聚类系数可调的无标度网络生成算法

0. BA无标度网络模型简单介绍：

实际网络的两个重要的特性：

      （1）增长性：即网络的规模是不断的增长的，ER随机图和WS小世界模型中的网络的大小是固定

      （2）优先连接（Preferential  attachment以下简称PA）：新的节点更倾向于和那些具有较高的连接度的hub节点相连。这种现象也叫作“富者更富（Rich get richer）”或者是“马太效应”

本文主要实现Petter Holme发表在PRE上的一篇paper《Growing scale-free networks with tunable clustering》中讲述的构造聚类系数可以调节的BA网络的算法。

1.论文简单介绍

       通过在标准的BA网络模型构造算法中添加一步“构造三角形”的方法，可以通过不同的参数形成具有不同的聚类系数的网络。通过该算法生成的网络在保持了经典的BA网络的幂律度分布特性的同时，还具有较高的聚类系数，而经典的BA网络在网络的规模越来越大的时候，网络的聚类系数趋于0.

2. 算法流程
          （1）：    初始时，网络中有 M_0 个全连接的节点。
          （2）：    每一步新增一个带有 M 条边的节点 i 。
          （3）：    PA。节点 i 先采用和BA网络模型相同的优先连接规则，和网络中的已经存在的节点 j 做一次优先连接。
          （4）：    TF（triad formation）。为了增加网络的聚类系数，接下来节点 i 将以概率 pt 随机的和节点 j 的邻居做三角连接，以 1 - pt 的概率做优先连接。直到所有的 m 条边都被添加完。 网络在添加下一个节点。

算法中引入triad formation可以理解为当我们认识一个人的时候，也会想要去认识他的朋友。

（BA无标度网络模型的构造算法可以参见我先前的文章）

3. 程序源码

程序中出现的由main函数带入的参数m_t为论文中介绍的控制参数，其中m_t = （M - 1）*Pt，公式中的M为每次添加一个节点的时候要带入网络中的边数，Pt为上述的算法流程中出现的概率。通过输入不通的参数m_t可以调控网络的聚类系数。

