Jeff Kubina > Graph-Centrality-Pagerank > Graph::Centrality::Pagerank

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Module Version: 1.05   Source  

NAME ^

Graph::Centrality::Pagerank - Computes pagerank of all nodes in a graph.

SYNOPSIS ^

  use Graph::Centrality::Pagerank;
  use Data::Dump qw(dump);
  my $ranker = Graph::Centrality::Pagerank->new();
  my $listOfEdges = [[1,2],[3,4]];
  dump $ranker->getPagerankOfNodes (listOfEdges => $listOfEdges);
  # dumps:
  # {
  #   1 => "0.175438596989046",
  #   2 => "0.324561403010954",
  #   3 => "0.175438596989046",
  #   4 => "0.324561403010954",
  # }

DESCRIPTION ^

Graph::Centrality::Pagerank computes the pagerank of the all nodes in a graph. The input can be a list of edges or a Graph. Graph::Centrality::Pagerank is written entirely in Perl and is not recommended for use in high performance applications.

CONSTRUCTOR ^

new

The method new creates an instance of the Graph::Centrality::Pagerank class with the following parameters:

dampeningFactor
 dampeningFactor => 0.85

dampeningFactor is the dampening factor used when computing pagerank. It must be a value ranging from zero to one; the default is 0.85. Note the incidence matrix generated from the graph is multiplied (scaled) by dampeningFactor, not by 1 - dampeningFactor.

maxRelError
 maxRelError => sqrt (machine-epsilon)

maxRelError is the maximum average relative error that is permitted between successive pagerank vectors before the iterative process to approximate the pagerank vector should be stopped. The default is the square root of the systems machine epsilon. Usually, most pagerank values computed will have -log10(maxRelError) digits of accuracy. maxRelError must be positive and less than or equal to 0.01.

minIterations
 minIterations => 0

minIterations is the minimum number of iterations that will be computed before the pagerank iterations are stopped, even if maxRelError is achieved. The default is zero.

maxIterations
 maxIterations => int (2 * ((maxRelError / ln (dampeningFactor) + 1))

maxIterations is the maximum number of iterations that can be performed to approximate the pagerank vector even if maxRelError is not achieved. The default is 2 * ((maxRelError / ln (dampeningFactor) + 1). If dampeningFactor is zero, then maxIterations is one. If dampeningFactor is one, then maxIterations is equal to the total nodes in the graph.

linkSinkNodes
 linkSinkNodes => 1

In a directed graph sink nodes are the nodes with no edges emanating out from them. In the pagerank algorithm these nodes are automatically linked to all other nodes in the graph. To prevent this set linkSinkNodes to zero; the default is one.

directed
 directed => 1

If directed is true, the pagerank computations are done with the graph edges being directed. If directed is false, the pageranks are computed treating the graph as undirected; the default value of directed is one.

useEdgeWeights
 useEdgeWeights => 0

If useEdgeWeights is true, then any weight associated with an edge is used in the computation of pagerank. The default weight for any edge without an assigned weight is one. The default value of useEdgeWeights is zero, which forces all edge weights to be one.

METHOD ^

getPagerankOfNodes

The method getPagerankOfNodes computes the pagerank of each node in the graph. The graph can be defined using the graph parameter or by supplying a list of edges. All the parameters used by the constructor new can also be set here and they will override the values used with new. getPagerankOfNodes returns a reference to a hash where the keys are the graph nodes and the values are the pageranks of the node.

graph
 graph => Graph

graph must be a Graph object. If the directed parameter was not set with the constructor new or with this method, then directed is set to the value of Graph->is_directed().

listOfEdges
 listOfEdges => [['a',10],[10,11]]

listOfEdges must be a list of edges, where an edge is a pair of strings of the form [from-node, to-node] or a triple of the form [from-node, to-node, numeric-edge-weight]. Note that graph and listOfEdges can both be defined, in which case the union of their list of edges is used to compute the pageranks of the nodes.

listOfNodes
 listOfNodes => ['a',10, 'b']

listOfNodes is optional but, must be the list of nodes in the graph when provided; it defaults to all the nodes comprising the edges in listOfEdges or graph.

nodeWeights
  nodeWeights => {}

nodeWeights is an optional hash reference that can provide a weight for the nodes. If nodeWeights is not defined for any node in the graph, then each node has a weight of 1/scale(@listOfNodes). If nodeWeights is defined for at least one node in the graph, then the default weight of any undefined node is zero.

