将用户行为表示为二分图模型。假设给用户\(u\)进行个性化推荐，要计算所有节点相对于用户\(u\)的相关度，则PersonalRank从用户\(u\)对应的节点开始游走，每到一个节点都以\(1-d\)的概率停止游走并从\(u\)重新开始，或者以\(d\)的概率继续游走，从当前节点指向的节点中按照均匀分布随机选择一个节点往下游走。这样经过很多轮游走之后，每个顶点被访问到的概率也会收敛趋于稳定，这个时候我们就可以用概率来进行排名了。
在执行算法之前，我们需要初始化每个节点的初始概率值。如果我们对用户\(u\)进行推荐，则令\(u\)对应的节点的初始访问概率为1，其他节点的初始访问概率为0，然后再使用迭代公式计算。
\[PR(i)=(1-d)r_i+d\sum_{j \in in(i)} \frac {PR(j)}{|out(i)|} \\
r_i =
\begin{cases}
1 \ \ i=u \\
0 \ \ i!=u
\end{cases}
\]
r_i =
\begin{cases}
1 \ \ i=u \\
0 \ \ i!=u
\end{cases}
\]
一般有两种算法实现，一种是矩阵化实现，一种是非矩阵化实现。
非矩阵化实现
根据userID与itemID建立二分图。在代码中，self.G代表全局有向图，为区分userID与itemID分别加了不同的前缀。另外，user-item对保存在图中，方向是相互的。接下来，就在图中根据概率进行转移。
import pandas as pd
import time
class PersonalRank:
def __init__(self,X,Y):
X,Y = ['user_'+str(x) for x in X],['item_'+str(y) for y in Y]
self.G = self.get_graph(X,Y)
def get_graph(self,X,Y):
"""
Args:
X: user id
Y: item id
Returns:
graph:dic['user_id1':{'item_id1':1}, ... ]
"""
item_user = dict()
for i in range(len(X)):
user = X[i]
item = Y[i]
if item not in item_user:
item_user[item] = {}
item_user[item][user]=1
user_item = dict()
for i in range(len(Y)):
user = X[i]
item = Y[i]
if user not in user_item:
user_item[user] = {}
user_item[user][item]=1
G = dict(item_user,**user_item)
return G
def recommend(self, alpha, userID, max_depth,K=10):
# rank = dict()
userID = 'user_' + str(userID)
rank = {x: 0 for x in self.G.keys()}
rank[userID] = 1
# 开始迭代
begin = time.time()
for k in range(max_depth):
tmp = {x: 0 for x in self.G.keys()}
# 取出节点i和他的出边尾节点集合ri
for i, ri in self.G.items():
# 取节点i的出边的尾节点j以及边E(i,j)的权重wij,边的权重都为1，归一化后就是1/len(ri)
for j, wij in ri.items():
tmp[j] += alpha * rank[i] / (1.0 * len(ri))
tmp[userID] += (1 - alpha)
rank = tmp
end = time.time()
print('use_time', end - begin)
lst = sorted(rank.items(), key=lambda x: x[1], reverse=True)[:K]
for ele in lst:
print("%s:%.3f, \t" % (ele[0], ele[1]))
if __name__ == '__main__':
moviesPath = '../data/ml-1m/movies.dat'
ratingsPath = '../data/ml-1m/ratings.dat'
usersPath = '../data/ml-1m/users.dat'
# usersDF = pd.read_csv(usersPath,index_col=None,sep='::',header=None,names=['user_id', 'gender', 'age', 'occupation', 'zip'])
# moviesDF = pd.read_csv(moviesPath,index_col=None,sep='::',header=None,names=['movie_id', 'title', 'genres'])
ratingsDF = pd.read_csv(ratingsPath, index_col=None, sep='::', header=None,names=['user_id', 'movie_id', 'rating', 'timestamp'])
X=ratingsDF['user_id'][:1000]
Y=ratingsDF['movie_id'][:1000]
PersonalRank(X,Y).recommend(alpha=0.8,userID=1,max_depth=50,K=30)#输出对用户1推荐的 top10 item
# print('PersonalRank result',rank)
矩阵化实现
\[r = (1-\alpha)r_o + \alpha M^T r
\]
\]
其中，\(r\)是\(m+n\)行，1列的矩阵，每一行代表该顶点对固定顶点的PR值；是\(m+n\)行，1列的矩阵，负责选取某一个顶点作为固定顶点，其数值只有1行为1，其余为0。\(M\)是m+n行，m+n列的矩阵，是转移矩阵，其值\(M_{ij}=\frac{1}{out(i)},j \in out(i) \ else \ 0\),即为顶点的出度倒数，若没有连接边则为0。上式可转换为：
\[r = (E-\alpha M^T)^{-1}(1-\alpha)r_o
\]
\]
其中，\((E-\alpha M^T)^{-1}\)可以看做所有顶点的推荐结果，每一列代表一个顶点项，对该顶点的PR值。
#-*-coding:utf-8-*-
"""
author:jamest
date:20190310
PersonalRank function with Matrix
"""
import pandas as pd
import numpy as np
import time
import operator
from scipy.sparse import coo_matrix
from scipy.sparse.linalg import gmres
class PersonalRank:
def __init__(self,X,Y):
X,Y = ['user_'+str(x) for x in X],['item_'+str(y) for y in Y]
