社区发现算法之标签传播（LPA）
标签传播算法（LPA）的做法比较简单：
第一步: 为所有节点指定一个唯一的标签；
第二步: 逐轮刷新所有节点的标签，直到达到收敛要求为止。对于每一轮刷新，节点标签刷新的规则如下:
对于某一个节点，考察其所有邻居节点的标签，并进行统计，将出现个数最多的那个标签赋给当前节点。当个数最多的标签不唯一时，随机选一个。
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注：算法中的记号 N_n^k 表示节点 n 的邻居中标签为 k 的所有节点构成的集合。
标签传播算法（label propagation）的核心思想非常简单：相似的数据应该具有相同的label。LP算法包括两大步骤：1）构造相似矩阵；2）勇敢的传播吧。
LP算法是基于Graph的，因此我们需要先构建一个图。我们为所有的数据构建一个图，图的节点就是一个数据点，包含labeled和unlabeled的数据。节点i和节点j的边表示他们的相似度。这个图的构建方法有很多，这里我们假设这个图是全连接的，节点i和节点j的边权重为： 
这里，α是超参。
还有个非常常用的图构建方法是knn图，也就是只保留每个节点的k近邻权重，其他的为0，也就是不存在边，因此是稀疏的相似矩阵。
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标签传播算法非常简单：通过节点之间的边传播label。边的权重越大，表示两个节点越相似，那么label越容易传播过去。我们定义一个NxN的概率转移矩阵P：
Pij表示从节点i转移到节点j的概率。假设有C个类和L个labeled样本，我们定义一个LxC的label矩阵YL，第i行表示第i个样本的标签指示向量，即如果第i个样本的类别是j，那么该行的第j个元素为1，其他为0。同样，我们也给U个unlabeled样本一个UxC的label矩阵YU。把他们合并，我们得到一个NxC的soft label矩阵F=[YL;YU]。soft label的意思是，我们保留样本i属于每个类别的概率，而不是互斥性的，这个样本以概率1只属于一个类。当然了，最后确定这个样本i的类别的时候，是取max也就是概率最大的那个类作为它的类别的。那F里面有个YU，它一开始是不知道的，那最开始的值是多少？无所谓，随便设置一个值就可以了。
千呼万唤始出来，简单的LP算法如下：
1）执行传播：F=PF
2）重置F中labeled样本的标签：FL=YL
3）重复步骤1）和2）直到F收敛。
步骤1）就是将矩阵P和矩阵F相乘，这一步，每个节点都将自己的label以P确定的概率传播给其他节点。如果两个节点越相似（在欧式空间中距离越近），那么对方的label就越容易被自己的label赋予，就是更容易拉帮结派。步骤2）非常关键，因为labeled数据的label是事先确定的，它不能被带跑，所以每次传播完，它都得回归它本来的label。随着labeled数据不断的将自己的label传播出去，最后的类边界会穿越高密度区域，而停留在低密度的间隔中。相当于每个不同类别的labeled样本划分了势力范围。
变身的LP算法
我们知道，我们每次迭代都是计算一个soft label矩阵F=[YL;YU]，但是YL是已知的，计算它没有什么用，在步骤2）的时候，还得把它弄回来。我们关心的只是YU，那我们能不能只计算YU呢？Yes。我们将矩阵P做以下划分：
代码如下：
import time
import numpy as np
# return k neighbors index
def navie_knn(dataSet, query, k):
numSamples = dataSet.shape[0]
## step 1: calculate Euclidean distance
diff = np.tile(query, (numSamples, 1)) - dataSet
squaredDiff = diff ** 2
squaredDist = np.sum(squaredDiff, axis = 1) # sum is performed by row
## step 2: sort the distance
sortedDistIndices = np.argsort(squaredDist)
if k > len(sortedDistIndices):
k = len(sortedDistIndices)
return sortedDistIndices[0:k]
# build a big graph (normalized weight matrix)
def buildGraph(MatX, kernel_type, rbf_sigma = None, knn_num_neighbors = None):
num_samples = MatX.shape[0]
affinity_matrix = np.zeros((num_samples, num_samples), np.float32)
if kernel_type == 'rbf':
if rbf_sigma == None:
raise ValueError('You should input a sigma of rbf kernel!')
