Asymptotic estimates for efficiency, vulnerability and cost for random networks

An investigation both analytical and numerical of the values of efficiency, vulnerability and cost functions is provided. Several comparisons among these functions are made following a pairwise strategy, and taking into account different efficiency, vulnerability and cost functions. It is shown in all cases that these magnitudes display strong correlations, which are typically piecewise linear in log-space as well as universal (i.e., independent of the number of nodes n) to a great extent. The analytical part of the work develops explicit bounds linking all pairs of such parameters. These results are numerically compared with the result of a random testing for Erdös–Rényi networks.