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.