# The PageRank Algorithm

The PageRank is an algorithm that measures the “importance” of the nodes in a graph. It assigns to each node a rank. This rank corresponds to the probability that a “random surfer” visits the node. The surfer goes from node to node in the following way: with probability d she chooses a random outgoing arc and with probability 1 - d she “teleports” to a random node (possibly not connected to the current). The probability d is called damping factor. By default it is 0.85 but it can be customized using the `setDampingFactor(double)` method. The ranks are real numbers between 0 and 1 and sum up to one.

## Usage

This implementation uses a variant of the power iteration algorithm to compute the node ranks. It computes the approximate ranks iteratively going closer to the exact values at each iteration. The accuracy can be controlled by a precision parameter . When the L1 norm of the difference between two consecutive rank vectors becomes less than this parameter, the result is considered precise enough and the computation stops.

This implementation works with both directed and undirected edges. An undirected edge acts as two directed arcs.

The graph dynamics is taken into account and the ranks are not computed from scratch at each modification in the structure of the graph. However, the ranks become less and less accurate after each modification. To establish the desired precision, one must either explicitly call `compute()` or ask for a rank of a node by calling `getRank(Node)`.

The computed ranks are stored in node attribute. The name of this attribute can be changed by a call to `setRankAttribute(String)` but only before the call to `init(Graph)`. Another way to obtain the ranks is to call `getRank(Node)`. The second method is preferable because it will update the ranks if needed and will always return values within the desired precision.

## Complexity

Each iteration takes O(m + n) time, where n is the number of nodes and m is the number of edges. The number of iterations needed to converge depends on the desired precision.

## Reference

• Lawrence Page, Sergey Brin, Rajeev Motwani and Terry Winograd. The PageRank citation ranking: Bringing order to the Web. 1999