Fast Random Projection

This section describes the Fast Random Projection (FastRP) node embedding algorithm in the Neo4j Graph Data Science library.

Supported algorithm traits:

Directed

1. Introduction

Fast Random Projection, or FastRP for short, is a node embedding algorithm in the family of random projection algorithms. These algorithms are theoretically backed by the Johnsson-Lindenstrauss lemma according to which one can project n vectors of arbitrary dimension into O(log(n)) dimensions and still approximately preserve pairwise distances among the points. In fact, a linear projection chosen in a random way satisfies this property.

Such techniques therefore allow for aggressive dimensionality reduction while preserving most of the distance information. The FastRP algorithm operates on graphs, in which case we care about preserving similarity between nodes and their neighbors. This means that two nodes that have similar neighborhoods should be assigned similar embedding vectors. Conversely, two nodes that are not similar should be not be assigned similar embedding vectors.

The FastRP algorithm initially assigns random vectors to all nodes using a technique called very sparse random projection, see (Achlioptas, 2003) below. Moreover, in GDS it is possible to use node properties for the creation of these initial random vectors in a way described below. We will also use projection of a node synonymously with the initial random vector of a node.

Starting with these random vectors and iteratively averaging over node neighborhoods, the algorithm constructs a sequence of intermediate embeddings e n to the ith for each node n. More precisely,

e n to the ith equals average of e m to the ith minus one

where m ranges over neighbors of n and e n to the zeroeth is the node’s initial random vector.

The embedding e n of node n, which is the output of the algorithm, is a combination of the vectors and embeddings defined above:

e n equals w zero times normalise r n plus sum from i equals 1 to k of w i times normalise e n to the ith

where normalize is the function which divides a vector with its L2 norm, the value of nodeSelfInfluence is w zero , and the values of iterationWeights are w 1 comma w 2 comma dot dot dot w k . We will return to Node Self Influence later on.

Therefore, each node’s embedding depends on a neighborhood of radius equal to the number of iterations. This way FastRP exploits higher-order relationships in the graph while still being highly scalable.

The present implementation extends the original algorithm to support weighted graphs, which computes weighted averages of neighboring embeddings using the relationship weights. In order to make use of this, the relationshipWeightProperty parameter should be set to an existing relationship property.

The original algorithm is intended only for undirected graphs. We support running on both on directed graphs and undirected graph. For directed graphs we consider only the outgoing neighbors when computing the intermediate embeddings for a node. Therefore, using the orientations NATURAL, REVERSE or UNDIRECTED will all give different embeddings. In general, it is recommended to first use UNDIRECTED as this is what the original algorithm was evaluated on.

For more information on this algorithm see:

1.1. Node properties

Most real-world graphs contain node properties which store information about the nodes and what they represent. The FastRP algorithm in the GDS library extends the original FastRP algorithm with a capability to take node properties into account. The resulting embeddings can therefore represent the graph more accurately.

The node property aware aspect of the algorithm is configured via the parameters featureProperties and propertyRatio. Each node property in featureProperties is associated with a randomly generated vector of dimension propertyDimension, where propertyDimension = embeddingDimension * propertyRatio. Each node is then initialized with a vector of size embeddingDimension formed by concatenation of two parts:

The first part is formed like in the standard FastRP algorithm,
The second one is a linear combination of the property vectors, using the property values of the node as weights.

The algorithm then proceeds with the same logic as the FastRP algorithm. Therefore, the algorithm will output arrays of size embeddingDimension. The last propertyDimension coordinates in the embedding captures information about property values of nearby nodes (the "property part" below), and the remaining coordinates (embeddingDimension - propertyDimension of them; "topology part") captures information about nearby presence of nodes.

[0, 1, ...        | ...,   N - 1, N]
 ^^^^^^^^^^^^^^^^ | ^^^^^^^^^^^^^^^
  topology part   |  property part
                  ^
           property ratio

2. Tuning algorithm parameters

In order to improve the embedding quality using FastRP on one of your graphs, it is possible to tune the algorithm parameters. This process of finding the best parameters for your specific use case and graph is typically referred to as hyperparameter tuning. We will go through each of the configuration parameters and explain how they behave.

