维基百科主特征量#

断言图中点相对重要性的一种经典方法是计算邻近矩阵的主特征向量，以便将第一个特征向每个点分配第一个特征向量的分量的值作为中心性得分：https://en.wikipedia.org/wiki/Eigenvector_centrality。在网页和链接的图表上，这些值被Google称为PageRank分数。

此示例的目标是分析维基百科文章内部链接的图表，根据该特征向量中心性按相对重要性对文章进行排名。

计算主特征量的传统方法是使用 power iteration method .这里的计算是通过scikit-learn中实现的Martinsson随机化的奇异值分解算法来实现的。

图形数据从DBpedia转储中获取。DBpedia是维基百科内容潜在结构化数据的提取。

# Authors: The scikit-learn developers
# SPDX-License-Identifier: BSD-3-Clause

import os
from bz2 import BZ2File
from datetime import datetime
from pprint import pprint
from time import time
from urllib.request import urlopen

import numpy as np
from scipy import sparse

from sklearn.decomposition import randomized_svd

下载数据（如果尚未在磁盘上）#

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)
        with open(filename, "wb") as f:
            f.write(opener.read())
        print()

加载重定向文件#

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

计算邻近矩阵#

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()}

使用随机奇异值分解计算主奇异量#

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:]])

计算中心性分数#

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, 1.0 / 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:]])