Triangle Count
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 Triangle Count algorithm counts the number of triangles for each node in the graph. A triangle is a set of three nodes where each node has a relationship to the other two. In graph theory terminology, this is sometimes referred to as a 3clique. The Triangle Count algorithm in the GDS library only finds triangles in undirected graphs.
Triangle counting has gained popularity in social network analysis, where it is used to detect communities and measure the cohesiveness of those communities. It can also be used to determine the stability of a graph, and is often used as part of the computation of network indices, such as clustering coefficients. The Triangle Count algorithm is also used to compute the Local Clustering Coefficient.
For more information on this algorithm, see:

Triangle count and clustering coefficient have been shown to be useful as features for classifying a given website as spam, or nonspam, content. This is described in "Efficient Semistreaming Algorithms for Local Triangle Counting in Massive Graphs".
2. Syntax
This section covers the syntax used to execute the Triangle Count 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.
The named graphs must be projected in the UNDIRECTED orientation for the Triangle Count algorithm.

CALL gds.triangleCount.stream(
graphName: String,
configuration: Map
)
YIELD
nodeId: Integer,
triangleCount: Integer
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. 

maxDegree 
Integer 

Yes 
If a node has a degree higher than this it will not be considered by the algorithm. The triangle count for these nodes will be 
Name  Type  Description 

nodeId 
Integer 
Node ID. 
triangleCount 
Integer 
Number of triangles the node is part of. Is 
CALL gds.triangleCount.stats(
graphName: String,
configuration: Map
)
YIELD
globalTriangleCount: Integer,
nodeCount: Integer,
preProcessingMillis: 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 

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. 

maxDegree 
Integer 

Yes 
If a node has a degree higher than this it will not be considered by the algorithm. The triangle count for these nodes will be 
Name  Type  Description 

globalTriangleCount 
Integer 
Total number of triangles in the graph. 
nodeCount 
Integer 
Number of nodes in the graph. 
preProcessingMillis 
Integer 
Milliseconds for preprocessing 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.triangleCount.mutate(
graphName: String,
configuration: Map
)
YIELD
globalTriangleCount: Integer,
nodeCount: Integer,
nodePropertiesWritten: Integer,
preProcessingMillis: 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 

mutateProperty 
String 

no 
The node property in the GDS graph to which the triangle count 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. 

maxDegree 
Integer 

Yes 
If a node has a degree higher than this it will not be considered by the algorithm. The triangle count for these nodes will be 
Name  Type  Description 

globalTriangleCount 
Integer 
Total number of triangles in the graph. 
nodeCount 
Integer 
Number of nodes in the graph. 
nodePropertiesWritten 
Integer 
Number of properties added to the projected graph. 
preProcessingMillis 
Integer 
Milliseconds for preprocessing 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 projected graph. 
configuration 
Map 
The configuration used for running the algorithm. 
CALL gds.triangleCount.write(
graphName: String,
configuration: Map
)
YIELD
globalTriangleCount: Integer,
nodeCount: Integer,
nodePropertiesWritten: Integer,
preProcessingMillis: 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 

mutateProperty 
String 

no 
The node property in the GDS graph to which the triangle count 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. 

maxDegree 
Integer 

Yes 
If a node has a degree higher than this it will not be considered by the algorithm. The triangle count for these nodes will be 
Name  Type  Description 

globalTriangleCount 
Integer 
Total number of triangles in the graph. 
nodeCount 
Integer 
Number of nodes in the graph. 
nodePropertiesWritten 
Integer 
Number of properties written to Neo4j. 
preProcessingMillis 
Integer 
Milliseconds for preprocessing 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. Triangles listing
In addition to the standard execution modes there is an alpha
procedure gds.alpha.triangles
that can be used to list all triangles in the graph.
This feature is in the alpha tier. For more information on feature tiers, see API Tiers.
CALL gds.alpha.triangles(
graphName: String,
configuration: Map
)
YIELD nodeA, nodeB, nodeC
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. 
Name  Type  Description 

nodeA 
Integer 
The ID of the first node in the given triangle. 
nodeB 
Integer 
The ID of the second node in the given triangle. 
nodeC 
Integer 
The ID of the third node in the given triangle. 
3. Examples
In this section we will show examples of running the Triangle Count 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 Triangle Count algorithm is defined only for undirected graphs.
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',
{
KNOWS: {
orientation: 'UNDIRECTED'
}
}
)
The Triangle Count algorithm requires the graph to be projected using the UNDIRECTED orientation for relationships.

