Node Similarity
This section describes the Node Similarity algorithm in the Neo4j Graph Data Science library. The algorithm is based on the Jaccard and Overlap similarity metrics.
Glossary
 Directed

Directed trait. The algorithm is welldefined on a directed graph.
 Undirected

Undirected trait. The algorithm is welldefined on an undirected graph.
 Homogeneous

Homogeneous trait. The algorithm will treat all nodes and relationships in its input graph(s) similarly, as if they were all of the same type. If multiple types of nodes or relationships exist in the graph, this must be taken into account when analysing the results of the algorithm.
 Heterogeneous

Heterogeneous trait. The algorithm has the ability to distinguish between nodes and/or relationships of different types.
 Weighted

Weighted trait. The algorithm supports configuration to set node and/or relationship properties to use as weights. These values can represent cost, time, capacity or some other domainspecific properties, specified via the nodeWeightProperty, nodeProperties and relationshipWeightProperty configuration parameters. The algorithm will by default consider each node and/or relationship as equally important.
1. Introduction
The Node Similarity algorithm compares a set of nodes based on the nodes they are connected to. Two nodes are considered similar if they share many of the same neighbors. Node Similarity computes pairwise similarities based on either the Jaccard metric, also known as the Jaccard Similarity Score, or the Overlap coefficient, also known as the Szymkiewicz–Simpson coefficient.
Given two sets A
and B
, the Jaccard Similarity is computed using the following formula:
The Overlap coefficient is computed using the following formula:
The input of this algorithm is a bipartite, connected graph containing two disjoint node sets. Each relationship starts from a node in the first node set and ends at a node in the second node set.
The Node Similarity algorithm compares each node that has outgoing relationships with each other such node.
For every node n
, we collect the outgoing neighborhood N(n)
of that node, that is, all nodes m
such that there is a relationship from n
to m
.
For each pair n
, m
, the algorithm computes a similarity for that pair that equals the outcome of the selected similarity metric for N(n)
and N(m)
.
Node Similarity has time complexity O(n^{3}) and space complexity O(n^{2}). We compute and store neighbour sets in time and space O(n^{2}), then compute pairwise similarity scores in time O(n^{3}).
In order to bound memory usage you can specify an explicit limit on the number of results to output per node, this is the 'topK' parameter. It can be set to any value, except 0. You will lose precision in the overall computation of course, and running time is unaffected  we still have to compute results before potentially throwing them away.
The output of the algorithm are new relationships between pairs of the first node set. Similarity scores are expressed via relationship properties.
For more information on this algorithm, see:
It is also possible to apply filtering on the source and/or target nodes in the produced similarity pairs. You can consider the filtered Node Similarity algorithm for this purpose.
Running this algorithm requires sufficient available memory. Before running this algorithm, we recommend that you read Memory Estimation. 
2. Syntax
This section covers the syntax used to execute the Node Similarity 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.
CALL gds.nodeSimilarity.stream(
graphName: String,
configuration: Map
) YIELD
node1: Integer,
node2: Integer,
similarity: Float
Name  Type  Default  Optional  Description 

graphName 
String 

no 
The name of a graph stored in the catalog. 
configuration 
Map 

yes 
Configuration for algorithmspecifics and/or graph filtering. 
Name  Type  Default  Optional  Description 

List of String 

yes 
Filter the named graph using the given node labels. 

List of String 

yes 
Filter the named graph using the given relationship types. 

Integer 

yes 
The number of concurrent threads used for running the algorithm. 

String 

yes 
An ID that can be provided to more easily track the algorithm’s progress. 

