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 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).

翻译：尽管由于复杂性，对环路的研究尚不充分，但已有研究发现，启发式地增强环路可通过降低度分布的方差来提升连接鲁棒性。尽管已知许多真实的无标度网络包含较短环路（如三角形），但在更广泛的网络类别中研究较长环路的分布仍有待深入。通过研究具有连续变化度分布（包括三种典型无标度网络、经典Erdős-Rényi随机图及规则网络）的各类网络中最短环路长度，我们发现了更窄的度分布与较长环路之间的关联。特别地，我们证明更窄的度分布包含更长的最短环路，这是广泛随机网络类别中的普适性质。我们提出通过构建长度为O(log N)的长环路可增强连接鲁棒性。