  The fault tolerance of random graphs with unbounded degrees with respect to
connectivity is investigated, which relates 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, we propose the
cavity method for finite systems. The analysis enables us to obtain an
approximation formula for random graphs with any number of nodes and an
arbitrary degree distribution. In addition, its asymptotic analysis reveals
that the phase transition occurs in semi-dense random graphs whose average
degree grows logarithmically. These 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.