For statistically sound results, it is a good idea to reserve a test set excluded from parameter tuning. After selecting a set of parameter values, the embedding quality can be evaluated using a downstream machine learning task on the test set. By varying the parameter values and studying the precision of the machine learning task, it is possible to deduce the parameter values that best fit the concrete dataset and use case. To construct such a set you may want to use a dedicated node label in the graph to denote a subgraph without the test data.

2.1. Embedding dimension

The embedding dimension is the length of the produced vectors. A greater dimension offers a greater precision, but is more costly to operate over.

The optimal embedding dimension depends on the number of nodes in the graph. Since the amount of information the embedding can encode is limited by its dimension, a larger graph will tend to require a greater embedding dimension. A typical value is a power of two in the range 128 - 1024. A value of at least 256 gives good results on graphs in the order of 10⁵ nodes, but in general increasing the dimension improves results. Increasing embedding dimension will however increase memory requirements and runtime linearly.

2.2. Normalization strength

The normalization strength is used to control how node degrees influence the embedding. Using a negative value will downplay the importance of high degree neighbors, while a positive value will instead increase their importance. The optimal normalization strength depends on the graph and on the task that the embeddings will be used for. In the original paper, hyperparameter tuning was done in the range of [-1,0] (no positive values), but we have found cases where a positive normalization strengths gives better results.

2.3. Iteration weights

The iteration weights parameter control two aspects: the number of iterations, and their relative impact on the final node embedding. The parameter is a list of numbers, indicating one iteration per number where the number is the weight applied to that iteration.

In each iteration, the algorithm will expand across all relationships in the graph. This has some implications:

With a single iteration, only direct neighbors will be considered for each node embedding.
With two iterations, direct neighbors and second-degree neighbors will be considered for each node embedding.
With three iterations, direct neighbors, second-degree neighbors, and third-degree neighbors will be considered for each node embedding. Direct neighbors may be reached twice, in different iterations.
In general, the embedding corresponding to the i:th iteration contains features depending on nodes reachable with paths of length i. If the graph is undirected, then a node reachable with a path of length L can also be reached with length L+2k, for any integer k.
In particular, a node may reach back to itself on each even iteration (depending on the direction in the graph).

It is good to have at least one non-zero weight in an even and in an odd position. Typically, using at least a few iterations, for example three, is recommended. However, a too high value will consider nodes far away and may not be informative or even be detrimental. The intuition here is that as the projections reach further away from the node, the less specific the neighborhood becomes. Of course, a greater number of iterations will also take more time to complete.

2.4. Node Self Influence

Node Self Influence is a variation of the original FastRP algorithm.

How much a node’s embedding is affected by the intermediate embedding at iteration i is controlled by the i'th element of iterationWeights. This can also be seen as how much the initial random vectors, or projections, of nodes that can be reached in i hops from a node affect the embedding of the node. Similarly, nodeSelfInfluence behaves like an iteration weight for a 0 th iteration, or the amount of influence the projection of a node has on the embedding of the same node.

A reason for setting this parameter to a non-zero value is if your graph has low connectivity or a significant amount of isolated nodes. Isolated nodes combined with using propertyRatio = 0.0 leads to embeddings that contain all zeros. However using node properties along with node self influence can thus produce more meaningful embeddings for such nodes. This can be seen as producing fallback features when graph structure is (locally) missing. Moreover, sometimes a node’s own properties are simply informative features and are good to include even if connectivity is high. Finally, node self influence can be used for pure dimensionality reduction to compress node properties used for node classification.

If node properties are not used, using nodeSelfInfluence may also have a positive effect, depending on other settings and on the problem.

2.5. Orientation

Choosing the right orientation when creating the graph may have the single greatest impact. The FastRP algorithm is designed to work with undirected graphs, and we expect this to be the best in most cases. If you expect only outgoing or incoming relationships to be informative for a prediction task, then you may want to try using the orientations NATURAL or REVERSE respectively.

