The rationality of four metrics of network robustness: A viewpoint of robust growth of generalized meshes

There are quite a number of different metrics of network robustness. This paper addresses the rationality of four metrics of network robustness (the algebraic connectivity, the effective resistance, the average edge betweenness, and the efficiency) by investigating the robust growth of generalized meshes (GMs). First, a heuristic growth algorithm (the Proximity-Growth algorithm) is proposed. The resulting proximity-optimal GMs are intuitively robust and hence are adopted as the benchmark. Then, a generalized mesh (GM) is grown up by stepwise optimizing a given measure of network robustness. The following findings are presented: (1) The algebraic connectivity-optimal GMs deviate quickly from the proximity-optimal GMs, yielding a number of less robust GMs. This hints that the rationality of the algebraic connectivity as a measure of network robustness is still in doubt. (2) The effective resistace-optimal GMs and the average edge betweenness-optimal GMs are in line with the proximity-optimal GMs. This partly justifies the two quantities as metrics of network robustness. (3) The efficiency-optimal GMs deviate gradually from the proximity-optimal GMs, yielding some less robust GMs. This suggests the limited utility of the efficiency as a measure of network robustness.