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

TL;DR

This paper develops a mean-field approximation for assessing the fault tolerance of semi-dense random graphs, particularly in wireless sensor networks, revealing phase transition behavior as network density increases.

Contribution

It introduces a cavity method-based mean-field approximation for semi-dense random graphs with arbitrary degree distributions, extending understanding of network robustness.

Findings

Phase transition occurs in semi-dense graphs with logarithmic average degree.

Mean-field approximation accurately predicts network disconnection probabilities.

Results are supported by numerical simulations aligning with mathematical analysis.

Abstract

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.