Boolean 

yes 
If disabled the progress percentage will not be logged. 
Name  Type  Default  Optional  Description 

similarityCutoff 
Float 

yes 
Lower limit for the similarity score to be present in the result. Values must be between 0 and 1. 
degreeCutoff 
Integer 

yes 
Lower limit on the node degree for a node to be considered in the comparisons. This value can not be lower than 1. 
topK 
Integer 

yes 
Limit on the number of scores per node. The K largest results are returned. This value cannot be lower than 1. 
bottomK 
Integer 

yes 
Limit on the number of scores per node. The K smallest results are returned. This value cannot be lower than 1. 
topN 
Integer 

yes 
Global limit on the number of scores computed. The N largest total results are returned. This value cannot be negative, a value of 0 means no global limit. 
bottomN 
Integer 

yes 
Global limit on the number of scores computed. The N smallest total results are returned. This value cannot be negative, a value of 0 means no global limit. 
String 

yes 
Name of the relationship property to use as weights. If unspecified, the algorithm runs unweighted. 

similarityMetric 
String 

yes 
The metric used to compute similarity.
Can be either 
Name  Type  Description 


Integer 
Node ID of the first node. 

Integer 
Node ID of the second node. 

Float 
Similarity score for the two nodes. 
CALL gds.nodeSimilarity.stats(
graphName: String,
configuration: Map
)
YIELD
preProcessingMillis: Integer,
computeMillis: Integer,
postProcessingMillis: Integer,
nodesCompared: Integer,
similarityPairs: Integer,
similarityDistribution: Map,
configuration: Map
Name  Type  Default  Optional  Description 

graphName 
String 

no 
The name of a graph stored in the catalog. 
configuration 
Map 

yes 
Configuration for algorithmspecifics and/or graph filtering. 
Name  Type  Default  Optional  Description 

List of String 

yes 
Filter the named graph using the given node labels. 

List of String 

yes 
Filter the named graph using the given relationship types. 

Integer 

yes 
The number of concurrent threads used for running the algorithm. 

String 

yes 
An ID that can be provided to more easily track the algorithm’s progress. 

Boolean 

yes 
If disabled the progress percentage will not be logged. 
Name  Type  Default  Optional  Description 

similarityCutoff 
Float 

yes 
Lower limit for the similarity score to be present in the result. Values must be between 0 and 1. 
degreeCutoff 
Integer 

yes 
Lower limit on the node degree for a node to be considered in the comparisons. This value can not be lower than 1. 
topK 
Integer 

yes 
Limit on the number of scores per node. The K largest results are returned. This value cannot be lower than 1. 
bottomK 
Integer 

yes 
Limit on the number of scores per node. The K smallest results are returned. This value cannot be lower than 1. 
topN 
Integer 

yes 
Global limit on the number of scores computed. The N largest total results are returned. This value cannot be negative, a value of 0 means no global limit. 
bottomN 
Integer 

yes 
Global limit on the number of scores computed. The N smallest total results are returned. This value cannot be negative, a value of 0 means no global limit. 
String 

yes 
Name of the relationship property to use as weights. If unspecified, the algorithm runs unweighted. 

similarityMetric 
String 

yes 
The metric used to compute similarity.
Can be either 
Name  Type  Description 

preProcessingMillis 
Integer 
Milliseconds for preprocessing the data. 
computeMillis 
Integer 
Milliseconds for running the algorithm. 
postProcessingMillis 
Integer 
Milliseconds for computing component count and distribution statistics. 
nodesCompared 
Integer 
The number of nodes for which similarity was computed. 
similarityPairs 
Integer 
The number of similarities in the result. 
similarityDistribution 
Map 
Map containing min, max, mean as well as p50, p75, p90, p95, p99 and p999 percentile values of the computed similarity results. 
configuration 
Map 
The configuration used for running the algorithm. 
CALL gds.nodeSimilarity.mutate(
graphName: String,
configuration: Map
)
YIELD
preProcessingMillis: Integer,
computeMillis: Integer,
mutateMillis: Integer,
postProcessingMillis: Integer,
relationshipsWritten: Integer,
nodesCompared: Integer,
similarityDistribution: Map,
configuration: Map
Name  Type  Default  Optional  Description 

graphName 
String 

no 
The name of a graph stored in the catalog. 
configuration 
Map 

yes 
Configuration for algorithmspecifics and/or graph filtering. 
Name  Type  Default  Optional  Description 

mutateRelationshipType 
String 

no 
The relationship type used for the new relationships written to the projected graph. 
mutateProperty 
String 

no 
The relationship property in the GDS graph to which the similarity score is written. 
List of String 

yes 
Filter the named graph using the given node labels. 

