page.rank {igraph} | R Documentation

## The Page Rank algorithm

### Description

Calculates the Google PageRank for the specified vertices.

### Usage

    page.rank (graph, vids = V(graph), directed = TRUE, damping = 0.85,
        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).

`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. If `weights` is a numerical vector then it used, even if the graph has a `weights` edge attribute. If this is `NA`, then no edge weights are used (even if the graph has a `weight` edge attribute.

`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:

`vector` | A numeric vector with the PageRank scores.
`value` | The eigenvalue corresponding to the eigenvector with the page rank scores. It should be always exactly one.
`options` | Some information about the underlying ARPACK calculation. See `arpack` for details.

For `page.rank.old` a numeric vector of Page Rank scores.

### Author(s)

Tamas Nepusz ntamas@rmki.kfki.hu and Gabor Csardi csardi@rmki.kfki.hu

### 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`

### Examples

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

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

[Package igraph version 0.5.2 Index]