self.G = self.get_graph(X,Y)
def get_graph(self,X,Y):
"""
Args:
X: user id
Y: item id
Returns:
graph:dic['user_id1':{'item_id1':1}, ... ]
"""
item_user = dict()
for i in range(len(X)):
user = X[i]
item = Y[i]
if item not in item_user:
item_user[item] = {}
item_user[item][user]=1
user_item = dict()
for i in range(len(Y)):
user = X[i]
item = Y[i]
if user not in user_item:
user_item[user] = {}
user_item[user][item]=1
G = dict(item_user,**user_item)
return G
def graph_to_m(self):
"""
Returns:
a coo_matrix sparse mat M
a list,total user item points
a dict,map all the point to row index
"""
graph = self.G
vertex = list(graph.keys())
address_dict = {}
total_len = len(vertex)
for index in range(len(vertex)):
address_dict[vertex[index]] = index
row = []
col = []
data = []
for element_i in graph:
weight = round(1/len(graph[element_i]),3)
row_index= address_dict[element_i]
for element_j in graph[element_i]:
col_index = address_dict[element_j]
row.append(row_index)
col.append(col_index)
data.append(weight)
row = np.array(row)
col = np.array(col)
data = np.array(data)
m = coo_matrix((data,(row,col)),shape=(total_len,total_len))
return m,vertex,address_dict
def mat_all_point(self,m_mat,vertex,alpha):
"""
get E-alpha*m_mat.T
Args:
m_mat
vertex:total item and user points
alpha:the prob for random walking
Returns:
a sparse
"""
total_len = len(vertex)
row = []
col = []
data = []
for index in range(total_len):
row.append(index)
col.append(index)
data.append(1)
row = np.array(row)
col = np.array(col)
data = np.array(data)
eye_t = coo_matrix((data,(row,col)),shape=(total_len,total_len))
return eye_t.tocsr()-alpha*m_mat.tocsr().transpose()
def recommend_use_matrix(self, alpha, userID, K=10,use_matrix=True):
"""
Args:
alpha:the prob for random walking
userID:the user to recom
K:recom item num
Returns:
a dic,key:itemid ,value:pr score
"""
m, vertex, address_dict = self.graph_to_m()
userID = 'user_' + str(userID)
print('add',address_dict)
if userID not in address_dict:
return []
score_dict = {}
recom_dict = {}
mat_all = self.mat_all_point(m,vertex,alpha)
index = address_dict[userID]
initial_list = [[0] for row in range(len(vertex))]
initial_list[index] = [1]
r_zero = np.array(initial_list)
res = gmres(mat_all,r_zero,tol=1e-8)[0]
for index in range(len(res)):
point = vertex[index]
if len(point.strip().split('_'))<2:
continue
if point in self.G[userID]:
continue
score_dict[point] = round(res[index],3)
for zuhe in sorted(score_dict.items(),key=operator.itemgetter(1),reverse=True)[:K]:
point,score = zuhe[0],zuhe[1]
recom_dict[point] = score
return recom_dict
if __name__ == '__main__':
moviesPath = '../data/ml-1m/movies.dat'
ratingsPath = '../data/ml-1m/ratings.dat'
usersPath = '../data/ml-1m/users.dat'
# usersDF = pd.read_csv(usersPath,index_col=None,sep='::',header=None,names=['user_id', 'gender', 'age', 'occupation', 'zip'])
# moviesDF = pd.read_csv(moviesPath,index_col=None,sep='::',header=None,names=['movie_id', 'title', 'genres'])
ratingsDF = pd.read_csv(ratingsPath, index_col=None, sep='::', header=None,names=['user_id', 'movie_id', 'rating', 'timestamp'])
X=ratingsDF['user_id'][:1000]
Y=ratingsDF['movie_id'][:1000]
rank = PersonalRank(X,Y).recommend_use_matrix(alpha=0.8,userID=1,K=30)
print('PersonalRank result',rank)