3. Syntax

This section covers the syntax used to execute the FastRP 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.

FastRP syntax per mode

Run FastRP in stream mode on a named graph.

CALL gds.fastRP.stream(
  graphName: String,
  configuration: Map
) YIELD
  nodeId: Integer,
  embedding: List of 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. General configuration for algorithm execution on a named graph.
Name	Type	Default	Optional	Description
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.

Table 3. Algorithm specific configuration
Name	Type	Default	Optional	Description
propertyRatio	Float	`0.0`	yes	The desired ratio of the property embedding dimension to the total `embeddingDimension`. A positive value requires `featureProperties` to be non-empty.
featureProperties	List of String	`[]`	yes	The names of the node properties that should be used as input features. All property names must exist in the in-memory graph and be of type Float or List of Float.
embeddingDimension	Integer	`n/a`	no	The dimension of the computed node embeddings. Minimum value is 1.
iterationWeights	List of Float	`[0.0, 1.0, 1.0]`	yes	Contains a weight for each iteration. The weight controls how much the intermediate embedding from the iteration contributes to the final embedding.
nodeSelfInfluence	Float	`0.0`	yes	Controls for each node how much its initial random vector contributes to its final embedding.
normalizationStrength	Float	`0.0`	yes	The initial random vector for each node is scaled by its degree to the power of `normalizationStrength`.
randomSeed	Integer	`n/a`	yes	A random seed which is used for all randomness in computing the embeddings.
relationshipWeightProperty	String	`null`	yes	Name of the relationship property to use for weighted random projection. If unspecified, the algorithm runs unweighted.
The number of iterations is equal to the length of `iterationWeights`.
It is required that `iterationWeights` is non-empty or `nodeSelfInfluence` is non-zero.

Table 4. Results
Name	Type	Description
nodeId	Integer	Node ID.
embedding	List of Float	FastRP node embedding.

Run FastRP in stats mode on a named graph.

CALL gds.fastRP.stats(
  graphName: String,
  configuration: Map
) YIELD
  nodeCount: Integer,
  createMillis: Integer,
  computeMillis: Integer,
  configuration: Map

Table 5. 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 6. General configuration for algorithm execution on a named graph.
Name	Type	Default	Optional	Description
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.

Table 7. Algorithm specific configuration
Name	Type	Default	Optional	Description
propertyRatio	Float	`0.0`	yes	The desired ratio of the property embedding dimension to the total `embeddingDimension`. A positive value requires `featureProperties` to be non-empty.
featureProperties	List of String	`[]`	yes	The names of the node properties that should be used as input features. All property names must exist in the in-memory graph and be of type Float or List of Float.
embeddingDimension	Integer	`n/a`	no	The dimension of the computed node embeddings. Minimum value is 1.
iterationWeights	List of Float	`[0.0, 1.0, 1.0]`	yes	Contains a weight for each iteration. The weight controls how much the intermediate embedding from the iteration contributes to the final embedding.
nodeSelfInfluence	Float	`0.0`	yes	Controls for each node how much its initial random vector contributes to its final embedding.
normalizationStrength	Float	`0.0`	yes	The initial random vector for each node is scaled by its degree to the power of `normalizationStrength`.
randomSeed	Integer	`n/a`	yes	A random seed which is used for all randomness in computing the embeddings.
relationshipWeightProperty	String	`null`	yes	Name of the relationship property to use for weighted random projection. If unspecified, the algorithm runs unweighted.
The number of iterations is equal to the length of `iterationWeights`.
It is required that `iterationWeights` is non-empty or `nodeSelfInfluence` is non-zero.

Table 8. Results
Name	Type	Description
nodeCount	Integer	Number of nodes processed.
createMillis	Integer	Milliseconds for creating the graph.
computeMillis	Integer	Milliseconds for running the algorithm.
configuration	Map	Configuration used for running the algorithm.

Run FastRP in mutate mode on a named graph.

