关于3-正则随机图与具有给定度序列的随机图的模块性
On the modularity of 3-regular random graphs and random graphs with given degree sequences
AI总结：
该研究证明了随机3-正则图模块性的渐近几乎必然下界，验证了相关猜想并改进了上界，同时针对给定有界度序列的随机图，分亚临界、超临界区域推导了模块性的相关结果。
摘要：
图的模块性是衡量其社区结构的参数，取值在0到1之间，数值越高说明图的聚类程度越强。\n 本文证明了随机3-正则图的模块性在渐近几乎必然（a.a.s.）意义下至少为0.667026，从而证明了McDiarmid和Skerman提出的一个猜想；同时将该研究中给出的渐近几乎必然上界改进至0.789998。\n 对于从给定有界度序列中均匀选取的图$G_n$，设其平均度为$d(G_n)$，连通分量数目为$|CC(G_n)|$，本文根据巨分量的存在性将情况分为两种区域。在亚临界区域，计算了模块性的第二项；在超临界区域，证明存在$\varepsilon > 0$，使得模块性在渐近几乎必然意义下至少为$\dfrac{2\left(1 - μ\right)}{d(G_n)}+\varepsilon$，其中$μ$是$\dfrac{|CC(G_n)|}{n}$的渐近几乎必然极限。
英文摘要：
The modularity of a graph is a parameter that measures its community structure; the higher its value (between $0$ and $1$), the more clustered the graph is. In this paper we show that the modularity of a random $3$-regular graph is at least $0.667026$ asymptotically almost surely (a.a.s.), thereby proving a conjecture of McDiarmid and Skerman. We also improve the a.a.s. upper bound given therein to $0.789998$. For a uniformly chosen graph $G_n$ over a given bounded degree sequence with average degree $d(G_n)$ and with $|CC(G_n)|$ many connected components, we distinguish two regimes with respect to the existence of a giant component. In the subcritical regime, we compute the second term of the modularity. In the supercritical regime, we prove that there is $\varepsilon > 0$, for which the modularity is a.a.s. at least \begin{equation*} \dfrac{2\left(1 - μ\right)}{d(G_n)}+\varepsilon, \end{equation*} where $μ$ is the asymptotically almost sure limit of $\dfrac{|CC(G_n)|}{n}$.
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