社团划分——Fast Unfolding算法

社团划分——Fast Unfolding算法

一、社区划分问题

1、社区以及社区划分

在社交网络中,用户相当于每一个点,用户之间通过互相的关注关系构成了整个网络的结构,在这样的网络中,有的用户之间的连接较为紧密,有的用户之间的连接关系较为稀疏,在这样的的网络中,连接较为紧密的部分可以被看成一个社区,其内部的节点之间有较为紧密的连接,而在两个社区间则相对连接较为稀疏,这便称为社团结构。

(Newman and Gievan 2004) A community is a subgraph containing nodes which are more densely linked to each other than to the rest of the graph or equivalently, a graph has a community structure if the number of links into any subgraph is higher than the number of links between those subgraphs.

如下图:

用红色的点和黑色的点对其进行标注,整个网络被划分成了两个部分,其中,这两个部分的内部连接较为紧密,而这两个社区之间的连接则较为稀疏。如何去划分上述的社区便称为社区划分的问题。

2、社区划分的算法

在社区划分问题中,存在着很多的算法,如由Newman和Gievan提出的GN算法,标签传播算法(Label Propagation Algorithm, LPA),这些算法都能一定程度的解决社区划分的问题,但是性能则是各不相同。总的来说,在社区划分中,主要分为两大类算法

  1. 凝聚方法(agglomerative method):添加边
  2. 分裂方法(divisive method):移除边

在后续的文章中,我们会继续关注不同的社区划分的算法,在这篇文章中,主要关注Fast Unfolding算法。

3、社区划分的评价标准

为了评价社区划分的优劣,Newman等人提出了模块度的概念,用模块度来衡量社区划分的好坏。简单来讲,就是将连接比较稠密的点划分在一个社区中,这样模块度的值会变大,最终,模块度最大的划分是最优的社区划分。

二、模块度的概念

1、模块度的公式

社区划分的目标是使得划分后的社区内部的连接较为紧密,而在社区之间的连接较为稀疏,通过模块度的可以刻画这样的划分的优劣,模块度越大,则社区划分的效果越好 ,模块度的公式如下所示:

Q=12mi,j[Ai,jkikj2m]δ(ci,cj)

Q=\frac{1}{2m}\sum_{i,j}\left [ A_{i,j}-\frac{k_ik_j}{2m} \right ]\delta \left ( c_i,c_j \right )

其中,m=12∑i,jAi,j m = \frac{1}{2}\sum_{i, j}A_{i,j} 表示的是网络中的所有的权重,Ai,j A_{i,j}表示的是节点i i和节点j j之间的权重,ki=∑jAi,j k_i=\sum_{j}A_{i,j}表示的是与顶点i i连接的边的权重,ci c_i表示的是顶点被分配到的社区,δ(ci,cj) \delta \left ( c_i,c_j \right )用于判断顶点i i与顶点j j是否被划分在同一个社区中,若是,则返回1 1,否则,返回0 0。

2、模块度公式的简化形式

上述的模块度的计算可以得到以下的简化形式:

Q=∑c⎡⎣∑in2m−(∑tot2m)2⎤⎦

Q=\sum_{c}\left [ \frac{\sum_{in}}{2m}-\left ( \frac{\sum_ {tot}}{2m} \right )^2 \right ]

其中,∑in \sum_{in}表示的是社区c c内部的权重,∑tot \sum_ {tot}表示的是与社区c c内部的点连接的边的权重,包括社区内部的边以及社区外部的边。

3、模块度公式的解释

模块度(modularity)指的是网络中连接社区结构内部顶点的边所占的比例,减去在同样的社团结构下任意连接这两个节点的比例的期望值。

三、Fast Unfolding算法

1、Fast Unfolding算法的思路

模块度成为度量社区划分优劣的重要标准,划分后的网络模块度值越大,说明社区划分的效果越好,Fast Unfolding算法便是基于模块度对社区划分的算法,Fast Unfolding算法是一种迭代的算法,主要目标是不断划分社区使得划分后的整个网络的模块度不断增大。

2、Fast Unfolding算法的过程

Fast Unfolding算法主要包括两个阶段,如下图所示:

第一阶段称为Modularity Optimization,主要是将每个节点划分到与其邻接的节点所在的社区中,以使得模块度的值不断变大;第二阶段称为Community Aggregation,主要是将第一步划分出来的社区聚合成为一个点,即根据上一步生成的社区结构重新构造网络。重复以上的过程,直到网络中的结构不再改变为止。

具体的算法过程如下所示:

  1. 初始化,将每个点划分在不同的社区中;
  2. 对每个节点,将每个点尝试划分到与其邻接的点所在的社区中,计算此时的模块度,判断划分前后的模块度的差值ΔQ \Delta Q是否为正数,若为正数,则接受本次的划分,若不为正数,则放弃本次的划分;
  3. 重复以上的过程,直到不能再增大模块度为止;
  4. 构造新图,新图中的每个点代表的是步骤3中划出来的每个社区,继续执行步骤2和步骤3,直到社区的结构不再改变为止。

