Calculates the Google PageRank for the specified vertices.

```
page_rank(
graph,
algo = c("prpack", "arpack", "power"),
vids = V(graph),
directed = TRUE,
damping = 0.85,
personalized = NULL,
weights = NULL,
options = NULL
)
```page_rank_old(
graph,
vids = V(graph),
directed = TRUE,
niter = 1000,
eps = 0.001,
damping = 0.85,
old = FALSE
)

graph

The graph object.

algo

Character scalar, which implementation to use to carry out the
calculation. The default is `"prpack"`

, which uses the PRPACK library
(https://github.com/dgleich/prpack). This is a new implementation in igraph
version 0.7, and the suggested one, as it is the most stable and the fastest
for all but small graphs. `"arpack"`

uses the ARPACK library, the
default implementation from igraph version 0.5 until version 0.7.
`power`

uses a simple implementation of the power method, this was the
default in igraph before version 0.5 and is the same as calling
`page_rank_old`

.

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 should contains an entry for each vertex and it will be rescaled to sum up to one.

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.
This function interprets edge weights as connection strengths. In the
random surfer model, an edge with a larger weight is more likely to be
selected by the surfer.

options

Either a named list, to override some ARPACK options. See
`arpack`

for details; or a named list to override the default
options for the power method (if `algo="power"`

). The default options
for the power method are `niter=1000`

and `eps=0.001`

. This
argument is ignored if the PRPACK implementation is used.

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.

For `page_rank`

a named list with entries:

A numeric vector with the PageRank scores.

The eigenvalue corresponding to the eigenvector with the page rank scores. It should be always exactly one.

Some information about the underlying
ARPACK calculation. See `arpack`

for details. This entry is
`NULL`

if not the ARPACK implementation was used.

For page_rank_old a numeric vector of Page Rank scores.

For the explanation of the PageRank algorithm, see the following webpage: http://infolab.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.

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.

Other centrality scores: `closeness`

,
`betweenness`

, `degree`

```
# NOT RUN {
g <- sample_gnp(20, 5/20, directed=TRUE)
page_rank(g)$vector
g2 <- make_star(10)
page_rank(g2)$vector
# Personalized PageRank
g3 <- make_ring(10)
page_rank(g3)$vector
reset <- seq(vcount(g3))
page_rank(g3, personalized=reset)$vector
# }
```

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