正则图和树型图的模块性
Modularity of regular and treelike graphs
摘要：
大型网络的聚类算法通常使用模块度值来测试顶点集的哪些分区更好地表示数据中的结构。图的模度是图的划分的最大模度。本文研究了两类图的模度。对于给定顶点数的$r$ -正则图，我们研究了最小可能模度、典型模度和最大可能模度。特别是，我们看到，随机三次图的模块通常是在间隔$(0.666, 0.804)$，和随机$r$ -正则图与大$r$它通常是顺序$1/\sqrt{r}$。这些结果有助于为正则图的统计检验建立基线。模块化的周期和低度树是已知的接近1：我们将这些结果的“树”图，其中的产品的树宽和最大程度是小于的边缘数。这产生例如上面提到的关于随机三次图的模块性的（确定性的）下界$0.666$。
英文摘要：
Clustering algorithms for large networks typically use modularity values to test which partitions of the vertex set better represent structure in the data. The modularity of a graph is the maximum modularity of a partition. We consider the modularity of two kinds of graphs. For $r$-regular graphs with a given number of vertices, we investigate the minimum possible modularity, the typical modularity, and the maximum possible modularity. In particular, we see that for random cubic graphs the modularity is usually in the interval $(0.666, 0.804)$, and for random $r$-regular graphs with large $r$ it usually is of order $1/\sqrt{r}$. These results help to establish baselines for statistical tests on regular graphs. The modularity of cycles and low degree trees is known to be close to 1: we extend these results to `treelike' graphs, where the product of treewidth and maximum degree is much less than the number of edges. This yields for example the (deterministic) lower bound $0.666$ mentioned above on the modularity of random cubic graphs.
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