for i in xrange(num_samples):
row_sum = 0.0
for j in xrange(num_samples):
diff = MatX[i, :] - MatX[j, :]
affinity_matrix[i][j] = np.exp(sum(diff**2) / (-2.0 * rbf_sigma**2))
row_sum += affinity_matrix[i][j]
affinity_matrix[i][:] /= row_sum
elif kernel_type == 'knn':
if knn_num_neighbors == None:
raise ValueError('You should input a k of knn kernel!')
for i in xrange(num_samples):
k_neighbors = navie_knn(MatX, MatX[i, :], knn_num_neighbors)
affinity_matrix[i][k_neighbors] = 1.0 / knn_num_neighbors
else:
raise NameError('Not support kernel type! You can use knn or rbf!')
return affinity_matrix
# label propagation
def labelPropagation(Mat_Label, Mat_Unlabel, labels, kernel_type = 'rbf', rbf_sigma = 1.5, \
knn_num_neighbors = 10, max_iter = 500, tol = 1e-3):
# initialize
num_label_samples = Mat_Label.shape[0]
num_unlabel_samples = Mat_Unlabel.shape[0]
num_samples = num_label_samples + num_unlabel_samples
labels_list = np.unique(labels)
num_classes = len(labels_list)
MatX = np.vstack((Mat_Label, Mat_Unlabel))
clamp_data_label = np.zeros((num_label_samples, num_classes), np.float32)
for i in xrange(num_label_samples):
clamp_data_label[i][labels[i]] = 1.0
label_function = np.zeros((num_samples, num_classes), np.float32)
label_function[0 : num_label_samples] = clamp_data_label
label_function[num_label_samples : num_samples] = -1
# graph construction
affinity_matrix = buildGraph(MatX, kernel_type, rbf_sigma, knn_num_neighbors)
# start to propagation
iter = 0; pre_label_function = np.zeros((num_samples, num_classes), np.float32)
changed = np.abs(pre_label_function - label_function).sum()
while iter < max_iter and changed > tol:
if iter % 1 == 0:
print "---> Iteration %d/%d, changed: %f" % (iter, max_iter, changed)
pre_label_function = label_function
iter += 1
# propagation
label_function = np.dot(affinity_matrix, label_function)
# clamp
label_function[0 : num_label_samples] = clamp_data_label
# check converge
changed = np.abs(pre_label_function - label_function).sum()
# get terminate label of unlabeled data
unlabel_data_labels = np.zeros(num_unlabel_samples)
for i in xrange(num_unlabel_samples):
unlabel_data_labels[i] = np.argmax(label_function[i+num_label_samples])
return unlabel_data_labelsJBQkFCMA==/dissolve/70/gravity/SouthEast)测试代码：
import time
import math
import numpy as np
from labelPropagation import labelPropagation
def show(Mat_Label, labels, Mat_Unlabel, unlabel_data_labels):
import matplotlib.pyplot as plt
for i in range(Mat_Label.shape[0]):
if int(labels[i]) == 0:
plt.plot(Mat_Label[i, 0], Mat_Label[i, 1], 'Dr')
elif int(labels[i]) == 1:
plt.plot(Mat_Label[i, 0], Mat_Label[i, 1], 'Db')
else:
plt.plot(Mat_Label[i, 0], Mat_Label[i, 1], 'Dy')
for i in range(Mat_Unlabel.shape[0]):
if int(unlabel_data_labels[i]) == 0:
plt.plot(Mat_Unlabel[i, 0], Mat_Unlabel[i, 1], 'or')
elif int(unlabel_data_labels[i]) == 1:
plt.plot(Mat_Unlabel[i, 0], Mat_Unlabel[i, 1], 'ob')
else:
plt.plot(Mat_Unlabel[i, 0], Mat_Unlabel[i, 1], 'oy')
plt.xlabel('X1'); plt.ylabel('X2')
plt.xlim(0.0, 12.)
plt.ylim(0.0, 12.)
