模块度与Louvain社区发现算法
来源:互联网 发布:datamap 软件 编辑:程序博客网 时间:2024/06/14 01:23
Louvain算法是基于模块度的社区发现算法,该算法在效率和效果上都表现较好,并且能够发现层次性的社区结构,其优化目标是最大化整个社区网络的模块度。
模块度(Modularity )
模块度是评估一个社区网络划分好坏的度量方法,它的物理含义是社区内节点的连边数与随机情况下的边数只差,它的取值范围是 [−1/2,1),其定义如下:
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/0051.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/003D.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/0031.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/0032.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/006D.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Size2/Regular/400/2211.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/0069.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/283/002C.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/006A.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/005B.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/0041.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/0069.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/006A.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/2212.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/006B.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/0069.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/006B.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/006A.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/0032.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/006D.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/005D.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/03B4.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/0028.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/0063.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/0069.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/002C.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/0063.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/006A.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/0029.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/03B4.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/0028.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/0075.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/002C.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/0076.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/0029.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/003D.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/007B.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/283/0031.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/0077.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/0068.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/0065.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/006E.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/283/00A0.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/0075.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/283/003D.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/283/003D.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/0076.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/283/0030.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/283/00A0.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/0065.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/006C.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/0073.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/0065.png?rev=2.6.1)
其中,
公式中
模块度的公式定义可以作如下简化:
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/0051.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/003D.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/0031.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/0032.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/006D.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Size2/Regular/400/2211.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/0069.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/283/002C.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/006A.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/005B.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/0041.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/0069.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/006A.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/2212.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/006B.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/0069.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/006B.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/006A.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/0032.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/006D.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/005D.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/03B4.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/0028.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/0063.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/0069.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/002C.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/0063.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/006A.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/0029.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/003D.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/0031.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/0032.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/006D.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/005B.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Size2/Regular/400/2211.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/0069.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/283/002C.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/006A.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/0041.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/0069.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/006A.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/2212.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Size1/Regular/400/2211.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/0069.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/006B.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/0069.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Size1/Regular/400/2211.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/006A.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/006B.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/006A.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/0032.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/006D.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/005D.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/03B4.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/0028.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/0063.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/0069.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/002C.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/0063.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/006A.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/0029.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/003D.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/0031.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/0032.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/006D.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Size2/Regular/400/2211.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/0063.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/005B.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/03A3.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/0069.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/006E.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/2212.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/0028.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/03A3.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/0074.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/006F.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/0074.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/0029.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/283/0032.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/0032.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/006D.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/005D.png?rev=2.6.1)
其中
上面的公式还可以进一步简化成:
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/0051.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/003D.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Size2/Regular/400/2211.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/0063.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/005B.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/03A3.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/0069.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/006E.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/0032.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/006D.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/2212.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/0028.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/03A3.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/0074.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/006F.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/0074.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/0032.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/400/006D.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/0029.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/283/0032.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/005D.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/003D.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Size2/Regular/400/2211.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Math/Italic/283/0063.png?rev=2.6.1)
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/005B.png?rev=2.6.1)
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这样模块度也可以理解是社区内部边的权重减去所有与社区节点相连的边的权重和,对无向图更好理解,即社区内部边的度数减去社区内节点的总度数。
基于模块度的社区发现算法,都是以最大化模块度Q为目标。
Louvain算法
Louvain算法的思想很简单:
1)将图中的每个节点看成一个独立的社区,次数社区的数目与节点个数相同;
2)对每个节点i,依次尝试把节点i分配到其每个邻居节点所在的社区,计算分配前与分配后的模块度变化
3)重复2),直到所有节点的所属社区不再变化;
4)对图进行压缩,将所有在同一个社区的节点压缩成一个新节点,社区内节点之间的边的权重转化为新节点的环的权重,社区间的边权重转化为新节点间的边权重;
5)重复1)直到整个图的模块度不再发生变化。
从流程来看,该算法能够产生层次性的社区结构,其中计算耗时较多的是最底一层的社区划分,节点按社区压缩后,将大大缩小边和节点数目,并且计算节点i分配到其邻居j的时模块度的变化只与节点i、j的社区有关,与其他社区无关,因此计算很快。在论文中,把节点i分配到邻居节点j所在的社区c时模块度变化为:
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/0394.png?rev=2.6.1)
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其中
在实现的时候模块度变化还可以简化,把上面的公式展开,很多项就抵消了,化简之和:
![](http://mathjax.cnblogs.com/2_6_1/fonts/HTML-CSS/TeX/png/Main/Regular/400/0394.png?rev=2.6.1)
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论文中指出,算法第2)步节点的顺序会对分群结果又一定影响,但分群效果差距不大,只是会影响算法的时间效率,还有论文指出按度数从到的小的顺序处理速度最快,不过我在1K边上的图测试,差距不大。
分布式实现
我实现了Louvain算法的Spark版本和单机版,单机版在算法的第2)步中,节点i的社区变更会在节点i+1的社区分配时可见,但分布式实现,这一点不能满足,因为在分布式环境下,节点i和节点i+1的社区变更可能是在不同机器上同时进行,不能进行实时传递,因此在分布式实现时,节点i+1看到只能是节点i的上一轮的社区,因为这个原因,会导致一些节点互换社区,比如1号节点分到了2号节点所在的社区,而2号节点又分配到了节点1所在的社区,解决这个问题我是参考淘宝的一个实现,用节点id和和社区id构成的边组成新图,再用联通图来调整节点的社区。
最后,工作了不能像上学的时候随意把代码放出来了,So.没有源代码分享~
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