PageRank

Glossary

Directed: Directed trait. The algorithm is well-defined on a directed graph.
Directed: Directed trait. The algorithm ignores the direction of the graph.
Directed: Directed trait. The algorithm does not run on a directed graph.
Undirected: Undirected trait. The algorithm is well-defined on an undirected graph.
Undirected: Undirected trait. The algorithm ignores the undirectedness of the graph.
Heterogeneous nodes: Heterogeneous nodes fully supported. The algorithm has the ability to distinguish between nodes of different types.
Heterogeneous nodes: Heterogeneous nodes allowed. The algorithm treats all selected nodes similarly regardless of their label.
Heterogeneous relationships: Heterogeneous relationships fully supported. The algorithm has the ability to distinguish between relationships of different types.
Heterogeneous relationships: Heterogeneous relationships allowed. The algorithm treats all selected relationships similarly regardless of their type.
Weighted relationships: Weighted trait. The algorithm supports a relationship property to be used as weight, specified via the relationshipWeightProperty configuration parameter.
Weighted relationships: Weighted trait. The algorithm treats each relationship as equally important, discarding the value of any relationship weight.

Introduction

The PageRank algorithm measures the importance of each node within the graph, based on the number incoming relationships and the importance of the corresponding source nodes. The underlying assumption roughly speaking is that a page is only as important as the pages that link to it.

PageRank is introduced in the original Google paper as a function that solves the following equation:

where,

we assume that a page A has pages T₁ to T_n which point to it.
d is a damping factor which can be set between 0 (inclusive) and 1 (exclusive). It is usually set to 0.85.
C(A) is defined as the number of links going out of page A.

This equation is used to iteratively update a candidate solution and arrive at an approximate solution to the same equation.

For more information on this algorithm, see:

Running this algorithm requires sufficient memory availability. Before running this algorithm, we recommend that you read Memory Estimation.

Considerations

There are some things to be aware of when using the PageRank algorithm:

If there are no relationships from within a group of pages to outside the group, then the group is considered a spider trap.
Rank sink can occur when a network of pages is forming an infinite cycle.
Dead-ends occur when pages have no outgoing relationship.

Changing the damping factor can help with all the considerations above. It can be interpreted as a probability of a web surfer to sometimes jump to a random page and therefore not getting stuck in sinks.

Syntax

This section covers the syntax used to execute the PageRank algorithm in each of its execution modes. We are describing the named graph variant of the syntax. To learn more about general syntax variants, see Syntax overview.

PageRank syntax per mode

Run PageRank in stream mode on a named graph.

CALL gds.pageRank.stream(
  graphName: String,
  configuration: Map
)
YIELD
  nodeId: Integer,
  score: Float

Table 1. Parameters
Name	Type	Default	Optional	Description
graphName	String	`n/a`	no	The name of a graph stored in the catalog.
configuration	Map	`{}`	yes	Configuration for algorithm-specifics and/or graph filtering.

Table 2. Configuration
Name	Type	Default	Optional	Description
nodeLabels	List of String	`['*']`	yes	Filter the named graph using the given node labels. Nodes with any of the given labels will be included.
relationshipTypes	List of String	`['*']`	yes	Filter the named graph using the given relationship types. Relationships with any of the given types will be included.
concurrency	Integer	`4`	yes	The number of concurrent threads used for running the algorithm.
jobId	String	`Generated internally`	yes	An ID that can be provided to more easily track the algorithm’s progress.
logProgress	Boolean	`true`	yes	If disabled the progress percentage will not be logged.
dampingFactor	Float	`0.85`	yes	The damping factor of the Page Rank calculation. Must be in [0, 1).
maxIterations	Integer	`20`	yes	The maximum number of iterations of Page Rank to run.
tolerance	Float	`0.0000001`	yes	Minimum change in scores between iterations. If all scores change less than the tolerance value the result is considered stable and the algorithm returns.
relationshipWeightProperty	String	`null`	yes	Name of the relationship property to use as weights. If unspecified, the algorithm runs unweighted.
sourceNodes	List of Node or Number	`[]`	yes	The nodes or node ids to use for computing Personalized Page Rank.
scaler	String or Map	`None`	yes	The name of the scaler applied for the final scores. Supported values are `None`, `MinMax`, `Max`, `Mean`, `Log`, and `StdScore`. To apply scaler-specific configuration, use the Map syntax: `{scaler: 'name', …}`.

