (1) it is not to be destroyed easily; (2) it can be easily reconstructed and continue its work after being destroyed. Thus, communication networks must be constructed to be as stable as possible, not only with respect to the initial disruption, but also with respect to the possible reconstruction of the network.A network can be modeled by a connected graph, a vertex in the graph represents a node of network, an edge represents the communication link between two nodes. It is obvious that the higher the connectivity of a graph is, the more stable the network which is represented by the above graph is. From above discussion, many graph theoretical parameters have been used in the past to describe the stability of communication networks. Most notably, the vertex-connectivity and edge-connectivity have been frequently used. Menger et al. have done many useful work about these two parameters. But, along with the research, we find that these two parameters have insufficiency in measuring the stability of networks, since these two parameters do not take into account what remains after the graph is disconnected. Consequently, a number of other parameters have been introduced that attempt to cope with this difficulty, including toughness and edge-toughness, integrity and edge-integrity, tenacity and edge-tenacity, scattering number and rupture degree. Unlike the connectivity measures, each of these parameters shows not only the difficulty to break down the network but also the damage that has been caused. We call these parameters the "standard stability parameters".