注意:在步骤2中计算节点的顺序对模块度的计算是没有影响的,而是对计算时间有影响

四、算法实现

针对上图表示的网络,最终的结果为:

可以使用下面的程序实现其基本的原理:

import string

def loadData(filePath):
    f = open(filePath)
    vector_dict = {}
    edge_dict = {}
    for line in f.readlines():
        lines = line.strip().split("\t")

        for i in xrange(2):
            if lines[i] not in vector_dict:
                #put the vector into the vector_dict
                vector_dict[lines[i]] = True
                #put the edges into the edge_dict
                edge_list = []
                if len(lines) == 3:
                    edge_list.append(lines[1-i]+":"+lines[2])
                else:
                    edge_list.append(lines[1-i]+":"+"1")
                edge_dict[lines[i]] = edge_list
            else:
                edge_list = edge_dict[lines[i]]
                if len(lines) == 3:
                    edge_list.append(lines[1-i]+":"+lines[2])
                else:
                    edge_list.append(lines[1-i]+":"+"1")
                edge_dict[lines[i]] = edge_list

return vector_dict, edge_dict

def modularity(vector_dict, edge_dict):
    Q = 0.0
    # m represents the total wight
    m = 0
    for i in edge_dict.keys():
        edge_list = edge_dict[i]
        for j in xrange(len(edge_list)):
            l = edge_list[j].strip().split(":")
            m += string.atof(l[1].strip())

    # cal community of every vector
    #find member in every community
    community_dict = {}
    for i in vector_dict.keys():
        if vector_dict[i] not in community_dict:
            community_list = []
        else:
            community_list = community_dict[vector_dict[i]]

        community_list.append(i)
        community_dict[vector_dict[i]] = community_list

    #cal inner link num and degree
    innerLink_dict = {}
    for i in community_dict.keys():
        sum_in = 0.0
        sum_tot = 0.0
        #vector num
        vector_list = community_dict[i]
        #print "vector_list : ", vector_list
        #two loop cal inner link
        if len(vector_list) == 1:
            tmp_list = edge_dict[vector_list[0]]
            tmp_dict = {}
            for link_mem in tmp_list:
                l = link_mem.strip().split(":")
                tmp_dict[l[0]] = l[1]
            if vector_list[0] in tmp_dict:
                sum_in = string.atof(tmp_dict[vector_list[0]])
            else:
                sum_in = 0.0
        else:
            for j in xrange(0,len(vector_list)):
                link_list = edge_dict[vector_list[j]]
                tmp_dict = {}
                for link_mem in link_list:
                    l = link_mem.strip().split(":")
                    #split the vector and weight
                    tmp_dict[l[0]] = l[1]
                for k in xrange(0, len(vector_list)):
                    if vector_list[k] in tmp_dict:
                        sum_in += string.atof(tmp_dict[vector_list[k]])

        #cal degree
        for vec in vector_list:
            link_list = edge_dict[vec]
            for i in link_list:
                l = i.strip().split(":")
                sum_tot += string.atof(l[1])        
        Q += ((sum_in / m) - (sum_tot/m)*(sum_tot/m))
    return Q

def chage_community(vector_dict, edge_dict, Q):
    vector_tmp_dict = {}
    for key in vector_dict:
        vector_tmp_dict[key] = vector_dict[key]

    #for every vector chose it's neighbor
    for key in vector_tmp_dict.keys():
        neighbor_vector_list = edge_dict[key]
        for vec in neighbor_vector_list:
            ori_com = vector_tmp_dict[key]
            vec_v = vec.strip().split(":")

            #compare the list_member with ori_com
            if ori_com != vector_tmp_dict[vec_v[0]]:
                vector_tmp_dict[key] = vector_tmp_dict[vec_v[0]]
                Q_new = modularity(vector_tmp_dict, edge_dict)
                #print Q_new
                if (Q_new - Q) > 0:
                    Q = Q_new
                else:
                    vector_tmp_dict[key] = ori_com
    return vector_tmp_dict, Q

def modify_community(vector_dict):
    #modify the community
    community_dict = {}
    community_num = 0
    for community_values in vector_dict.values():
        if community_values not in community_dict:
            community_dict[community_values] = community_num
            community_num += 1
    for key in vector_dict.keys():
        vector_dict[key] = community_dict[vector_dict[key]]
    return community_num

def rebuild_graph(vector_dict, edge_dict, community_num):
    vector_new_dict = {}
    edge_new_dict = {}
    # cal the inner connection in every community
    community_dict = {}
    for key in vector_dict.keys():
        if vector_dict[key] not in community_dict:
            community_list = []
        else:
            community_list = community_dict[vector_dict[key]]

        community_list.append(key)
        community_dict[vector_dict[key]] = community_list