Table 3. Results
Name	Type	Description
nodeId	Integer	Node ID.
score	Float	PageRank score.

Run PageRank in stats mode on a named graph.

CALL gds.pageRank.stats(
  graphName: String,
  configuration: Map
)
YIELD
  ranIterations: Integer,
  didConverge: Boolean,
  preProcessingMillis: Integer,
  computeMillis: Integer,
  postProcessingMillis: Integer,
  centralityDistribution: Map,
  configuration: Map

Table 4. Parameters
Name	Type	Default	Optional	Description
graphName	String	`n/a`	no	The name of a graph stored in the catalog.
configuration	Map	`{}`	yes	Configuration for algorithm-specifics and/or graph filtering.

Table 5. Configuration
Name	Type	Default	Optional	Description
nodeLabels	List of String	`['*']`	yes	Filter the named graph using the given node labels. Nodes with any of the given labels will be included.
relationshipTypes	List of String	`['*']`	yes	Filter the named graph using the given relationship types. Relationships with any of the given types will be included.
concurrency	Integer	`4`	yes	The number of concurrent threads used for running the algorithm.
jobId	String	`Generated internally`	yes	An ID that can be provided to more easily track the algorithm’s progress.
logProgress	Boolean	`true`	yes	If disabled the progress percentage will not be logged.
dampingFactor	Float	`0.85`	yes	The damping factor of the Page Rank calculation. Must be in [0, 1).
maxIterations	Integer	`20`	yes	The maximum number of iterations of Page Rank to run.
tolerance	Float	`0.0000001`	yes	Minimum change in scores between iterations. If all scores change less than the tolerance value the result is considered stable and the algorithm returns.
relationshipWeightProperty	String	`null`	yes	Name of the relationship property to use as weights. If unspecified, the algorithm runs unweighted.
sourceNodes	List of Node or Number	`[]`	yes	The nodes or node ids to use for computing Personalized Page Rank.
scaler	String or Map	`None`	yes	The name of the scaler applied for the final scores. Supported values are `None`, `MinMax`, `Max`, `Mean`, `Log`, and `StdScore`. To apply scaler-specific configuration, use the Map syntax: `{scaler: 'name', …}`.

Table 6. Results
Name	Type	Description
ranIterations	Integer	The number of iterations run.
didConverge	Boolean	Indicates if the algorithm converged.
preProcessingMillis	Integer	Milliseconds for preprocessing the graph.
computeMillis	Integer	Milliseconds for running the algorithm.
postProcessingMillis	Integer	Milliseconds for computing the `centralityDistribution`.
centralityDistribution	Map	Map containing min, max, mean as well as p50, p75, p90, p95, p99 and p999 percentile values of centrality values.
configuration	Map	The configuration used for running the algorithm.

Run PageRank in mutate mode on a named graph.

CALL gds.pageRank.mutate(
  graphName: String,
  configuration: Map
)
YIELD
  nodePropertiesWritten: Integer,
  ranIterations: Integer,
  didConverge: Boolean,
  preProcessingMillis: Integer,
  computeMillis: Integer,
  postProcessingMillis: Integer,
  mutateMillis: Integer,
  centralityDistribution: Map,
  configuration: Map

Table 7. Parameters
Name	Type	Default	Optional	Description
graphName	String	`n/a`	no	The name of a graph stored in the catalog.
configuration	Map	`{}`	yes	Configuration for algorithm-specifics and/or graph filtering.