CALL gds.fastRP.mutate(
  graphName: String,
  configuration: Map
) YIELD
  nodeCount: Integer,
  nodePropertiesWritten: Integer,
  createMillis: Integer,
  computeMillis: Integer,
  mutateMillis: Integer,
  configuration: Map

Table 9. 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 10. General configuration for algorithm execution on a named graph.
Name	Type	Default	Optional	Description
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.
mutateProperty	String	`n/a`	no	The node property in the GDS graph to which the embedding is written.

Table 11. Algorithm specific configuration
Name	Type	Default	Optional	Description
propertyRatio	Float	`0.0`	yes	The desired ratio of the property embedding dimension to the total `embeddingDimension`. A positive value requires `featureProperties` to be non-empty.
featureProperties	List of String	`[]`	yes	The names of the node properties that should be used as input features. All property names must exist in the in-memory graph and be of type Float or List of Float.
embeddingDimension	Integer	`n/a`	no	The dimension of the computed node embeddings. Minimum value is 1.
iterationWeights	List of Float	`[0.0, 1.0, 1.0]`	yes	Contains a weight for each iteration. The weight controls how much the intermediate embedding from the iteration contributes to the final embedding.
nodeSelfInfluence	Float	`0.0`	yes	Controls for each node how much its initial random vector contributes to its final embedding.
normalizationStrength	Float	`0.0`	yes	The initial random vector for each node is scaled by its degree to the power of `normalizationStrength`.
randomSeed	Integer	`n/a`	yes	A random seed which is used for all randomness in computing the embeddings.
relationshipWeightProperty	String	`null`	yes	Name of the relationship property to use for weighted random projection. If unspecified, the algorithm runs unweighted.
The number of iterations is equal to the length of `iterationWeights`.
It is required that `iterationWeights` is non-empty or `nodeSelfInfluence` is non-zero.

Table 12. Results
Name	Type	Description
nodeCount	Integer	Number of nodes processed.
nodePropertiesWritten	Integer	Number of node properties written.
createMillis	Integer	Milliseconds for creating the graph.
computeMillis	Integer	Milliseconds for running the algorithm.
mutateMillis	Integer	Milliseconds for adding properties to the in-memory graph.
configuration	Map	Configuration used for running the algorithm.

Run FastRP in write mode on a named graph.

CALL gds.fastRP.write(
  graphName: String,
  configuration: Map
) YIELD
  nodeCount: Integer,
  nodePropertiesWritten: Integer,
  createMillis: Integer,
  computeMillis: Integer,
  writeMillis: Integer,
  configuration: Map

Table 13. 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 14. General configuration for algorithm execution on a named graph.
Name	Type	Default	Optional	Description
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. Also provides the default value for 'writeConcurrency'.
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 embedding is written.

Table 15. Algorithm specific configuration
Name	Type	Default	Optional	Description
propertyRatio	Float	`0.0`	yes	The desired ratio of the property embedding dimension to the total `embeddingDimension`. A positive value requires `featureProperties` to be non-empty.
featureProperties	List of String	`[]`	yes	The names of the node properties that should be used as input features. All property names must exist in the in-memory graph and be of type Float or List of Float.
embeddingDimension	Integer	`n/a`	no	The dimension of the computed node embeddings. Minimum value is 1.
iterationWeights	List of Float	`[0.0, 1.0, 1.0]`	yes	Contains a weight for each iteration. The weight controls how much the intermediate embedding from the iteration contributes to the final embedding.
nodeSelfInfluence	Float	`0.0`	yes	Controls for each node how much its initial random vector contributes to its final embedding.
normalizationStrength	Float	`0.0`	yes	The initial random vector for each node is scaled by its degree to the power of `normalizationStrength`.
randomSeed	Integer	`n/a`	yes	A random seed which is used for all randomness in computing the embeddings.
relationshipWeightProperty	String	`null`	yes	Name of the relationship property to use for weighted random projection. If unspecified, the algorithm runs unweighted.
The number of iterations is equal to the length of `iterationWeights`.
It is required that `iterationWeights` is non-empty or `nodeSelfInfluence` is non-zero.

