Local Clustering Coefficient
This section describes the Local Clustering Coefficient algorithm in the Neo4j Graph Data Science library.
1. Introduction
The Local Clustering Coefficient algorithm computes the local clustering coefficient for each node in the graph. The local clustering coefficient C_{n} of a node n describes the likelihood that the neighbours of n are also connected. To compute C_{n} we use the number of triangles a node is a part of T_{n}, and the degree of the node d_{n}. The formula to compute the local clustering coefficient is as follows:
As we can see the triangle count is required to compute the local clustering coefficient. To do this the Triangle Count algorithm is utilised.
Additionally, the algorithm can compute the average clustering coefficient for the whole graph. This is the normalised sum over all the local clustering coefficients.
For more information, see Clustering Coefficient.
2. Syntax
This section covers the syntax used to execute the Local Clustering Coefficient 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.localClusteringCoefficient.stream(
graphName: String,
configuration: Map
)
YIELD
nodeId: Integer,
localClusteringCoefficient: Double
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 

nodeLabels 
String[] 

yes 
Filter the named graph using the given node labels. 
String[] 

yes 
Filter the named graph using the given relationship types. 

Integer 

yes 
The number of concurrent threads used for running the algorithm. 
Name  Type  Default  Optional  Description 

triangleCountProperty 
String 

Yes 
Node property that contains precomputed triangle count. 
Name  Type  Description 

nodeId 
Integer 
Node ID. 
localClusteringCoefficient 
Double 
Local clustering coefficient. 
CALL gds.localClusteringCoefficient.stats(
graphName: String,
configuration: Map
)
YIELD
averageClusteringCoefficient: Double,
nodeCount: Integer,
createMillis: Integer,
computeMillis: Integer,
postProcessingMillis: Integer,
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 

nodeLabels 
String[] 

yes 
Filter the named graph using the given node labels. 
String[] 

yes 
Filter the named graph using the given relationship types. 

Integer 

yes 
The number of concurrent threads used for running the algorithm. 
Name  Type  Default  Optional  Description 

triangleCountProperty 
String 

Yes 
Node property that contains precomputed triangle count. 
Name  Type  Description 

averageClusteringCoefficient 
Double 
The average clustering coefficient. 
nodeCount 
Integer 
Number of nodes in the graph. 
createMillis 
Integer 
Milliseconds for creating the graph. 
computeMillis 
Integer 
Milliseconds for running the algorithm. 
postProcessingMillis 
Integer 
Milliseconds for computing the global metrics. 
configuration 
Map 
The configuration used for running the algorithm. 
CALL gds.localClusteringCoefficient.mutate(
graphName: String,
configuration: Map
)
YIELD
averageClusteringCoefficient: Double,
nodeCount: Integer,
nodePropertiesWritten: Integer,
createMillis: Integer,
computeMillis: Integer,
postProcessingMillis: Integer,
mutateMillis: Integer,
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 

nodeLabels 
String[] 

yes 
Filter the named graph using the given node labels. 
String[] 

yes 
Filter the named graph using the given relationship types. 

Integer 

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

mutateProperty 
String 

no 
The node property in the GDS graph to which the local clustering coefficient is written. 
Name  Type  Default  Optional  Description 

triangleCountProperty 
String 

Yes 
Node property that contains precomputed triangle count. 
Name  Type  Description 

averageClusteringCoefficient 
Double 
The average clustering coefficient. 
nodeCount 
Integer 
Number of nodes in the graph. 
nodePropertiesWritten 
Integer 
Number of properties added to the inmemory graph. 
createMillis 
Integer 
Milliseconds for creating the graph. 
computeMillis 
Integer 
Milliseconds for running the algorithm. 
postProcessingMillis 
Integer 
Milliseconds for computing the global metrics. 
mutateMillis 
Integer 
Milliseconds for adding properties to the inmemory graph. 
configuration 
Map 
The configuration used for running the algorithm. 
CALL gds.localClusteringCoefficient.write(
graphName: String,
configuration: Map
)
YIELD
averageClusteringCoefficient: Double,
nodeCount: Integer,
nodePropertiesWritten: Integer,
createMillis: Integer,
computeMillis: Integer,
postProcessingMillis: Integer,
writeMillis: Integer,
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 

nodeLabels 
String[] 

yes 
Filter the named graph using the given node labels. 
String[] 

yes 
Filter the named graph using the given relationship types. 