#include<stdio.h>#include<stdlib.h>#include<math.h>struct Node;typedef struct Node* NodePtr;typedef struct Node{int degree;double weight;double probabilityDistribution;}Node;Node* decisionMaking;int** adjacentMatrix;int* initalNetwork;int NETWORK_SIZE, M, M_0;/* * NETWORK_SIZE:网络的大小 * M:每次向已有的网络中添加一个节点M条边 * M_0:初始的全联通网络的大小 * */double m_t;/* * 用于调控网络的聚类系数 m_t = (M-1) * Pt */void initial();void initalNetwork_M0_connected();void updateDecisionMakingData();void generateFreeScaleNetwork();void saveNetwork();double calClusteringCoefficient();double calClusteringCoefficientOfNode();void cal_averageDegree();int main(int argc, char** argv){if( 5 != argc ){printf("this algorithm requires 4 user-specify parameters\n");printf("\t1.the size of network.\n");printf("\t2.the initial size of network.\n");printf("\t3.the size of add edges per times.\n");printf("\t4.the control parameter m_t.\n");printf("\texample: \"a.exe 100 3 3 0.9\"\n");exit(0);}srand((unsigned)time(NULL));NETWORK_SIZE = atoi(argv[1]);M_0 = atoi(argv[2]);M = atoi(argv[3]);m_t = atof(argv[4]);initial();initalNetwork_M0_connected();generateFreeScaleNetwork();saveNetwork();printf("ClusteringCoefficient : %f\n", calClusteringCoefficient());cal_averageDegree();//write2file(adjacentMatrix, NETWORK_SIZE, "freeScaleWithCluster_edges.data");return 0;}/* * 算法中使用到的数据结构的分配 * */void initial(){if( !(decisionMaking = (NodePtr)malloc(sizeof(Node) * (NETWORK_SIZE + 1))) ){printf("decisionMaking* malloc error\n");exit(0);}if( !(adjacentMatrix = (int**)malloc(sizeof(int*) * (NETWORK_SIZE + 1))) ){printf("adjacentMatrix** malloc error\n");exit(0);}int i;for( i = 1; i <= NETWORK_SIZE; i++ ){if( !(adjacentMatrix[i] = (int*)malloc(sizeof(int) * (NETWORK_SIZE + 1))) ){printf("adjacentMatrix[%d]* malloc error\n", i);exit(0);}}//存放初始网络中的节点的编号if( !(initalNetwork = (int*)malloc(sizeof(int) * (M_0 + 1))) ){printf("initalNetwork* malloc error\n");exit(0);}}/* * 初始化：在NETWORK_SIZE中随机选择M_0个节点构成连通的网络。 * */void initalNetwork_M0_connected(){int i, j, randomFirst, randomSecond;for( i = 1; i <= NETWORK_SIZE; i++ )for( j = 1; j <= NETWORK_SIZE; j++ )adjacentMatrix[i][j] = 0;// 随机产生M_0个节点, 构成初始的网络for( i = 1; i <= M_0; i++ ){initalNetwork[i] = rand() % NETWORK_SIZE + 1;for( j = 1; j < i; j++ )if( initalNetwork[i] == initalNetwork[j] ){i--;break;}}for( i = 1; i < M_0; i++ )adjacentMatrix[initalNetwork[i]][initalNetwork[i+1]] = adjacentMatrix[initalNetwork[i+1]][initalNetwork[i]] = 1;adjacentMatrix[initalNetwork[M_0]][initalNetwork[1]] = adjacentMatrix[initalNetwork[1]][initalNetwork[M_0]] = 1;updateDecisionMakingData();}/* * 通过adjacentMatrix更新decisionMaking数组, 也就是要给上面定义的结构体赋值 * */void updateDecisionMakingData(){int i, j, totalDegree = 0;for( i = 1; i <= NETWORK_SIZE; i++ )decisionMaking[i].degree = 0;for( i = 1; i <= NETWORK_SIZE; i++ )for( j = 1; j <= NETWORK_SIZE; j++ )decisionMaking[i].degree += adjacentMatrix[i][j];for( i = 1; i <= NETWORK_SIZE; i++ )totalDegree += decisionMaking[i].degree;for( i = 1; i <= NETWORK_SIZE; i++ )decisionMaking[i].weight = decisionMaking[i].degree/(double)totalDegree;decisionMaking[1].probabilityDistribution = decisionMaking[1].weight;for( i = 2; i <= NETWORK_SIZE; i++ )decisionMaking[i].probabilityDistribution = decisionMaking[i - 1].probabilityDistribution + decisionMaking[i].weight;}/* * 构造聚类系数可以调节的BA无标度网络模型 * */void generateFreeScaleNetwork(){int i, k, j = 1, length = 0;int random_auxiliary_old[NETWORK_SIZE + 1];int random_auxiliary[NETWORK_SIZE + 1 - M_0];/* * 要保证每次引入一个<新的>的节点，所以要随机选择不重复的节点加入，并且把初始网络中的M_0个节点先删除 * */for( i = 1; i <= NETWORK_SIZE; i++ )random_auxiliary_old[i] = i;for( i = 1; i <= M_0; i++ )random_auxiliary_old[initalNetwork[i]] = 0;for( i = 1; i <= NETWORK_SIZE; i++ )if( random_auxiliary_old[i] != 0 )random_auxiliary[j++] = random_auxiliary_old[i];/* * 添加新的节点构造无标度网络 * */int new_node_index, new_node_value;double random_decision = 0.0;int targetNode;//表示找到的已经在网络中的将要连接的节点length = NETWORK_SIZE - M_0;int flag;int connectedNodeSet[M];for( i = 1; i <= NETWORK_SIZE - M_0; i++ )//需要添加到网络中的节点的个数为NETWORK_SIZE - M_0{//得到随机的不重复的加入节点 i ，将其添加到网络中new_node_index = rand() % length + 1;new_node_value = random_auxiliary[new_node_index];random_auxiliary[new_node_index] = random_auxiliary[length--];/* * one BA step * *///对于选择到的节点new_node_value(i)做一次优先连接, 连接到一个网络中已经存在的节点。