Table 8. Configuration
Name	Type	Default	Optional	Description
mutateProperty	String	`n/a`	no	The node property in the GDS graph to which the score is written.
nodeLabels	List of String	`['*']`	yes	Filter the named graph using the given node labels.
relationshipTypes	List of String	`['*']`	yes	Filter the named graph using the given relationship types.
concurrency	Integer	`4`	yes	The number of concurrent threads used for running the algorithm.
jobId	String	`Generated internally`	yes	An ID that can be provided to more easily track the algorithm’s progress.
dampingFactor	Float	`0.85`	yes	The damping factor of the Page Rank calculation. Must be in [0, 1).
maxIterations	Integer	`20`	yes	The maximum number of iterations of Page Rank to run.
tolerance	Float	`0.0000001`	yes	Minimum change in scores between iterations. If all scores change less than the tolerance value the result is considered stable and the algorithm returns.
relationshipWeightProperty	String	`null`	yes	Name of the relationship property to use as weights. If unspecified, the algorithm runs unweighted.
sourceNodes	List of Node or Number	`[]`	yes	The nodes or node ids to use for computing Personalized Page Rank.
scaler	String or Map	`None`	yes	The name of the scaler applied for the final scores. Supported values are `None`, `MinMax`, `Max`, `Mean`, `Log`, and `StdScore`. To apply scaler-specific configuration, use the Map syntax: `{scaler: 'name', …}`.

Table 9. Results
Name	Type	Description
ranIterations	Integer	The number of iterations run.
didConverge	Boolean	Indicates if the algorithm converged.
preProcessingMillis	Integer	Milliseconds for preprocessing the graph.
computeMillis	Integer	Milliseconds for running the algorithm.
postProcessingMillis	Integer	Milliseconds for computing the `centralityDistribution`.
mutateMillis	Integer	Milliseconds for adding properties to the projected graph.
nodePropertiesWritten	Integer	The number of properties that were written to the projected graph.
centralityDistribution	Map	Map containing min, max, mean as well as p50, p75, p90, p95, p99 and p999 percentile values of centrality values.
configuration	Map	The configuration used for running the algorithm.

Run PageRank in write mode on a named graph.

CALL gds.pageRank.write(
  graphName: String,
  configuration: Map
)
YIELD
  nodePropertiesWritten: Integer,
  ranIterations: Integer,
  didConverge: Boolean,
  preProcessingMillis: Integer,
  computeMillis: Integer,
  postProcessingMillis: Integer,
  writeMillis: Integer,
  centralityDistribution: Map,
  configuration: Map

Table 10. Parameters
Name	Type	Default	Optional	Description
graphName	String	`n/a`	no	The name of a graph stored in the catalog.
configuration	Map	`{}`	yes	Configuration for algorithm-specifics and/or graph filtering.

Table 11. Configuration
Name	Type	Default	Optional	Description
nodeLabels	List of String	`['*']`	yes	Filter the named graph using the given node labels. Nodes with any of the given labels will be included.
relationshipTypes	List of String	`['*']`	yes	Filter the named graph using the given relationship types. Relationships with any of the given types will be included.
concurrency	Integer	`4`	yes	The number of concurrent threads used for running the algorithm.
jobId	String	`Generated internally`	yes	An ID that can be provided to more easily track the algorithm’s progress.
logProgress	Boolean	`true`	yes	If disabled the progress percentage will not be logged.
writeConcurrency	Integer	`value of 'concurrency'`	yes	The number of concurrent threads used for writing the result to Neo4j.
writeProperty	String	`n/a`	no	The node property in the Neo4j database to which the score is written.
dampingFactor	Float	`0.85`	yes	The damping factor of the Page Rank calculation. Must be in [0, 1).
maxIterations	Integer	`20`	yes	The maximum number of iterations of Page Rank to run.
tolerance	Float	`0.0000001`	yes	Minimum change in scores between iterations. If all scores change less than the tolerance value the result is considered stable and the algorithm returns.
relationshipWeightProperty	String	`null`	yes	Name of the relationship property to use as weights. If unspecified, the algorithm runs unweighted.
sourceNodes	List of Node or Number	`[]`	yes	The nodes or node ids to use for computing Personalized Page Rank.
scaler	String or Map	`None`	yes	The name of the scaler applied for the final scores. Supported values are `None`, `MinMax`, `Max`, `Mean`, `Log`, and `StdScore`. To apply scaler-specific configuration, use the Map syntax: `{scaler: 'name', …}`.

