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The fault tolerance of random graphs with unbounded degrees with respect toconnectivity is investigated, which relates to the reliability of wirelesssensor networks with unreliable relay nodes. The model evaluates the networkbreakdown probability that a graph is disconnected after stochastic noderemoval. To establish a mean-field approximation for the model, we propose thecavity method for finite systems. The analysis enables us to obtain anapproximation formula for random graphs with any number of nodes and anarbitrary degree distribution. In addition, its asymptotic analysis revealsthat the phase transition occurs in semi-dense random graphs whose averagedegree grows logarithmically. These results, which are supported by numericalsimulations, coincide with the mathematical results, indicating successfulpredictions by mean-field approximation for unbounded but not dense randomgraphs.

Fault Tolerance of Random Graphs with respect to Connectivity: Mean-field Approximation for Semi-dense Random Graphs