flag = 0;random_decision = (rand() % NETWORK_SIZE) / (double)NETWORK_SIZE;for( k = 1; k <= NETWORK_SIZE; k++ )if( decisionMaking[k].probabilityDistribution >= random_decision && decisionMaking[k].degree != 0 && adjacentMatrix[new_node_value][k] != 1 ){targetNode = k;flag = 1;break;}if( flag == 0 )for( k = 1; k <= NETWORK_SIZE; k++ )if( decisionMaking[k].degree != 0 && adjacentMatrix[new_node_value][k] != 1 ){targetNode = k;break;}adjacentMatrix[new_node_value][targetNode] = adjacentMatrix[targetNode][new_node_value] = 1;/* * TF step * */int remainEdges = M - 1;while( remainEdges > 0 ){//以概率pt做三角连接double exeTFConnect = (rand()%NETWORK_SIZE)/(double)NETWORK_SIZE;if( exeTFConnect < ( m_t / (double)(M - 1) ) ){int neighbors_Number = 0;for( k = 1; k <= NETWORK_SIZE; k++ )if( k != new_node_value && adjacentMatrix[targetNode][k] && !adjacentMatrix[k][new_node_value] )neighbors_Number++;if( neighbors_Number != 0 ){int neighborSet[neighbors_Number];int temp_count = 0;for( k = 1; k <= NETWORK_SIZE; k++ )if( k != new_node_value && adjacentMatrix[targetNode][k] && !adjacentMatrix[k][new_node_value] )neighborSet[temp_count++] = k;int random_neighbor_index = rand() % neighbors_Number;adjacentMatrix[neighborSet[random_neighbor_index]][new_node_value] = adjacentMatrix[new_node_value][neighborSet[random_neighbor_index]] = 1;remainEdges--;}}if( remainEdges <= 0 )continue;//以概率1-pt做优先连接double exeBAconnect = (rand()%NETWORK_SIZE)/(double)NETWORK_SIZE;if( exeBAconnect < 1 - (m_t / (double)(M - 1)) ){flag = 0;random_decision = (rand() % 1000) / (double)1000;for( k = 1; k <= NETWORK_SIZE; k++ )if( decisionMaking[k].probabilityDistribution >= random_decision && decisionMaking[k].degree != 0 && adjacentMatrix[new_node_value][k] != 1 ){targetNode = k;flag = 1;break;}if( flag == 0 )for( k = 1; k <= NETWORK_SIZE; k++ )if( decisionMaking[k].degree != 0 && adjacentMatrix[new_node_value][k] != 1 ){targetNode = k;break;}adjacentMatrix[new_node_value][targetNode] = adjacentMatrix[targetNode][new_node_value] = 1;remainEdges--;}}updateDecisionMakingData();}}void saveNetwork(){FILE* fwrite;int i, j;if( NULL == (fwrite = fopen("SF_tunableClustering.data", "w")) ){printf("open file error");exit(0);}for( i = 1; i <= NETWORK_SIZE; i++ ){for( j = 1; j <= NETWORK_SIZE; j++ ){fprintf(fwrite, "%d ", adjacentMatrix[i][j]);}fprintf(fwrite, "\n");}fclose(fwrite);}/**************************************************************************************************** * * 计算网络的聚类系数 * *****************************************************************************************************/double calClusteringCoefficient(){int i;double clusteringCoefficient = 0.0;for( i = 1; i <= NETWORK_SIZE; i++ )clusteringCoefficient += calClusteringCoefficientOfNode(i);return clusteringCoefficient / NETWORK_SIZE;}/* * 计算每一个节点的聚类系数 * */double calClusteringCoefficientOfNode(int node){int neighbors, triad;int i, j;triad = neighbors = 0;for( i = 1; i <= NETWORK_SIZE; i++ ){for( j = i + 1; j <= NETWORK_SIZE; j++ ){if( adjacentMatrix[node][i] && adjacentMatrix[node][j] && adjacentMatrix[i][j] ){triad++;}}}for( i = 1; i <= NETWORK_SIZE; i++ )if( adjacentMatrix[node][i] )neighbors++;return (double)(triad * 2)/(double)(neighbors * (neighbors - 1));}/* * 计算网络的平均度 * */void cal_averageDegree(){int i, j;int degree = 0;for( i = 1; i <= NETWORK_SIZE; i++ )for( j = 1; j <= NETWORK_SIZE; j++ )if( adjacentMatrix[i][j] )degree++;printf("the average degree is %f\n", degree/(double)NETWORK_SIZE);}

4.论文结果复现

在论文中给出了控制参数m_t对于网络的聚类系数的影响。下图分别为论文给出的结果和自己复现的结果，有一定的差异，但是总体的应该是趋势是正确的。

5.一个该算法生成的网络的可视化结果