List of String 

yes 
Filter the named graph using the given relationship types. 

Integer 

yes 
The number of concurrent threads used for running the algorithm. 

String 

yes 
An ID that can be provided to more easily track the algorithm’s progress. 
Name  Type  Default  Optional  Description 

similarityCutoff 
Float 

yes 
Lower limit for the similarity score to be present in the result. Values must be between 0 and 1. 
degreeCutoff 
Integer 

yes 
Lower limit on the node degree for a node to be considered in the comparisons. This value can not be lower than 1. 
topK 
Integer 

yes 
Limit on the number of scores per node. The K largest results are returned. This value cannot be lower than 1. 
bottomK 
Integer 

yes 
Limit on the number of scores per node. The K smallest results are returned. This value cannot be lower than 1. 
topN 
Integer 

yes 
Global limit on the number of scores computed. The N largest total results are returned. This value cannot be negative, a value of 0 means no global limit. 
bottomN 
Integer 

yes 
Global limit on the number of scores computed. The N smallest total results are returned. This value cannot be negative, a value of 0 means no global limit. 
String 

yes 
Name of the relationship property to use as weights. If unspecified, the algorithm runs unweighted. 

similarityMetric 
String 

yes 
The metric used to compute similarity.
Can be either 
Name  Type  Description 

preProcessingMillis 
Integer 
Milliseconds for preprocessing the data. 
computeMillis 
Integer 
Milliseconds for running the algorithm. 
mutateMillis 
Integer 
Milliseconds for adding properties to the projected graph. 
postProcessingMillis 
Integer 
Milliseconds for computing percentiles. 
nodesCompared 
Integer 
The number of nodes for which similarity was computed. 
relationshipsWritten 
Integer 
The number of relationships created. 
similarityDistribution 
Map 
Map containing min, max, mean, stdDev and p1, p5, p10, p25, p75, p90, p95, p99, p100 percentile values of the computed similarity results. 
configuration 
Map 
The configuration used for running the algorithm. 
CALL gds.nodeSimilarity.write(
graphName: String,
configuration: Map
)
YIELD
preProcessingMillis: Integer,
computeMillis: Integer,
writeMillis: Integer,
postProcessingMillis: Integer,
nodesCompared: Integer,
relationshipsWritten: Integer,
similarityDistribution: Map,
configuration: Map
Name  Type  Default  Optional  Description 

graphName 
String 

no 
The name of a graph stored in the catalog. 
configuration 
Map 

yes 
Configuration for algorithmspecifics and/or graph filtering. 
Name  Type  Default  Optional  Description 

List of String 

yes 
Filter the named graph using the given node labels. 

List of String 

yes 
Filter the named graph using the given relationship types. 

Integer 

yes 
The number of concurrent threads used for running the algorithm. 

String 

yes 
An ID that can be provided to more easily track the algorithm’s progress. 

Boolean 

yes 
If disabled the progress percentage will not be logged. 

Integer 

yes 
The number of concurrent threads used for writing the result to Neo4j. 

writeRelationshipType 
String 

no 
The relationship type used to persist the computed relationships in the Neo4j database. 
String 

no 
The relationship property in the Neo4j database to which the similarity score is written. 
Name  Type  Default  Optional  Description 

similarityCutoff 
Float 

yes 
Lower limit for the similarity score to be present in the result. Values must be between 0 and 1. 
degreeCutoff 
Integer 

yes 
Lower limit on the node degree for a node to be considered in the comparisons. This value can not be lower than 1. 
topK 
Integer 

yes 
Limit on the number of scores per node. The K largest results are returned. This value cannot be lower than 1. 
bottomK 
Integer 

yes 
Limit on the number of scores per node. The K smallest results are returned. This value cannot be lower than 1. 
topN 
Integer 

yes 
Global limit on the number of scores computed. The N largest total results are returned. This value cannot be negative, a value of 0 means no global limit. 
bottomN 
Integer 

yes 
Global limit on the number of scores computed. The N smallest total results are returned. This value cannot be negative, a value of 0 means no global limit. 
String 

yes 
Name of the relationship property to use as weights. If unspecified, the algorithm runs unweighted. 