Table 12. Results
Name	Type	Description
ranIterations	Integer	The number of iterations run.
didConverge	Boolean	Indicates if the algorithm converged.
preProcessingMillis	Integer	Milliseconds for preprocessing the graph.
computeMillis	Integer	Milliseconds for running the algorithm.
postProcessingMillis	Integer	Milliseconds for computing the `centralityDistribution`.
writeMillis	Integer	Milliseconds for writing result data back.
nodePropertiesWritten	Integer	The number of properties that were written to Neo4j.
centralityDistribution	Map	Map containing min, max, mean as well as p50, p75, p90, p95, p99 and p999 percentile values of centrality values.
configuration	Map	The configuration used for running the algorithm.

Examples

All the examples below should be run in an empty database.

The examples use native projections as the norm, although Cypher projections can be used as well.

In this section we will show examples of running the PageRank algorithm on a concrete graph. The intention is to illustrate what the results look like and to provide a guide in how to make use of the algorithm in a real setting. We will do this on a small web network graph of a handful nodes connected in a particular pattern. The example graph looks like this:

The following Cypher statement will create the example graph in the Neo4j database:

CREATE
  (home:Page {name:'Home'}),
  (about:Page {name:'About'}),
  (product:Page {name:'Product'}),
  (links:Page {name:'Links'}),
  (a:Page {name:'Site A'}),
  (b:Page {name:'Site B'}),
  (c:Page {name:'Site C'}),
  (d:Page {name:'Site D'}),

  (home)-[:LINKS {weight: 0.2}]->(about),
  (home)-[:LINKS {weight: 0.2}]->(links),
  (home)-[:LINKS {weight: 0.6}]->(product),
  (about)-[:LINKS {weight: 1.0}]->(home),
  (product)-[:LINKS {weight: 1.0}]->(home),
  (a)-[:LINKS {weight: 1.0}]->(home),
  (b)-[:LINKS {weight: 1.0}]->(home),
  (c)-[:LINKS {weight: 1.0}]->(home),
  (d)-[:LINKS {weight: 1.0}]->(home),
  (links)-[:LINKS {weight: 0.8}]->(home),
  (links)-[:LINKS {weight: 0.05}]->(a),
  (links)-[:LINKS {weight: 0.05}]->(b),
  (links)-[:LINKS {weight: 0.05}]->(c),
  (links)-[:LINKS {weight: 0.05}]->(d);

This graph represents eight pages, linking to one another. Each relationship has a property called weight, which describes the importance of the relationship.

The following statement will project a graph using a native projection and store it in the graph catalog under the name 'myGraph'.

CALL gds.graph.project(
  'myGraph',
  'Page',
  'LINKS',
  {
    relationshipProperties: 'weight'
  }
)

Memory Estimation

First off, we will estimate the cost of running the algorithm using the estimate procedure. This can be done with any execution mode. We will use the write mode in this example. Estimating the algorithm is useful to understand the memory impact that running the algorithm on your graph will have. When you later actually run the algorithm in one of the execution modes the system will perform an estimation. If the estimation shows that there is a very high probability of the execution going over its memory limitations, the execution is prohibited. To read more about this, see Automatic estimation and execution blocking.

For more details on estimate in general, see Memory Estimation.

The following will estimate the memory requirements for running the algorithm:

CALL gds.pageRank.write.estimate('myGraph', {
  writeProperty: 'pageRank',
  maxIterations: 20,
  dampingFactor: 0.85
})
YIELD nodeCount, relationshipCount, bytesMin, bytesMax, requiredMemory

Table 13. Results
nodeCount	relationshipCount	bytesMin	bytesMax	requiredMemory
8	14	696	696	"696 Bytes"

Stream

In the stream execution mode, the algorithm returns the score for each node. This allows us to inspect the results directly or post-process them in Cypher without any side effects. For example, we can order the results to find the nodes with the highest PageRank score.

For more details on the stream mode in general, see Stream.