Table 16. Results
Name	Type	Description
nodeCount	Integer	Number of nodes processed.
nodePropertiesWritten	Integer	Number of node properties written.
createMillis	Integer	Milliseconds for creating the graph.
computeMillis	Integer	Milliseconds for running the algorithm.
writeMillis	Integer	Milliseconds for writing result data back to Neo4j.
configuration	Map	Configuration used for running the algorithm.

3.1. Anonymous graphs

It is also possible to execute the algorithm on a graph that is projected in conjunction with the algorithm execution. In this case, the graph does not have a name, and we call it anonymous. When executing over an anonymous graph the configuration map contains a graph projection configuration as well as an algorithm configuration. All execution modes support execution on anonymous graphs, although we only show syntax and mode-specific configuration for the write mode for brevity.

For more information on syntax variants, see Syntax overview.

Run FastRP in write mode on an anonymous graph.

CALL gds.fastRP.write(
  configuration: Map
)
YIELD
  nodeCount: Integer,
  nodePropertiesWritten: Integer,
  createMillis: Integer,
  computeMillis: Integer,
  writeMillis: Integer,
  configuration: Map

Table 17. General configuration for algorithm execution on an anonymous graph.
Name	Type	Default	Optional	Description
nodeProjection	String, List of String or Map	`null`	yes	The node projection used for anonymous graph creation via a Native projection.
relationshipProjection	String, List of String or Map	`null`	yes	The relationship projection used for anonymous graph creation a Native projection.
nodeQuery	String	`null`	yes	The Cypher query used to select the nodes for anonymous graph creation via a Cypher projection.
relationshipQuery	String	`null`	yes	The Cypher query used to select the relationships for anonymous graph creation via a Cypher projection.
nodeProperties	String, List of String or Map	`null`	yes	The node properties to project during anonymous graph creation.
relationshipProperties	String, List of String or Map	`null`	yes	The relationship properties to project during anonymous graph creation.
concurrency	Integer	`4`	yes	The number of concurrent threads used for running the algorithm. Also provides the default value for 'readConcurrency' and 'writeConcurrency'.
readConcurrency	Integer	`value of 'concurrency'`	yes	The number of concurrent threads used for creating the graph.
writeConcurrency	Integer	`value of 'concurrency'`	yes	WRITE mode only: The number of concurrent threads used for writing the result.
writeProperty	String	`n/a`	no	WRITE mode only: The node property to which the embedding is written to.

Table 18. Algorithm specific configuration
Name	Type	Default	Optional	Description
propertyRatio	Float	`0.0`	yes	The desired ratio of the property embedding dimension to the total `embeddingDimension`. A positive value requires `featureProperties` to be non-empty.
featureProperties	List of String	`[]`	yes	The names of the node properties that should be used as input features. All property names must exist in the in-memory graph and be of type Float or List of Float.
embeddingDimension	Integer	`n/a`	no	The dimension of the computed node embeddings. Minimum value is 1.
iterationWeights	List of Float	`[0.0, 1.0, 1.0]`	yes	Contains a weight for each iteration. The weight controls how much the intermediate embedding from the iteration contributes to the final embedding.
nodeSelfInfluence	Float	`0.0`	yes	Controls for each node how much its initial random vector contributes to its final embedding.
normalizationStrength	Float	`0.0`	yes	The initial random vector for each node is scaled by its degree to the power of `normalizationStrength`.
randomSeed	Integer	`n/a`	yes	A random seed which is used for all randomness in computing the embeddings.
relationshipWeightProperty	String	`null`	yes	Name of the relationship property to use for weighted random projection. If unspecified, the algorithm runs unweighted.
The number of iterations is equal to the length of `iterationWeights`.
It is required that `iterationWeights` is non-empty or `nodeSelfInfluence` is non-zero.

The results are the same as for running write mode with a named graph, see the write mode syntax above.