Integer 

yes 
The number of concurrent threads used for running the algorithm. Also provides the default value for 'writeConcurrency'. 

Integer 

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

String 

no 
The node property in the Neo4j database to which the local clustering coefficient is written. 
Name  Type  Default  Optional  Description 

triangleCountProperty 
String 

Yes 
Node property that contains precomputed triangle count. 
Name  Type  Description 

averageClusteringCoefficient 
Double 
The average clustering coefficient. 
nodeCount 
Integer 
Number of nodes in the graph. 
nodePropertiesWritten 
Integer 
Number of properties written to Neo4j. 
createMillis 
Integer 
Milliseconds for creating the graph. 
computeMillis 
Integer 
Milliseconds for running the algorithm. 
postProcessingMillis 
Integer 
Milliseconds for computing the global metrics. 
writeMillis 
Integer 
Milliseconds for writing results back to Neo4j. 
configuration 
Map 
The configuration used for running the algorithm. 
2.1. 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 modespecific configuration for the write
mode for brevity.
For more information on syntax variants, see Syntax overview.
CALL gds.localClusteringCoefficient.write(
configuration: Map
)
YIELD
averageClusteringCoefficient: Double,
nodeCount: Integer,
nodePropertiesWritten: Integer,
createMillis: Integer,
computeMillis: Integer,
writeMillis: Integer,
configuration: Map
Name  Type  Default  Optional  Description 

nodeProjection 
String, String[] or Map 

yes 
The node projection used for anonymous graph creation via a Native projection. 
relationshipProjection 
String, String[] or Map 

yes 
The relationship projection used for anonymous graph creation a Native projection. 
nodeQuery 
String 

yes 
The Cypher query used to select the nodes for anonymous graph creation via a Cypher projection. 
relationshipQuery 
String 

yes 
The Cypher query used to select the relationships for anonymous graph creation via a Cypher projection. 
nodeProperties 
String, String[] or Map 

yes 
The node properties to project during anonymous graph creation. 
relationshipProperties 
String, String[] or Map 

yes 
The relationship properties to project during anonymous graph creation. 
Integer 

yes 
The number of concurrent threads used for running the algorithm. Also provides the default value for 'readConcurrency' and 'writeConcurrency'. 

readConcurrency 
Integer 

yes 
The number of concurrent threads used for creating the graph. 
Integer 

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

String 

no 
The node property in the Neo4j database to which the local clustering coefficient is written. 
Name  Type  Default  Optional  Description 

triangleCountProperty 
String 

Yes 
Node property that contains precomputed triangle count. 
The results are the same as for running write mode with a named graph, see the write mode syntax above.
3. Examples
In this section we will show examples of running the Local Clustering Coefficient 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:
CREATE
(alice:Person {name: 'Alice'}),
(michael:Person {name: 'Michael'}),
(karin:Person {name: 'Karin'}),
(chris:Person {name: 'Chris'}),
(will:Person {name: 'Will'}),
(mark:Person {name: 'Mark'}),
(michael)[:KNOWS]>(karin),
(michael)[:KNOWS]>(chris),
(will)[:KNOWS]>(michael),
(mark)[:KNOWS]>(michael),
(mark)[:KNOWS]>(will),
(alice)[:KNOWS]>(michael),
(will)[:KNOWS]>(chris),
(chris)[:KNOWS]>(karin)
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 must use the UNDIRECTED
orientation.
This is because the Local Clustering Coefficient algorithm is defined only for undirected graphs.
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. 
CALL gds.graph.create(
'myGraph',
'Person',
{
KNOWS: {
orientation: 'UNDIRECTED'
}
}
)
The Local Clustering Coefficient algorithm requires the graph to be created using the UNDIRECTED orientation for relationships.

