# Fast Random Projection

Supported algorithm traits:

## 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.

 FastRP is featured in the end-to-end example Jupyter notebooks:

## 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 for each node n. More precisely, where m ranges over neighbors of n and is the node’s initial random vector.

The embedding of node n, which is the output of the algorithm, is a combination of the vectors and embeddings defined above: where `normalize` is the function which divides a vector with its L2 norm, the value of `nodeSelfInfluence` is , and the values of `iterationWeights` are . 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.

### 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:

1. The first part is formed like in the standard FastRP algorithm,

2. 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```

### 1.2. Usage in machine learning pipelines

It may be useful to generate node embeddings with FastRP as a node property step in a machine learning pipeline (like Link prediction pipelines and Node property prediction). Since FastRP is a random algorithm and inductive only for `propertyRatio=1.0`, there are some things to have in mind.

In order for a machine learning model to be able to make useful predictions, it is important that features produced during prediction are of a similar distribution to the features produced during training of the model. Moreover, node property steps (whether FastRP or not) added to a pipeline are executed both during training, and during the prediction by the trained model. It is therefore problematic when a pipeline contains an embedding step which yields all too dissimilar embeddings during training and prediction.

This has some implications on how to use FastRP as a node property step. In general, if a pipeline is trained using FastRP as a node property step on some graph "g", then the resulting trained model should only be applied to graphs that are not too dissimilar to "g".

If `propertyRatio<1.0`, most of the nodes in the graph that a prediction is being run on, must be the same nodes (in the database sense) as in the original graph "g" that was used during training. The reason for this is that FastRP is a random algorithm, and in this case is seeded based on the nodes' ids in the Neo4j database from whence the nodes came.

If `propertyRatio=1.0` however, the random initial node embeddings are derived from node property vectors only, so there is no random seeding based on node ids.

Additionally, in order for the initial random vectors (independent of `propertyRatio` used) to be consistent between runs (training and prediction calls), a value for the `randomSeed` configuration parameter must be provided when adding the FastRP node property step to the training pipeline.

## 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. Configuration
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.

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.

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 projected 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 3. 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,
preProcessingMillis: Integer,
computeMillis: Integer,
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.

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.

logProgress

Boolean

`true`

yes

If disabled the progress percentage will not be logged.

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 projected 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 6. Results
Name Type Description

nodeCount

Integer

Number of nodes processed.

preProcessingMillis

Integer

Milliseconds for preprocessing 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,
preProcessingMillis: Integer,
computeMillis: Integer,
mutateMillis: Integer,
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 embedding 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.

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 projected 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 9. Results
Name Type Description

nodeCount

Integer

Number of nodes processed.

nodePropertiesWritten

Integer

Number of node properties written.

preProcessingMillis

Integer

Milliseconds for preprocessing 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,
preProcessingMillis: Integer,
computeMillis: Integer,
writeMillis: Integer,
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.

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.

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 embedding is written.

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 projected 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.

preProcessingMillis

Integer

Milliseconds for preprocessing 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.

## 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, Cypher projections can also be used.
The following statement will project a graph using a native projection and store it in the graph catalog under the name 'persons'.
``````CALL gds.graph.project(
'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 13. Results
nodeCount relationshipCount bytesMin bytesMax requiredMemory

7

18

11320

11320

"11320 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 14. 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 15. 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 16. 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 17. 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 18. 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 19. 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).