plt.show()
def loadCircleData(num_data):
center = np.array([5.0, 5.0])
radiu_inner = 2
radiu_outer = 4
num_inner = num_data / 3
num_outer = num_data - num_inner
data = []
theta = 0.0
for i in range(num_inner):
pho = (theta % 360) * math.pi / 180
tmp = np.zeros(2, np.float32)
tmp[0] = radiu_inner * math.cos(pho) + np.random.rand(1) + center[0]
tmp[1] = radiu_inner * math.sin(pho) + np.random.rand(1) + center[1]
data.append(tmp)
theta += 2
theta = 0.0
for i in range(num_outer):
pho = (theta % 360) * math.pi / 180
tmp = np.zeros(2, np.float32)
tmp[0] = radiu_outer * math.cos(pho) + np.random.rand(1) + center[0]
tmp[1] = radiu_outer * math.sin(pho) + np.random.rand(1) + center[1]
data.append(tmp)
theta += 1
Mat_Label = np.zeros((2, 2), np.float32)
Mat_Label[0] = center + np.array([-radiu_inner + 0.5, 0])
Mat_Label[1] = center + np.array([-radiu_outer + 0.5, 0])
labels = [0, 1]
Mat_Unlabel = np.vstack(data)
return Mat_Label, labels, Mat_Unlabel
def loadBandData(num_unlabel_samples):
#Mat_Label = np.array([[5.0, 2.], [5.0, 8.0]])
#labels = [0, 1]
#Mat_Unlabel = np.array([[5.1, 2.], [5.0, 8.1]])
Mat_Label = np.array([[5.0, 2.], [5.0, 8.0]])
labels = [0, 1]
num_dim = Mat_Label.shape[1]
Mat_Unlabel = np.zeros((num_unlabel_samples, num_dim), np.float32)
Mat_Unlabel[:num_unlabel_samples/2, :] = (np.random.rand(num_unlabel_samples/2, num_dim) - 0.5) * np.array([3, 1]) + Mat_Label[0]
Mat_Unlabel[num_unlabel_samples/2 : num_unlabel_samples, :] = (np.random.rand(num_unlabel_samples/2, num_dim) - 0.5) * np.array([3, 1]) + Mat_Label[1]
return Mat_Label, labels, Mat_Unlabel
if name == “main“:
num_unlabel_samples = 800
#Mat_Label, labels, Mat_Unlabel = loadBandData(num_unlabel_samples)
Mat_Label, labels, Mat_Unlabel = loadCircleData(num_unlabel_samples)
## Notice: when use 'rbf' as our kernel, the choice of hyper parameter 'sigma' is very import! It should be
## chose according to your dataset, specific the distance of two data points. I think it should ensure that
## each point has about 10 knn or w_i,j is large enough. It also influence the speed of converge. So, may be
## 'knn' kernel is better!
#unlabel_data_labels = labelPropagation.labelPropagation(Mat_Label, Mat_Unlabel, labels, kernel_type = 'rbf', rbf_sigma = 0.2)
unlabel_data_labels = labelPropagation(Mat_Label, Mat_Unlabel, labels, kernel_type = 'knn', knn_num_neighbors = 10, max_iter = 400)
show(Mat_Label, labels, Mat_Unlabel, unlabel_data_labels)
结果如下：
利用networkx：
#coding:utf-8
'''
Created on 2017-1-4
@author: 刘帅
'''
import collections
import random
import networkx as nx
class LPA():
def __init__(self, G, max_iter = 20):
self._G = G
self._n = len(G.node) #number of nodes
self._max_iter = max_iter
def can_stop(self):
# all node has the label same with its most neighbor
for i in range(self._n):
node = self._G.node[i]
label = node["label"]
max_labels = self.get_max_neighbor_label(i)
if(label not in max_labels):
return False
return True
def get_max_neighbor_label(self,node_index):
m = collections.defaultdict(int)
for neighbor_index in self._G.neighbors(node_index):
neighbor_label = self._G.node[neighbor_index]["label"]
m[neighbor_label] += 1
max_v = max(m.itervalues())
return [item[0] for item in m.items() if item[1] == max_v]
'''asynchronous update'''
def populate_label(self):
#random visit
visitSequence = random.sample(self._G.nodes(),len(self._G.nodes()))
for i in visitSequence:
node = self._G.node[i]
label = node["label"]
max_labels = self.get_max_neighbor_label(i)
if(label not in max_labels):
newLabel = random.choice(max_labels)
node["label"] = newLabel
def get_communities(self):
communities = collections.defaultdict(lambda:list())
for node in self._G.nodes(True):
label = node[1]["label"]
communities[label].append(node[0])
return communities.values()
def execute(self):
#initial label
for i in range(self._n):
self._G.node[i]["label"] = i
iter_time = 0
#populate label
while(not self.can_stop() and iter_time结果如下：
[0, 1, 3, 7, 11, 12, 13, 17, 19, 21]
[5, 6, 16]
[4, 10]
[2, 8, 9, 14, 15, 18, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]