EXAMPLES ^

A rather dull example with one node and no edges:

  use Graph;
  use Graph::Centrality::Pagerank;
  use Data::Dump qw(dump);
  my $ranker = Graph::Centrality::Pagerank->new();
  my $listOfNodes = [1];
  dump $ranker->getPagerankOfNodes (listOfNodes => $listOfNodes);
  # dumps:
  # {
  #   1 => 1
  # }

An example of a graph with two components:

  use Graph::Centrality::Pagerank;
  use Data::Dump qw(dump);
  my $ranker = Graph::Centrality::Pagerank->new();
  my $listOfEdges = [[1,2],[3,4]];
  dump $ranker->getPagerankOfNodes (listOfEdges => $listOfEdges);
  # dumps:
  # {
  #   1 => "0.175438596989046",
  #   2 => "0.324561403010954",
  #   3 => "0.175438596989046",
  #   4 => "0.324561403010954",
  # }

In this case the edges are placed in a Graph:

  use Graph;
  use Graph::Centrality::Pagerank;
  use Data::Dump qw(dump);
  my $listOfEdges = [[1,2],[3,4]];
  my $graph = Graph->new (edges => $listOfEdges);
  my $ranker = Graph::Centrality::Pagerank->new();
  dump $ranker->getPagerankOfNodes (graph => $graph);
  # dumps:
  # {
  #   1 => "0.175438596989046",
  #   2 => "0.324561403010954",
  #   3 => "0.175438596989046",
  #   4 => "0.324561403010954",
  # }

Below is the first example in the paper How Google Finds Your Needle in the Web's Haystack by David Austin.

  use Graph::Centrality::Pagerank;
  use Data::Dump qw(dump);
  my $ranker = Graph::Centrality::Pagerank->new();
  my $listOfEdges = [[1,2],[1,3],[2,4],[3,2],[3,5],[4,2],[4,5],[4,6],[5,6],
    [5,7],[5,8],[6,8],[7,5],[7,1],[7,8],[8,6],[8,7]];
  dump $ranker->getPagerankOfNodes (listOfEdges => $listOfEdges,
    dampeningFactor => 1);
  # dumps:
  # {
  #   1 => "0.0599999994835539",
  #   2 => "0.0675000002254998",
  #   3 => "0.0300000002967361",
  #   4 => "0.0674999997408677",
  #   5 => "0.0974999994123176",
  #   6 => "0.202500001447512",
  #   7 => "0.180000001348251",
  #   8 => "0.294999998045262",
  # }

Below is the second example in the paper. Notice linkSinkNodes is set to zero.

  use Graph::Centrality::Pagerank;
  use Data::Dump qw(dump);
  my $ranker = Graph::Centrality::Pagerank->new();
  my $listOfEdges = [[1,2]];
  dump $ranker->getPagerankOfNodes (listOfEdges => $listOfEdges,
    dampeningFactor => 1, linkSinkNodes => 0);
  # dumps:
  # { 1 => 0, 2 => 0 }

Below is the third example in the paper. Notice in this case linkSinkNodes is set to one, the default value.

  use Graph::Centrality::Pagerank;
  use Data::Dump qw(dump);
  my $ranker = Graph::Centrality::Pagerank->new();
  my $listOfEdges = [[1,2]];
  dump $ranker->getPagerankOfNodes (listOfEdges => $listOfEdges,
    dampeningFactor => 1, linkSinkNodes => 1);
  # dumps:
  # { 1 => "0.33333333209157", 2 => "0.66666666790843" }

Below is the fourth example in the paper. The result is different from the paper since the starting vector for Graph::Centrality::Pagerank is

  { 1 => "0.2", 2 => "0.2", 3 => "0.2", 4 => "0.2", 5 => "0.2" }

while the starting vector in the paper is

  { 1 => 1, 2 => 0, 3 => 0, 4 => 0, 5 => 0 }.

  use Graph::Centrality::Pagerank;
  use Data::Dump qw(dump);
  my $ranker = Graph::Centrality::Pagerank->new();
  my $listOfEdges = [[1,2],[2,3],[3,4],[4,5],[5,1]];
  dump $ranker->getPagerankOfNodes (listOfEdges => $listOfEdges,
    dampeningFactor => 1, linkSinkNodes => 0);
  # dumps:
  # { 1 => "0.2", 2 => "0.2", 3 => "0.2", 4 => "0.2", 5 => "0.2" }

