We prove new lower bounds on the modularity of graphs. Specifically, the modularity of a graph $G$ with average degree $\bar d$ is $\Omega(\bar{d}^{-1/2})$, under some mild assumptions on the degree sequence of $G$. The lower bound $\Omega(\bar{d}^{-1/2})$ applies, for instance, to graphs with a power-law degree sequence or a near-regular degree sequence. It has been suggested that the relatively high modularity of the Erd\H{o}s-R\'enyi random graph $G_{n,p}$ stems from the random fluctuations in its edge distribution, however our results imply high modularity for any graph with a degree sequence matching that typically found in $G_{n,p}$. The proof of the new lower bound relies on certain weight-balanced bisections with few cross-edges, which build on ideas of Alon [Combinatorics, Probability and Computing (1997)] and may be of independent interest.

翻译：我们证明了图模度的新下界。具体地，在图的度序列满足某些温和假设的条件下，平均度为$\bar d$的图$G$的模度为$\Omega(\bar{d}^{-1/2})$。下界$\Omega(\bar{d}^{-1/2})$适用于例如具有幂律度序列或近似正则度序列的图。此前有观点认为，Erd\H{o}s-R\'enyi随机图$G_{n,p}$相对较高的模度源于其边分布的随机波动，然而我们的结果表明，对于任何具有与$G_{n,p}$典型度序列相匹配的度序列的图，都存在高模度。新下界的证明依赖于具有少量交叉边的特定权重平衡二分划分，该划分基于Alon [组合学、概率与计算 (1997)] 的思想，且可能具有独立意义。