# page.rank

##### 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:

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.

##### 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`

##### 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)*