    # cal vector_new_dict
    for key in community_dict.keys():
        vector_new_dict[str(key)] = str(key)

    # put the community_list into vector_new_dict

    #cal inner link num
    innerLink_dict = {}
    for i in community_dict.keys():
        sum_in = 0.0
        #vector num
        vector_list = community_dict[i]
        #two loop cal inner link
        if len(vector_list) == 1:
            sum_in = 0.0
        else:
            for j in xrange(0,len(vector_list)):
                link_list = edge_dict[vector_list[j]]
                tmp_dict = {}
                for link_mem in link_list:
                    l = link_mem.strip().split(":")
                    #split the vector and weight
                    tmp_dict[l[0]] = l[1]
                for k in xrange(0, len(vector_list)):
                    if vector_list[k] in tmp_dict:
                        sum_in += string.atof(tmp_dict[vector_list[k]])

        inner_list = []
        inner_list.append(str(i) + ":" + str(sum_in))
        edge_new_dict[str(i)] = inner_list

    #cal outer link num
    community_list = community_dict.keys()
    for i in xrange(len(community_list)):
        for j in xrange(len(community_list)):
            if i != j:
                sum_outer = 0.0
                member_list_1 = community_dict[community_list[i]]
                member_list_2 = community_dict[community_list[j]]

                for i_1 in xrange(len(member_list_1)):
                    tmp_dict = {}
                    tmp_list = edge_dict[member_list_1[i_1]]

                    for k in xrange(len(tmp_list)):
                        tmp = tmp_list[k].strip().split(":");
                        tmp_dict[tmp[0]] = tmp[1]
                    for j_1 in xrange(len(member_list_2)):
                        if member_list_2[j_1] in tmp_dict:
                            sum_outer += string.atof(tmp_dict[member_list_2[j_1]])

                if sum_outer != 0:
                    inner_list = edge_new_dict[str(community_list[i])]
                    inner_list.append(str(j) + ":" + str(sum_outer))
                    edge_new_dict[str(community_list[i])] = inner_list
    return vector_new_dict, edge_new_dict, community_dict

def fast_unfolding(vector_dict, edge_dict):
    #1. initilization:put every vector into different communities
    #   the easiest way:use the vector num as the community num
    for i in vector_dict.keys():
        vector_dict[i] = i

    #print "vector_dict : ", vector_dict
    #print "edge_dict : ", edge_dict

    Q = modularity(vector_dict, edge_dict)  

    #2. for every vector, chose the community
    Q_new = 0.0
    while (Q_new != Q):
        Q_new = Q
        vector_dict, Q = chage_community(vector_dict, edge_dict, Q)
    community_num = modify_community(vector_dict)
    print "Q = ", Q
    print "vector_dict.key : ", vector_dict.keys()
    print "vector_dict.value : ", vector_dict.values()
    Q_best = Q
    while (True):
        #3. rebulid new graph, re_run the second step
        print "edge_dict : ",edge_dict
        print "vector_dict : ",vector_dict
        print "\n rebuild"
        vector_dict, edge_new_dict, community_dict = rebuild_graph(vector_dict, edge_dict, community_num)
        #print vector_dict
        print "community_dict : ", community_dict

        Q_new = 0.0
        while (Q_new != Q):
            Q_new = Q
            vector_dict, Q = chage_community(vector_dict, edge_new_dict, Q)
        community_num = modify_community(vector_dict)
        print "Q = ", Q
        if (Q_best == Q):
            break
        Q_best = Q
        vector_result = {}
        for key in community_dict.keys():
            value_of_vector = community_dict[key]
            for i in xrange(len(value_of_vector)):
                vector_result[value_of_vector[i]] = str(vector_dict[str(key)])
        for key in vector_result.keys():
            vector_dict[key] = vector_result[key]
        print "vector_dict.key : ", vector_dict.keys()
        print "vector_dict.value : ", vector_dict.values()

    #get the final result
    vector_result = {}
    for key in community_dict.keys():
        value_of_vector = community_dict[key]
        for i in xrange(len(value_of_vector)):
            vector_result[value_of_vector[i]] = str(vector_dict[str(key)])
    for key in vector_result.keys():
        vector_dict[key] = vector_result[key]
    print "Q_best : ", Q_best
    print "vector_result.key : ", vector_dict.keys()
    print "vector_result.value : ", vector_dict.values()

if __name__ == "__main__":
    vector_dict, edge_dict=loadData("./cd_data.txt")

    fast_unfolding(vector_dict, edge_dict)

参考文献

  1. Vincent D Blondel, Jean-Loup Guillaume, Renaud Lambiotte, Etienne Lefebvre, Fast unfolding of communities in large networks, in Journal of Statistical Mechanics: Theory and Experiment 2008 (10), P1000
  2. 社区发现算法FastUnfolding的GraphX实现 http://www.tuicool.com/articles/Jrieue

本文参与腾讯云自媒体分享计划,欢迎正在阅读的你也加入,一起分享。

发表于

我来说两句

0 条评论
登录 后参与评论

扫码关注云+社区

领取腾讯云代金券