In the following examples we will demonstrate using the Local Clustering Coefficient algorithm on 'myGraph'.
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.localClusteringCoefficient.write.estimate('myGraph', {
writeProperty: 'localClusteringCoefficient'
})
YIELD nodeCount, relationshipCount, bytesMin, bytesMax, requiredMemory
nodeCount  relationshipCount  bytesMin  bytesMax  requiredMemory 

6 
16 
288 
288 
"288 Bytes" 
Note that the relationship count is 16 although we only created 8 relationships in the original Cypher statement.
This is because we used the UNDIRECTED
orientation, which will project each relationship in each direction, effectively doubling the number of relationships.
3.2. Stream
In the stream
execution mode, the algorithm returns the local clustering coefficient for each node.
This allows us to inspect the results directly or postprocess them in Cypher without any side effects.
For example, we can order the results to find the nodes with the highest local clustering coefficient.
For more details on the stream
mode in general, see Stream.
stream
mode:CALL gds.localClusteringCoefficient.stream('myGraph')
YIELD nodeId, localClusteringCoefficient
RETURN gds.util.asNode(nodeId).name AS name, localClusteringCoefficient
ORDER BY localClusteringCoefficient DESC
name  localClusteringCoefficient 

"Karin" 
1.0 
"Mark" 
1.0 
"Chris" 
0.6666666666666666 
"Will" 
0.6666666666666666 
"Michael" 
0.3 
"Alice" 
0.0 
From the results we can see that the nodes 'Karin' and 'Mark' have the highest local clustering coefficients. This shows that they are the best at introducing their friends  all the people who know them, know each other! This can be verified in the example graph.
3.3. Stats
In the stats
execution mode, the algorithm returns a single row containing a summary of the algorithm result.
The summary result contains the avearage clustering coefficient of the graph, which is the normalised sum over all local clustering coefficients.
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.
stats
mode:CALL gds.localClusteringCoefficient.stats('myGraph')
YIELD averageClusteringCoefficient, nodeCount
averageClusteringCoefficient  nodeCount 

0.6055555555555555 
6 
The result shows that on average each node of our example graph has approximately 60% of its neighbours connected.
3.4. 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 local clustering coefficient 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.
mutate
mode:CALL gds.localClusteringCoefficient.mutate('myGraph', {
mutateProperty: 'localClusteringCoefficient'
})
YIELD averageClusteringCoefficient, nodeCount
averageClusteringCoefficient  nodeCount 

0.6055555555555555 
6 
The returned result is the same as in the stats
example.
Additionally, the graph 'myGraph' now has a node property localClusteringCoefficient
which stores the local clustering coefficient for each node.
To find out how to inspect the new schema of the inmemory graph, see :/managementops/graphcatalogops/index.adoc#cataloggraphlist.
3.5. Write
The write
execution mode extends the stats
mode with an important side effect: writing the local clustering coefficient 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.
write
mode:CALL gds.localClusteringCoefficient.write('myGraph', {
writeProperty: 'localClusteringCoefficient'
})
YIELD averageClusteringCoefficient, nodeCount
averageClusteringCoefficient  nodeCount 

0.6055555555555555 
6 
The returned result is the same as in the stats
example.
Additionally, each of the six nodes now has a new property localClusteringCoefficient
in the Neo4j database, containing the local clustering coefficient for that node.
3.6. Precomputed Counts
By default, the Local Clustering Coefficient algorithm executes Triangle Count as part of its computation.
It is also possible to avoid the triangle count computation by configuring the Local Clustering Coefficient algorithm to read the triangle count from a node property.
In order to do that we specify the triangleCountProperty
configuration parameter.
Please note that the Local Clustering Coefficient algorithm depends on the property holding actual triangle counts and not another number for the results to be actual local clustering coefficients.
To illustrate this we make use of the Triangle Count algorithm
in mutate
mode.
The Triangle Count algorithm is going to store its result back into 'myGraph'.
It is also possible to obtain the property value from the Neo4j database using a graph projection with a node property when creating the inmemory graph.
CALL gds.triangleCount.mutate('myGraph', {
mutateProperty: 'triangles'
})
stream
mode using precomputed triangle counts:CALL gds.localClusteringCoefficient.stream('myGraph', {
triangleCountProperty: 'triangles'
})
YIELD nodeId, localClusteringCoefficient
RETURN gds.util.asNode(nodeId).name AS name, localClusteringCoefficient
ORDER BY localClusteringCoefficient DESC
name  localClusteringCoefficient 

"Karin" 
1.0 
"Mark" 
1.0 
"Chris" 
0.6666666666666666 
"Will" 
0.6666666666666666 
"Michael" 
0.3 
"Alice" 
0.0 
As we can see the results are the same as in the stream
example where we did not specify a triangleCountProperty
.
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