The following will run the algorithm in stream mode:

CALL gds.pageRank.stream('myGraph')
YIELD nodeId, score
RETURN gds.util.asNode(nodeId).name AS name, score
ORDER BY score DESC, name ASC

Table 14. Results
name	score
"Home"	3.215681999884452
"About"	1.0542700552146722
"Links"	1.0542700552146722
"Product"	1.0542700552146722
"Site A"	0.3278578964488539
"Site B"	0.3278578964488539
"Site C"	0.3278578964488539
"Site D"	0.3278578964488539

The above query is running the algorithm in stream mode as unweighted and the returned scores are not normalized. Below, one can find an example for weighted graphs. Another example shows the application of a scaler to normalize the final scores.

While we are using the stream mode to illustrate running the algorithm as weighted or unweighted, all the algorithm modes support this configuration parameter.

Stats

In the stats execution mode, the algorithm returns a single row containing a summary of the algorithm result. For example PageRank stats returns centrality histogram which can be used to monitor the distribution of PageRank score values across all computed nodes. This execution mode does not have any side effects. It can be useful for evaluating algorithm performance by inspecting the computeMillis return item. In the examples below we will omit returning the timings. The full signature of the procedure can be found in the syntax section.

For more details on the stats mode in general, see Stats.

The following will run the algorithm and returns the result in form of statistical and measurement values

CALL gds.pageRank.stats('myGraph', {
  maxIterations: 20,
  dampingFactor: 0.85
})
YIELD centralityDistribution
RETURN centralityDistribution.max AS max

Table 15. Results
max
3.2156810760498047

The centrality histogram can be useful for inspecting the computed scores or perform normalizations.

Mutate

The mutate execution mode extends the stats mode with an important side effect: updating the named graph with a new node property containing the score for that node. The name of the new property is specified using the mandatory configuration parameter mutateProperty. The result is a single summary row, similar to stats, but with some additional metrics. The mutate mode is especially useful when multiple algorithms are used in conjunction.

For more details on the mutate mode in general, see Mutate.

The following will run the algorithm in mutate mode:

CALL gds.pageRank.mutate('myGraph', {
  maxIterations: 20,
  dampingFactor: 0.85,
  mutateProperty: 'pagerank'
})
YIELD nodePropertiesWritten, ranIterations

Table 16. Results
nodePropertiesWritten	ranIterations
`8`	`20`

Write

The write execution mode extends the stats mode with an important side effect: writing the score for each node as a property to the Neo4j database. The name of the new property is specified using the mandatory configuration parameter writeProperty. The result is a single summary row, similar to stats, but with some additional metrics. The write mode enables directly persisting the results to the database.

For more details on the write mode in general, see Write.

The following will run the algorithm in write mode:

CALL gds.pageRank.write('myGraph', {
  maxIterations: 20,
  dampingFactor: 0.85,
  writeProperty: 'pagerank'
})
YIELD nodePropertiesWritten, ranIterations

Table 17. Results
nodePropertiesWritten	ranIterations
`8`	`20`

Weighted

By default, the algorithm is considering the relationships of the graph to be unweighted, to change this behaviour we can use configuration parameter called relationshipWeightProperty. In the weighted case, the previous score of a node send to its neighbors, is multiplied by the relationship weight and then divided by the sum of the weights of its outgoing relationships. If the value of the relationship property is negative it will be ignored during computation. Below is an example of running the algorithm using the relationship property.

The following will run the algorithm in stream mode using relationship weights:

CALL gds.pageRank.stream('myGraph', {
  maxIterations: 20,
  dampingFactor: 0.85,
  relationshipWeightProperty: 'weight'
})
YIELD nodeId, score
RETURN gds.util.asNode(nodeId).name AS name, score
ORDER BY score DESC, name ASC

Table 18. Results
name	score
"Home"	3.53751028396339
"Product"	1.9357838291651097
"About"	0.7452612763883698
"Links"	0.7452612763883698
"Site A"	0.18152677135466103
"Site B"	0.18152677135466103
"Site C"	0.18152677135466103
"Site D"	0.18152677135466103

We are using stream mode to illustrate running the algorithm as weighted or unweighted, all the algorithm modes support this configuration parameter.

Tolerance

The tolerance configuration parameter denotes the minimum change in scores between iterations. If all scores change less than the configured tolerance value the result stabilises, and the algorithm returns.