4. Examples

In this section we will show examples of running the FastRP node embedding 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 social 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
  (dan:Person {name: 'Dan', age: 18}),
  (annie:Person {name: 'Annie', age: 12}),
  (matt:Person {name: 'Matt', age: 22}),
  (jeff:Person {name: 'Jeff', age: 51}),
  (brie:Person {name: 'Brie', age: 45}),
  (elsa:Person {name: 'Elsa', age: 65}),
  (john:Person {name: 'John', age: 64}),

  (dan)-[:KNOWS {weight: 1.0}]->(annie),
  (dan)-[:KNOWS {weight: 1.0}]->(matt),
  (annie)-[:KNOWS {weight: 1.0}]->(matt),
  (annie)-[:KNOWS {weight: 1.0}]->(jeff),
  (annie)-[:KNOWS {weight: 1.0}]->(brie),
  (matt)-[:KNOWS {weight: 3.5}]->(brie),
  (brie)-[:KNOWS {weight: 1.0}]->(elsa),
  (brie)-[:KNOWS {weight: 2.0}]->(jeff),
  (john)-[:KNOWS {weight: 1.0}]->(jeff);

This graph represents seven people who know one another. A relationship property weight denotes the strength of the knowledge between two persons.

With the graph in Neo4j we can now project it into the graph catalog to prepare it for algorithm execution. We do this using a native projection targeting the Person nodes and the KNOWS relationships. For the relationships we will use the UNDIRECTED orientation. This is because the FastRP algorithm has been measured to compute more predictive node embeddings in undirected graphs. We will also add the weight relationship property which we will make use of when running the weighted version of FastRP.

In the examples below we will use named graphs and native projections as the norm. However, anonymous graphs and/or Cypher projections can also be used.

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

CALL gds.graph.create(
  'persons',
  'Person',
  {
    KNOWS: {
      orientation: 'UNDIRECTED',
      properties: 'weight'
    }
  },
  { nodeProperties: ['age'] }
)

4.1. 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 stream 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.fastRP.stream.estimate('persons', {embeddingDimension: 128})
YIELD nodeCount, relationshipCount, bytesMin, bytesMax, requiredMemory

Table 19. Results
nodeCount	relationshipCount	bytesMin	bytesMax	requiredMemory
7	18	11424	11424	"11424 Bytes"

4.2. Stream

In the stream execution mode, the algorithm returns the embedding 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 collect the results and pass them into a similarity algorithm.

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

The following will run the algorithm, and stream results:

CALL gds.fastRP.stream('persons',
  {
    embeddingDimension: 4,
    randomSeed: 42
  }
)
YIELD nodeId, embedding

Table 20. Results
nodeId	embedding
0	[0.4774002134799957, -0.6602408289909363, -0.36686956882476807, -1.7089111804962158]
1	[0.7989360094070435, -0.4918718934059143, -0.41281944513320923, -1.6314401626586914]
2	[0.47275322675704956, -0.49587157368659973, -0.3340468406677246, -1.7141895294189453]
3	[0.8290714025497437, -0.3260476291179657, -0.3317275643348694, -1.4370529651641846]
4	[0.7749264240264893, -0.4773247539997101, 0.0675133764743805, -1.5248265266418457]
5	[0.8408374190330505, -0.37151476740837097, 0.12121132016181946, -1.530960202217102]
6	[1.0, -0.11054422706365585, -0.3697933852672577, -0.9225144982337952]

The results of the algorithm are not very intuitively interpretable, as the node embedding format is a mathematical abstraction of the node within its neighborhood, designed for machine learning programs. What we can see is that the embeddings have four elements (as configured using embeddingDimension) and that the numbers are relatively small (they all fit in the range of [-2, 2]). The magnitude of the numbers is controlled by the embeddingDimension, the number of nodes in the graph, and by the fact that FastRP performs euclidean normalization on the intermediate embedding vectors.

Due to the random nature of the algorithm the results will vary between the runs. However, this does not necessarily mean that the pairwise distances of two node embeddings vary as much.

4.3. Stats

In the stats execution mode, the algorithm returns a single row containing a summary of the algorithm result. 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.fastRP.stats('persons', { embeddingDimension: 8 })
YIELD nodeCount

Table 21. Results
nodeCount
7

The stats mode does not currently offer any statistical results for the embeddings themselves. We can however see that the algorithm has successfully processed all seven nodes in our example graph.