similarityMetric 
String 

yes 
The metric used to compute similarity.
Can be either 
Name  Type  Description 

preProcessingMillis 
Integer 
Milliseconds for preprocessing data. 
computeMillis 
Integer 
Milliseconds for running the algorithm. 
writeMillis 
Integer 
Milliseconds for writing result data back to Neo4j. 
postProcessingMillis 
Integer 
Milliseconds for computing percentiles. 
nodesCompared 
Integer 
The number of nodes for which similarity was computed. 
relationshipsWritten 
Integer 
The number of relationships created. 
similarityDistribution 
Map 
Map containing min, max, mean, stdDev and p1, p5, p10, p25, p75, p90, p95, p99, p100 percentile values of the computed similarity results. 
configuration 
Map 
The configuration used for running the algorithm. 
3. Examples
In this section we will show examples of running the Node Similarity 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 knowledge graph of a handful nodes connected in a particular pattern. The example graph looks like this:
CREATE
(alice:Person {name: 'Alice'}),
(bob:Person {name: 'Bob'}),
(carol:Person {name: 'Carol'}),
(dave:Person {name: 'Dave'}),
(eve:Person {name: 'Eve'}),
(guitar:Instrument {name: 'Guitar'}),
(synth:Instrument {name: 'Synthesizer'}),
(bongos:Instrument {name: 'Bongos'}),
(trumpet:Instrument {name: 'Trumpet'}),
(alice)[:LIKES]>(guitar),
(alice)[:LIKES]>(synth),
(alice)[:LIKES {strength: 0.5}]>(bongos),
(bob)[:LIKES]>(guitar),
(bob)[:LIKES]>(synth),
(carol)[:LIKES]>(bongos),
(dave)[:LIKES]>(guitar),
(dave)[:LIKES {strength: 1.5}]>(trumpet),
(dave)[:LIKES]>(bongos);
This bipartite graph has two node sets, Person nodes and Instrument nodes. The two node sets are connected via LIKES relationships. Each relationship starts at a Person node and ends at an Instrument node.
In the example, we want to use the Node Similarity algorithm to compare people based on the instruments they like.
The Node Similarity algorithm will only compute similarity for nodes that have a degree of at least 1. In the example graph, the Eve node will not be compared to other Person nodes.
In the examples below we will use named graphs and native projections as the norm. However, Cypher projections can also be used. 
CALL gds.graph.project(
'myGraph',
['Person', 'Instrument'],
{
LIKES: {
properties: {
strength: {
property: 'strength',
defaultValue: 1.0
}
}
}
}
);
In the following examples we will demonstrate using the Node Similarity algorithm on this graph.
3.1. 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 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.
CALL gds.nodeSimilarity.write.estimate('myGraph', {
writeRelationshipType: 'SIMILAR',
writeProperty: 'score'
})
YIELD nodeCount, relationshipCount, bytesMin, bytesMax, requiredMemory
nodeCount  relationshipCount  bytesMin  bytesMax  requiredMemory 

9 
9 
2384 
2600 
"[2384 Bytes ... 2600 Bytes]" 
3.2. Stream
In the stream
execution mode, the algorithm returns the similarity score for each relationship.
This allows us to inspect the results directly or postprocess them in Cypher without any side effects.
For more details on the stream
mode in general, see Stream.
CALL gds.nodeSimilarity.stream('myGraph')
YIELD node1, node2, similarity
RETURN gds.util.asNode(node1).name AS Person1, gds.util.asNode(node2).name AS Person2, similarity
ORDER BY similarity DESCENDING, Person1, Person2
Person1  Person2  similarity 