The following will run the algorithm in stream mode using bigger tolerance value:

CALL gds.pageRank.stream('myGraph', {
  maxIterations: 20,
  dampingFactor: 0.85,
  tolerance: 0.1
})
YIELD nodeId, score
RETURN gds.util.asNode(nodeId).name AS name, score
ORDER BY score DESC, name ASC

Table 19. Results
name	score
"Home"	1.5812450669583336
"About"	0.5980194356381945
"Links"	0.5980194356381945
"Product"	0.5980194356381945
"Site A"	0.23374955154166668
"Site B"	0.23374955154166668
"Site C"	0.23374955154166668
"Site D"	0.23374955154166668

In this example we are using tolerance: 0.1, so the results are a bit different compared to the ones from stream example which is using the default value of tolerance. Note that the nodes 'About', 'Link' and 'Product' now have the same score, while with the default value of tolerance the node 'Product' has higher score than the other two.

Damping Factor

The damping factor configuration parameter accepts values between 0 (inclusive) and 1 (exclusive). If its value is too high then problems of sinks and spider traps may occur, and the values may oscillate so that the algorithm does not converge. If it’s too low then all scores are pushed towards 1, and the result will not sufficiently reflect the structure of the graph.

The following will run the algorithm in stream mode using smaller dampingFactor value:

CALL gds.pageRank.stream('myGraph', {
  maxIterations: 20,
  dampingFactor: 0.05
})
YIELD nodeId, score
RETURN gds.util.asNode(nodeId).name AS name, score
ORDER BY score DESC, name ASC

Table 20. Results
name	score
"Home"	1.2487309425844906
"About"	0.9708121818724536
"Links"	0.9708121818724536
"Product"	0.9708121818724536
"Site A"	0.9597081216238426
"Site B"	0.9597081216238426
"Site C"	0.9597081216238426
"Site D"	0.9597081216238426

Compared to the results from the stream example which is using the default value of dampingFactor the score values are closer to each other when using dampingFactor: 0.05. Also, note that the nodes 'About', 'Link' and 'Product' now have the same score, while with the default value of dampingFactor the node 'Product' has higher score than the other two.

Personalised PageRank

Personalized PageRank is a variation of PageRank which is biased towards a set of sourceNodes. This variant of PageRank is often used as part of recommender systems.

The following examples show how to run PageRank centered around 'Site A'.

The following will run the algorithm and stream results:

MATCH (siteA:Page {name: 'Site A'})
CALL gds.pageRank.stream('myGraph', {
  maxIterations: 20,
  dampingFactor: 0.85,
  sourceNodes: [siteA]
})
YIELD nodeId, score
RETURN gds.util.asNode(nodeId).name AS name, score
ORDER BY score DESC, name ASC

Table 21. Results
name	score
"Home"	0.39902290442518784
"Site A"	0.16890325301726694
"About"	0.11220151747374331
"Links"	0.11220151747374331
"Product"	0.11220151747374331
"Site B"	0.01890325301726691
"Site C"	0.01890325301726691
"Site D"	0.01890325301726691

Comparing these results to the ones from the stream example (which is not using sourceNodes configuration parameter) shows that the 'Site A' node that we used in the sourceNodes list now scores second instead of fourth.

Scaling centrality scores

To normalize the final scores as part of the algorithm execution, one can use the scaler configuration parameter. A description of all available scalers can be found in the documentation for the scaleProperties procedure.

The following will run the algorithm in stream mode and returns normalized results:

CALL gds.pageRank.stream('myGraph', {
  scaler: "MEAN"
})
YIELD nodeId, score
RETURN gds.util.asNode(nodeId).name AS name, score
ORDER BY score DESC, name ASC

Table 22. Results
name	score
"Home"	0.780671346390832
"About"	0.032214422681946
"Links"	0.032214422681946
"Product"	0.032214422681946
"Site A"	-0.219328653609168
"Site B"	-0.219328653609168
"Site C"	-0.219328653609168
"Site D"	-0.219328653609168

Comparing the results with the stream example, we can see that the relative order of scores is the same.