Wikipedia PageRank With Randomized SVD

Introduction

In this lab, we will be analyzing the graph of links inside Wikipedia articles to rank articles by relative importance according to the eigenvector centrality. The traditional way to compute the principal eigenvector is to use the power iteration method. Here we will be using Martinsson's Randomized SVD algorithm implemented in scikit-learn.

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Skills Graph

Download data, if not already on disk

We will download the data from the DBpedia dumps which is an extraction of the latent structured data of the Wikipedia content.

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

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()

Load the redirect files

We will parse the redirections and build a transitively closed map out of it.

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 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))


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


## Loading the redirect files
redirects = get_redirects(redirects_filename)

Computing the Adjacency matrix

We will 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).

import numpy as np
from scipy import sparse


def get_adjacency_matrix(redirects_filename, page_links_filename, limit=None):
    """Extract the adjacency graph as a scipy sparse matrix"""
    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()}

Computing Principal Singular Vector using Randomized SVD

We will compute the principal singular vectors using the randomized_svd method implemented in scikit-learn.

from sklearn.decomposition import randomized_svd

print("Computing the principal singular vectors using randomized_svd")
U, s, V = randomized_svd(X, 5, n_iter=3)

Computing Centrality scores

We will compute the principal eigenvector score using a power iteration method.

def centrality_scores(X, alpha=0.85, max_iter=100, tol=1e-10):
    """Power iteration computation of the principal eigenvector"""
    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")
scores = centrality_scores(X, max_iter=100)

Summary

In this lab, we used Martinsson's Randomized SVD algorithm implemented in scikit-learn to analyze the graph of links inside Wikipedia articles to rank articles by relative importance according to the eigenvector centrality. We also computed the principal eigenvector score using a power iteration method.