Fault Tolerance of Random Graphs with respect to Connectivity: Phase Transition in Logarithmic Average Degree

The fault tolerance of random graphs with unbounded degrees with respect to connectivity is investigated. It is related to the reliability of wireless sensor networks with unreliable relay nodes. The model evaluates the network breakdown probability that a graph is disconnected after stochastic node removal. To establish a mean-field approximation for the model, the cavity method for finite systems is proposed. Then the asymptotic analysis is applied. As a result, the former enables us to obtain an approximation formula for any number of nodes and an arbitrary and degree distribution. In addition, the latter reveals that the phase transition occurs on random graphs with logarithmic average degrees. Those results, which are supported by numerical simulations, coincide with the mathematical results, indicating successful predictions by mean-field approximation for unbounded but not dense random graphs.