page.rank

0th

Percentile

The Page Rank algorithm

Calculates the Google PageRank for the specified vertices.

Keywords
graphs
Usage
page.rank (graph, vids = V(graph), directed = TRUE, damping = 0.85, 
    personalized = NULL, weights = NULL,
    options = igraph.arpack.default) 
page.rank.old (graph, vids = V(graph), directed = TRUE, niter = 1000, 
    eps = 0.001, damping = 0.85, old = FALSE)
Arguments
graph
The graph object.
vids
The vertices of interest.
directed
Logical, if true directed paths will be considered for directed graphs. It is ignored for undirected graphs.
damping
The damping factor (d in the original paper).
personalized
Optional vector giving a probability distribution to calculate personalized PageRank. For personalized PageRank, the probability of jumping to a node when abandoning the random walk is not uniform, but it is given by this vector. The vector sh
weights
A numerical vector or NULL. This argument can be used to give edge weights for calculating the weighted PageRank of vertices. If this is NULL and the graph has a weight edge attribute then that is used. I
options
A named list, to override some ARPACK options. See arpack for details.
niter
The maximum number of iterations to perform.
eps
The algorithm will consider the calculation as complete if the difference of PageRank values between iterations change less than this value for every node.
old
A logical scalar, whether the old style (pre igraph 0.5) normalization to use. See details below.
Details

For the explanation of the PageRank algorithm, see the following webpage: http://www-db.stanford.edu/~backrub/google.html, or the following reference:

Sergey Brin and Larry Page: The Anatomy of a Large-Scale Hypertextual Web Search Engine. Proceedings of the 7th World-Wide Web Conference, Brisbane, Australia, April 1998.

igraph 0.5 (and later) contains two PageRank calculation implementations. The page.rank function uses ARPACK to perform the calculation, see also arpack.

The page.rank.old function performs a simple power method, this is the implementation that was available under the name page.rank in pre 0.5 igraph versions. Note that page.rank.old has an argument called old. If this argument is FALSE (the default), then the proper PageRank algorithm is used, i.e. $(1-d)/n$ is added to the weighted PageRank of vertices to calculate the next iteration. If this argument is TRUE then $(1-d)$ is added, just like in the PageRank paper; $d$ is the damping factor, and $n$ is the total number of vertices. A further difference is that the old implementation does not renormalize the page rank vector after each iteration. Note that the old=FALSE method is not stable, is does not necessarily converge to a fixed point. It should be avoided for new code, it is only included for compatibility with old igraph versions. Please note that the PageRank of a given vertex depends on the PageRank of all other vertices, so even if you want to calculate the PageRank for only some of the vertices, all of them must be calculated. Requesting the PageRank for only some of the vertices does not result in any performance increase at all.

Since the calculation is an iterative process, the algorithm is stopped after a given count of iterations or if the PageRank value differences between iterations are less than a predefined value.

Value

  • For page.rank a named list with entries:
  • vectorA numeric vector with the PageRank scores.
  • valueThe eigenvalue corresponding to the eigenvector with the page rank scores. It should be always exactly one.
  • optionsSome information about the underlying ARPACK calculation. See arpack for details.
  • For page.rank.old a numeric vector of Page Rank scores.

concept

Page rank

References

Sergey Brin and Larry Page: The Anatomy of a Large-Scale Hypertextual Web Search Engine. Proceedings of the 7th World-Wide Web Conference, Brisbane, Australia, April 1998.

See Also

Other centrality scores: closeness, betweenness, degree

Aliases
  • page.rank
  • page.rank.old
Examples
g <- random.graph.game(20, 5/20, directed=TRUE)
page.rank(g)$vector

g2 <- graph.star(10)
page.rank(g2)$vector

# Personalized PageRank
g3 <- graph.ring(10)
page.rank(g3)$vector
reset <- seq(vcount(g3))
page.rank(g3, personalized=reset)$vector
Documentation reproduced from package igraph, version 0.6.5-2, License: GPL (>= 2)

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