Below is the fifth example in the paper.

  use Graph::Centrality::Pagerank;
  use Data::Dump qw(dump);
  my $ranker = Graph::Centrality::Pagerank->new();
  my $listOfEdges = [[1,3],[1,2],[2,4],[3,2],[3,5],[4,2],[4,5],[4,6],[5,6],
    [5,7],[5,8],[6,8],[7,5],[7,8],[8,6],[8,7]];
  dump $ranker->getPagerankOfNodes (listOfEdges => $listOfEdges,
    dampeningFactor => 1, linkSinkNodes => 0);
  # dumps:
  # {
  #   1 => 0,
  #   2 => "2.39601089109228e-54",
  #   3 => 0,
  #   4 => "5.47659632249665e-54",
  #   5 => "0.119999999997811",
  #   6 => "0.240000000003975",
  #   7 => "0.240000000003975",
  #   8 => "0.399999999994238",
  # }

An example of the effect of including edge weights:

  use Graph::Centrality::Pagerank;
  use Data::Dump qw(dump);
  my $ranker = Graph::Centrality::Pagerank->new();
  my $listOfEdges = [[2,1],[2,3]];
  dump $ranker->getPagerankOfNodes (listOfEdges => $listOfEdges);
  $listOfEdges = [[2,1,2],[2,3,1]];
  dump $ranker->getPagerankOfNodes (listOfEdges => $listOfEdges,
    useEdgeWeights => 1);

  # dumps:
  # all edges have weight 1.
  # {
  #   1 => "0.370129870353883",
  #   2 => "0.259740259292235",
  #   3 => "0.370129870353883",
  # }
  # edge [2, 1] has twice the weight of edge [2,3].
  # {
  #   1 => "0.406926407374432",
  #   2 => "0.259740259292235",
  #   3 => "0.333333333333333",
  # }

An example of the effect of including node weights:

  use Graph;
  use Graph::Centrality::Pagerank;
  use Data::Dump qw(dump);
  my $ranker = Graph::Centrality::Pagerank->new();
  my $listOfEdges = [[1,2],[2,3]];
  dump $ranker->getPagerankOfNodes (listOfEdges => $listOfEdges);
  dump $ranker->getPagerankOfNodes (listOfEdges => $listOfEdges,
    nodeWeights => {2 => .9, 3 => .1 });

  # dumps:
  # {
  #   1 => "0.184416783248514",
  #   2 => "0.341171047056969",
  #   3 => "0.474412169694517",
  # }
  # {
  #   1 => "0.135592438389592",
  #   2 => "0.385846009631034",
  #   3 => "0.478561551979374",
  # }

A example of the modules speed, or lack of.

  use Graph::Centrality::Pagerank;
  use Data::Dump qw(dump);
  my $ranker = Graph::Centrality::Pagerank->new();
  my @listOfEdges;
  for (my $i = 0; $i < 1000000; $i++)
    { push @listOfEdges, [int rand 10000, int rand 10000]; }
  my $startTime = time;
  my $pageranks = $ranker->getPagerankOfNodes (listOfEdges => \@listOfEdges);
  print time()-$startTime . "\n";
  # prints:
  # a non-negative integer after a long time.

INSTALLATION ^

To install the module run the following commands:

  perl Makefile.PL
  make
  make test
  make install

If you are on a windows box you should use 'nmake' rather than 'make'.

BUGS ^

Please email bugs reports or feature requests to bug-graph-centrality-pagerank@rt.cpan.org, or through the web interface at http://rt.cpan.org/NoAuth/ReportBug.html?Queue=Graph-Centrality-Pagerank. The author will be notified and you can be automatically notified of progress on the bug fix or feature request.

AUTHOR ^

 Jeff Kubina<jeff.kubina@gmail.com>

COPYRIGHT ^

Copyright (c) 2009 Jeff Kubina. All rights reserved. This program is free software; you can redistribute it and/or modify it under the same terms as Perl itself.

The full text of the license can be found in the LICENSE file included with this module.

KEYWORDS ^

centrality measure, eigenvector centrality, graph, network, pagerank

SEE ALSO ^

A good tutorial on the pagerank algorithm is the article How Google Finds Your Needle in the Web's Haystack by David Austin.

Graph

centrality measure, eigenvector centrality, graph, network, pagerank
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