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# Wikipedia principal eigenvector

A classical way to assert the relative importance of vertices in a graph is to compute the principal eigenvector of the adjacency matrix so as to assign to each vertex the values of the components of the first eigenvector as a centrality score:

> https://en.wikipedia.org/wiki/Eigenvector_centrality

On the graph of webpages and links those values are called the PageRank scores by Google.

The goal of this example is to analyze the graph of links inside wikipedia articles to rank articles by relative importance according to this eigenvector centrality.

The traditional way to compute the principal eigenvector is to use the power iteration method:

> https://en.wikipedia.org/wiki/Power_iteration

Here the computation is achieved thanks to Martinsson’s Randomized SVD algorithm implemented in scikit-learn.

The graph data is fetched from the DBpedia dumps. DBpedia is an extraction of the latent structured data of the Wikipedia content.

    # Author: Olivier Grisel <olivier.grisel@ensta.org>
    # License: 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

## Download 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)
            with open(filename, "wb") as f:
                f.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

## Computing the Adjacency matrix

    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()}

## Computing Principal Singular Vector using Randomized SVD

    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:]])

## Computing Centrality scores

    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:]])

Total running time of the script: (0 minutes 0.000 seconds)

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