維基百科主特征向量?
判斷圖中頂點相對重要性的一種經典方法是計算鄰接矩陣的主特征向量，以便將第一個特征向量的分量作為中心分數分配給每個頂點：
https://en.wikipedia.org/wiki/Eigenvector_centrality
在網頁和鏈接的圖表上，這些值被谷歌稱為PageRank分數。
本例的目標是分析維基百科文章中的鏈接圖，根據特征向量中心性對文章的相對重要性進行排序。
計算主特征向量的傳統方法是使用冪迭代法：
https://en.wikipedia.org/wiki/Power_iteration
這里的計算是通過Martinsson的隨機SVD算法實現， 在scikit -learn中。
圖形數據從DBpedia轉儲中獲取。DBpedia是對維基百科內容的潛在結構化數據的提取。
# Author: Olivier Grisel <olivier.grisel@ensta.org>
# License: BSD 3 clause
from bz2 import BZ2File
import os
from datetime import datetime
from pprint import pprint
from time import time
import numpy as np
from scipy import sparse
from sklearn.decomposition import randomized_svd
from urllib.request import urlopen
print(__doc__)
# #############################################################################
# Where to download the data, if not already on disk
redirects_url = "http://downloads.dbpedia.org/3.5.1/en/redirects_en.nt.bz2"
redirects_filename = redirects_url.rsplit("/", 1)[1]
page_links_url = "http://downloads.dbpedia.org/3.5.1/en/page_links_en.nt.bz2"
page_links_filename = page_links_url.rsplit("/", 1)[1]
resources = [
(redirects_url, redirects_filename),
(page_links_url, page_links_filename),
]
for url, filename in resources:
if not os.path.exists(filename):
print("Downloading data from '%s', please wait..." % url)
opener = urlopen(url)
open(filename, 'wb').write(opener.read())
print()
# #############################################################################
# Loading the redirect files
def index(redirects, index_map, k):
"""Find the index of an article name after redirect resolution"""
k = redirects.get(k, k)
return index_map.setdefault(k, len(index_map))
DBPEDIA_RESOURCE_PREFIX_LEN = len("http://dbpedia.org/resource/")
SHORTNAME_SLICE = slice(DBPEDIA_RESOURCE_PREFIX_LEN + 1, -1)
def short_name(nt_uri):
"""Remove the < and > URI markers and the common URI prefix"""
return nt_uri[SHORTNAME_SLICE]
def get_redirects(redirects_filename):
"""Parse the redirections and build a transitively closed map out of it"""
redirects = {}
print("Parsing the NT redirect file")
for l, line in enumerate(BZ2File(redirects_filename)):
split = line.split()
if len(split) != 4:
print("ignoring malformed line: " + line)
continue
redirects[short_name(split[0])] = short_name(split[2])
if l % 1000000 == 0:
print("[%s] line: %08d" % (datetime.now().isoformat(), l))
# compute the transitive closure
print("Computing the transitive closure of the redirect relation")
for l, source in enumerate(redirects.keys()):
transitive_target = None
target = redirects[source]
seen = {source}
while True:
transitive_target = target
target = redirects.get(target)
if target is None or target in seen:
break
seen.add(target)
redirects[source] = transitive_target
if l % 1000000 == 0:
print("[%s] line: %08d" % (datetime.now().isoformat(), l))
return redirects
def get_adjacency_matrix(redirects_filename, page_links_filename, limit=None):
"""Extract the adjacency graph as a scipy sparse matrix
Redirects are resolved first.
Returns X, the scipy sparse adjacency matrix, redirects as python
dict from article names to article names and index_map a python dict
from article names to python int (article indexes).
"""
print("Computing the redirect map")
redirects = get_redirects(redirects_filename)
print("Computing the integer index map")
index_map = dict()
links = list()
for l, line in enumerate(BZ2File(page_links_filename)):
split = line.split()
if len(split) != 4:
print("ignoring malformed line: " + line)
continue
i = index(redirects, index_map, short_name(split[0]))
j = index(redirects, index_map, short_name(split[2]))
links.append((i, j))
if l % 1000000 == 0:
print("[%s] line: %08d" % (datetime.now().isoformat(), l))
if limit is not None and l >= limit - 1:
break
print("Computing the adjacency matrix")
X = sparse.lil_matrix((len(index_map), len(index_map)), dtype=np.float32)
for i, j in links:
X[i, j] = 1.0
del links
print("Converting to CSR representation")
X = X.tocsr()
print("CSR conversion done")
return X, redirects, index_map
# stop after 5M links to make it possible to work in RAM
X, redirects, index_map = get_adjacency_matrix(
redirects_filename, page_links_filename, limit=5000000)
names = {i: name for name, i in index_map.items()}
print("Computing the principal singular vectors using randomized_svd")
t0 = time()
U, s, V = randomized_svd(X, 5, n_iter=3)
print("done in %0.3fs" % (time() - t0))
# print the names of the wikipedia related strongest components of the
# principal singular vector which should be similar to the highest eigenvector
print("Top wikipedia pages according to principal singular vectors")
pprint([names[i] for i in np.abs(U.T[0]).argsort()[-10:]])
pprint([names[i] for i in np.abs(V[0]).argsort()[-10:]])
def centrality_scores(X, alpha=0.85, max_iter=100, tol=1e-10):
"""Power iteration computation of the principal eigenvector
This method is also known as Google PageRank and the implementation
is based on the one from the NetworkX project (BSD licensed too)
with copyrights by:
Aric Hagberg <hagberg@lanl.gov>
Dan Schult <dschult@colgate.edu>
Pieter Swart <swart@lanl.gov>
"""
n = X.shape[0]
X = X.copy()
incoming_counts = np.asarray(X.sum(axis=1)).ravel()
print("Normalizing the graph")
for i in incoming_counts.nonzero()[0]:
X.data[X.indptr[i]:X.indptr[i + 1]] *= 1.0 / incoming_counts[i]
dangle = np.asarray(np.where(np.isclose(X.sum(axis=1), 0),
1.0 / n, 0)).ravel()
scores = np.full(n, 1. / n, dtype=np.float32) # initial guess
for i in range(max_iter):
print("power iteration #%d" % i)
prev_scores = scores
scores = (alpha * (scores * X + np.dot(dangle, prev_scores))
+ (1 - alpha) * prev_scores.sum() / n)
# check convergence: normalized l_inf norm
scores_max = np.abs(scores).max()
if scores_max == 0.0:
scores_max = 1.0
err = np.abs(scores - prev_scores).max() / scores_max
print("error: %0.6f" % err)
if err < n * tol:
return scores
return scores
print("Computing principal eigenvector score using a power iteration method")
t0 = time()
scores = centrality_scores(X, max_iter=100)
print("done in %0.3fs" % (time() - t0))
pprint([names[i] for i in np.abs(scores).argsort()[-10:]])
腳本的總運行時間：(0分0.000秒)
Download Python source code:wikipedia_principal_eigenvector.py
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