"Alice" 
"Bob" 
0.6666666666666666 
"Bob" 
"Alice" 
0.6666666666666666 
"Alice" 
"Dave" 
0.5 
"Dave" 
"Alice" 
0.5 
"Alice" 
"Carol" 
0.3333333333333333 
"Carol" 
"Alice" 
0.3333333333333333 
"Carol" 
"Dave" 
0.3333333333333333 
"Dave" 
"Carol" 
0.3333333333333333 
"Bob" 
"Dave" 
0.25 
"Dave" 
"Bob" 
0.25 
We use default values for the procedure configuration parameter. TopK is set to 10, topN is set to 0. Because of that the result set contains the top 10 similarity scores for each node.
If we would like to instead compare the Instruments to each other, we would then project the 
3.3. 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.
CALL gds.nodeSimilarity.stats('myGraph')
YIELD nodesCompared, similarityPairs
nodesCompared  similarityPairs 

4 
10 
3.4. Mutate
The mutate
execution mode extends the stats
mode with an important side effect: updating the named graph with a new relationship property containing the similarity score for that relationship.
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.
CALL gds.nodeSimilarity.mutate('myGraph', {
mutateRelationshipType: 'SIMILAR',
mutateProperty: 'score'
})
YIELD nodesCompared, relationshipsWritten
nodesCompared  relationshipsWritten 

4 
10 
As we can see from the results, the number of created relationships is equal to the number of rows in the streaming example.
The relationships that are produced by the mutation are always directed, even if the input graph is undirected.
If 
3.5. Write
The write
execution mode for each pair of nodes creates a relationship with their similarity score as a property to the Neo4j database.
The type of the new relationship is specified using the mandatory configuration parameter writeRelationshipType
.
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.
For more details on the write
mode in general, see Write.
CALL gds.nodeSimilarity.write('myGraph', {
writeRelationshipType: 'SIMILAR',
writeProperty: 'score'
})
YIELD nodesCompared, relationshipsWritten
nodesCompared  relationshipsWritten 

4 
10 
As we can see from the results, the number of created relationships is equal to the number of rows in the streaming example.
The relationships that are written are always directed, even if the input graph is undirected.
If 
3.6. Limit results
There are four limits that can be applied to the similarity results. Top limits the result to the highest similarity scores. Bottom limits the result to the lowest similarity scores. Both top and bottom limits can apply to the result as a whole ("N"), or to the result per node ("K").
There must always be a "K" limit, either bottomK or topK, which is a positive number. The default value for topK and bottomK is 10. 
total results  results per node  

highest score 
topN 
topK 
lowest score 
bottomN 
bottomK 
3.6.1. topK and bottomK
TopK and bottomK are limits on the number of scores computed per node. For topK, the K largest similarity scores per node are returned. For bottomK, the K smallest similarity scores per node are returned. TopK and bottomK cannot be 0, used in conjunction, and the default value is 10. If neither is specified, topK is used.
CALL gds.nodeSimilarity.stream('myGraph', { topK: 1 })
YIELD node1, node2, similarity
RETURN gds.util.asNode(node1).name AS Person1, gds.util.asNode(node2).name AS Person2, similarity
ORDER BY Person1
Person1  Person2  similarity 

"Alice" 
"Bob" 
0.666666666666667 
"Bob" 
"Alice" 
0.666666666666667 
"Carol" 
"Alice" 
0.333333333333333 
"Dave" 
"Alice" 
0.5 
CALL gds.nodeSimilarity.stream('myGraph', { bottomK: 1 })
YIELD node1, node2, similarity
RETURN gds.util.asNode(node1).name AS Person1, gds.util.asNode(node2).name AS Person2, similarity
ORDER BY Person1
Person1  Person2  similarity 