4.4. 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 embedding 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.fastRP.mutate(
  'persons',
  {
    embeddingDimension: 8,
    mutateProperty: 'fastrp-embedding'
  }
)
YIELD nodePropertiesWritten

Table 22. Results
nodePropertiesWritten
7

The returned result is similar to the stats example. Additionally, the graph 'persons' now has a node property fastrp-embedding which stores the node embedding for each node. To find out how to inspect the new schema of the in-memory graph, see Listing graphs.

4.5. Write

The write execution mode extends the stats mode with an important side effect: writing the embedding 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.fastRP.write(
  'persons',
  {
    embeddingDimension: 8,
    writeProperty: 'fastrp-embedding'
  }
)
YIELD nodePropertiesWritten

Table 23. Results
nodePropertiesWritten
7

The returned result is similar to the stats example. Additionally, each of the seven nodes now has a new property fastrp-embedding in the Neo4j database, containing the node embedding for that node.

4.6. 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. Below is an example of running the weighted variant of algorithm.

The following will run the algorithm, and stream results:

CALL gds.fastRP.stream(
  'persons',
  {
    embeddingDimension: 4,
    randomSeed: 42,
    relationshipWeightProperty: 'weight'
  }
)
YIELD nodeId, embedding

Table 24. Results
nodeId	embedding
0	[0.10945529490709305, -0.5032674074172974, 0.464673787355423, -1.7539862394332886]
1	[0.3639600872993469, -0.39210301637649536, 0.46271592378616333, -1.829423427581787]
2	[0.12314096093177795, -0.3213110864162445, 0.40100979804992676, -1.471055269241333]
3	[0.30704641342163086, -0.24944794178009033, 0.3947891891002655, -1.3463698625564575]
4	[0.23112300038337708, -0.30148714780807495, 0.584831714630127, -1.2822188138961792]
5	[0.14497177302837372, -0.2312137484550476, 0.5552002191543579, -1.2605633735656738]
6	[0.5139184594154358, -0.07954332232475281, 0.3690345287322998, -0.9176374077796936]

Since the initial state of the algorithm is randomised, it isn’t possible to intuitively analyse the effect of the relationship weights.

4.7. Using node properties as features

To explain the novel initialization using node properties, let us consider an example where embeddingDimension is 10, propertyRatio is 0.2. The dimension of the embedded properties, propertyDimension is thus 2. Assume we have a property f1 of scalar type, and a property f2 storing arrays of length 2. This means that there are 3 features which we order like f1 followed by the two values of f2. For each of these three features we sample a two dimensional random vector. Let’s say these are p1=[0.0, 2.4], p2=[-2.4, 0.0] and p3=[2.4, 0.0]. Consider now a node (n {f1: 0.5, f2: [1.0, -1.0]}). The linear combination mentioned above, is in concrete terms 0.5 * p1 + 1.0 * p2 - 1.0 * p3 = [-4.8, 1.2]. The initial random vector for the node n contains first 8 values sampled as in the original FastRP paper, and then our computed values -4.8 and 1.2, totalling 10 entries.

In the example below, we again set the embedding dimension to 2, but we set propertyRatio to 1, which means the embedding is computed from node properties only.

The following will run FastRP with feature properties:

CALL gds.fastRP.stream('persons', {
    randomSeed: 42,
    embeddingDimension: 2,
    propertyRatio: 1.0,
    featureProperties: ['age'],
    iterationWeights: [1.0]
}) YIELD nodeId, embedding

Table 25. Results
nodeId	embedding
0	[0.0, -1.0]
1	[0.0, -1.0]
2	[0.0, -0.9999999403953552]
3	[0.0, -1.0]
4	[0.0, -0.9999999403953552]
5	[0.0, -1.0]
6	[0.0, -1.0]

In this example, the embeddings are based on the age property. Because of L2 normalization which is applied to each iteration (here only one iteration), all nodes have the same embedding despite having different age values (apart from rounding errors).

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