Reliability Study Of 2-terminal And 3-terminal Networks Under Edge Failure

The whole complex system may not operate normally due to the failure of the vertices or edges of the network in life,which will cause great inconvenience and loss to human life,such as traffic networks,computer networks,communication networks,military command networks and so on.The research on network reliability provides theoretical guarantee for the normal operation of the network.Therefore,it is of great significance and wider application value to analyze the reliability of the network and design a more reliable network structure.At present,researchers have studied the reliability of networks from the perspectives of analysis and design.The former prefers to calculate the probability that the network can operate normally when the vertex or line fails,and the latter prefers to design the structure that the network can operate normally with the maximum probability when the fault occurs.In fact,in many real networks,due to the special purpose of the network,it is only necessary to ensure the normal operation of some key vertices in the network.For example,in the transportation network,it is only necessary to ensure the mutual accessibility between the shipping source and the receiving source.Therefore,the reliability analysis and reliability design of k-terminal network is of practical significance.This thesis mainly investigates the locally most reliability and uniformly most reliability of 2-terminal and 3-terminal networks if all vertices survive and each edge of the network fails independently with a fixed probability.The main research work and results are as follows:(1)When the edge failure probabilities are large and small,according to the results of the locally most reliable 2-terminal graphs,many classes of 2-terminal graphs without uniformly most reliable structure are constructed.If any two vertices in the target vertex layer are directly connected,and each target vertex and each non target vertex are also connected by an edge,the first non-zero coefficient of the 2-terminal unreliability polynomial is given.In addition,based on the above complete connection,when the induced subgraphs of non-target vertices are connected,the study of the locally most reliable 2-terminal graph can be transformed into finding the all-terminal graph with maximum edge connectivity and the minimum first non-zero coefficient of the all-terminal unreliability polynomial.(2)When the failure probability of each edge is large,four local reliability comparison criteria are given,and the theoretical framework is established.An effective search method is given by these criteria,which reduces the research scope and characterizes some locally most reliable 3-terminal graphs.A simple simulation experiment is carried out for the construction of the locally most reliable structure.(3)When the edge failure probability is very small,three local reliability comparison criteria are given,and the theoretical framework is established.An effective search method is given by these criteria,which reduces the research scope and characterizes some locally most reliable 3-terminal graphs.A simple simulation experiment is carried out for the construction of the locally most reliable structure.(4)The uniformly most reliable 3-terminal graphs with edge failure is studied,and many classes of 3-terminal graphs without uniformly most reliable structure in case of edge failure are determined.By constructing the injectivity between corresponding connected subgraphs,it is proved that deleting an edge between non-target vertices of a complete graph is the uniformly most reliable 3-terminal graph.A simple simulation experiment is carried out to construct the uniformly most reliable structure.