"Alice" 
"Carol" 
0.3333333333333333 
"Bob" 
"Dave" 
0.25 
"Carol" 
"Alice" 
0.3333333333333333 
"Dave" 
"Bob" 
0.25 
3.6.2. topN and bottomN
TopN and bottomN limit the number of similarity scores across all nodes. This is a limit on the total result set, in addition to the topK or bottomK limit on the results per node. For topN, the N largest similarity scores are returned. For bottomN, the N smallest similarity scores are returned. A value of 0 means no global limit is imposed and all results from topK or bottomK are returned.
CALL gds.nodeSimilarity.stream('myGraph', { topK: 1, topN: 3 })
YIELD node1, node2, similarity
RETURN gds.util.asNode(node1).name AS Person1, gds.util.asNode(node2).name AS Person2, similarity
ORDER BY similarity DESC, Person1, Person2
Person1  Person2  similarity 

"Alice" 
"Bob" 
0.666666666666667 
"Bob" 
"Alice" 
0.666666666666667 
"Dave" 
"Alice" 
0.5 
3.7. Degree cutoff and similarity cutoff
Degree cutoff is a lower limit on the node degree for a node to be considered in the comparisons. This value can not be lower than 1.
CALL gds.nodeSimilarity.stream('myGraph', { degreeCutoff: 3 })
YIELD node1, node2, similarity
RETURN gds.util.asNode(node1).name AS Person1, gds.util.asNode(node2).name AS Person2, similarity
ORDER BY Person1
Person1  Person2  similarity 

"Alice" 
"Dave" 
0.5 
"Dave" 
"Alice" 
0.5 
Similarity cutoff is a lower limit for the similarity score to be present in the result.
The default value is very small (1E42
) to exclude results with a similarity score of 0.
Setting similarity cutoff to 0 may yield a very large result set, increased runtime and memory consumption. 
CALL gds.nodeSimilarity.stream('myGraph', { similarityCutoff: 0.5 })
YIELD node1, node2, similarity
RETURN gds.util.asNode(node1).name AS Person1, gds.util.asNode(node2).name AS Person2, similarity
ORDER BY Person1
Person1  Person2  similarity 

"Alice" 
"Bob" 
0.666666666666667 
"Alice" 
"Dave" 
0.5 
"Bob" 
"Alice" 
0.6666666666666666 
"Dave" 
"Alice" 
0.5 
3.8. Weighted Similarity
Relationship properties can be used to modify the similarity induced by certain relationships. Weighted node similarity has as default the weighted Jaccard similarty, according to the formula:
Formally, given two nodes and their weighted neighbour lists A'
and B'
, we extend the lists to A
and B
, index over the union of their neighbours A' ∪ B'
by setting weight = 0 for any nonneighbour, and then apply the weighted Jaccard similarity.
It also supports weighted Overlap similarity, according to the formula:
Weighted similarity metrics are only defined for values greater or equal to 0. 
CALL gds.nodeSimilarity.stream('myGraph', { relationshipWeightProperty: 'strength', similarityCutoff: 0.3 })
YIELD node1, node2, similarity
RETURN gds.util.asNode(node1).name AS Person1, gds.util.asNode(node2).name AS Person2, similarity
ORDER BY Person1
Person1  Person2  similarity 

"Alice" 
"Bob" 
0.8 
"Alice" 
"Dave" 
0.333333333333333 
"Bob" 
"Alice" 
0.8 
"Dave" 
"Alice" 
0.333333333333333 
It can be seen that the similarity between Alice and Dave decreased (from 0.5 to 0.33) compared to the nonweighted version of this algorithm.
Alice likes Guitar, Synthesize and Bongos with strengths (1, 1, 0.5)
.
Dave likes Guitar, Bongos and Trumpet with strengths (1, 1, 1.5)
.
Therefore, taking Alice and Dave’s neighbours, we have list of strengths for Alice as A = (1, 1, 0.5, 0)
and for Dave B = (1, 0, 1, 1.5)
, indexed as Guitar, Synthesizer, Bongos, Trumpet
.
The weighted (Jaccard) node similarity of Alice and Dave is hence:
Analogously, the similarity between Alice and Bob increased (from 0.66 to 0.8) as the missing liked instrument has a lower impact on the similarity score.
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