Larger holes as narrower degree distributions in complex networks

Although the analysis of loops is not so much because of the complications, it has already been found that heuristically enhancing loops decreases the variance of degree distributions for improving the robustness of connectivity. While many real scale-free networks are known to contain shorter loops such as triangles, it remains to investigate the distributions of longer loops in more wide class of networks. We find a relation between narrower degree distributions and longer loops in investigating the lengths of the shortest loops in various networks with continuously changing degree distributions, including three typical types of realistic scale-free networks, classical Erd\"os-R\'enyi random graphs, and regular networks. In particular, we show that narrower degree distributions contain longer shortest loops, as a universal property in a wide class of random networks. We suggest that the robustness of connectivity is enhanced by constructing long loops of O(log N).