In the following examples we will demonstrate using the Triangle Count 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.triangleCount.write.estimate('myGraph', { writeProperty: 'triangleCount' })
YIELD nodeCount, relationshipCount, bytesMin, bytesMax, requiredMemory
nodeCount  relationshipCount  bytesMin  bytesMax  requiredMemory 

6 
16 
152 
152 
"152 Bytes" 
Note that the relationship count is 16, although we only projected 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 triangle count 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 triangle count.
For more details on the stream
mode in general, see Stream.
stream
mode:CALL gds.triangleCount.stream('myGraph')
YIELD nodeId, triangleCount
RETURN gds.util.asNode(nodeId).name AS name, triangleCount
ORDER BY triangleCount DESC
name  triangleCount 

"Michael" 
3 
"Chris" 
2 
"Will" 
2 
"Karin" 
1 
"Mark" 
1 
"Alice" 
0 
Here we find that the 'Michael' node has the most triangles.
This can be verified in the example graph.
Since the 'Alice' node only KNOWS
one other node, it can not be part of any triangle, and indeed the algorithm reports a count of zero.
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 global triangle count, which is the total number of triangles in the entire graph.
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.triangleCount.stats('myGraph')
YIELD globalTriangleCount, nodeCount
globalTriangleCount  nodeCount 

3 
6 
Here we can see that the graph has six nodes with a total number of three triangles. Comparing this to the stream example we can see that the 'Michael' node has a triangle count equal to the global triangle count. In other words, that node is part of all of the triangles in the graph and thus has a very central position in the graph.
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 triangle count 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 example, using the triangle count to compute the local clustering coefficient.
For more details on the mutate
mode in general, see Mutate.
mutate
mode:CALL gds.triangleCount.mutate('myGraph', {
mutateProperty: 'triangles'
})
YIELD globalTriangleCount, nodeCount
globalTriangleCount  nodeCount 

3 
6 
The returned result is the same as in the stats
example.
Additionally, the graph 'myGraph' now has a node property triangles
which stores the triangle count for each node.
To find out how to inspect the new schema of the inmemory graph, see Listing graphs.
3.5. Write
The write
execution mode extends the stats
mode with an important side effect: writing the triangle count 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.triangleCount.write('myGraph', {
writeProperty: 'triangles'
})
YIELD globalTriangleCount, nodeCount
globalTriangleCount  nodeCount 

3 
6 
The returned result is the same as in the stats
example.
Additionally, each of the six nodes now has a new property triangles
in the Neo4j database, containing the triangle count for that node.
3.6. Maximum Degree
The Triangle Count algorithm supports a maxDegree
configuration parameter that can be used to exclude nodes from processing if their degree is greater than the configured value.
This can be useful to speed up the computation when there are nodes with a very high degree (socalled super nodes) in the graph.
Super nodes have a great impact on the performance of the Triangle Count algorithm.
To learn about the degree distribution of your graph, see Listing graphs.
The nodes excluded from the computation get assigned a triangle count of 1
.
stream
mode with the maxDegree
parameter:CALL gds.triangleCount.stream('myGraph', {
maxDegree: 4
})
YIELD nodeId, triangleCount
RETURN gds.util.asNode(nodeId).name AS name, triangleCount
ORDER BY name ASC
name  triangleCount 

"Alice" 
0 
"Chris" 
0 
"Karin" 
0 
"Mark" 
0 
"Michael" 
1 
"Will" 
0 
Running the algorithm on the example graph with maxDegree: 4
excludes the 'Michael' node from the computation, as it has a degree of 5.
As this node is part of all the triangles in the example graph excluding it results in no triangles.
4. Triangles listing
It is also possible to list all the triangles in the graph.
To do this we make use of the alpha
procedure gds.alpha.triangles
.
This feature is in the alpha tier. For more information on feature tiers, see API Tiers.
CALL gds.alpha.triangles('myGraph')
YIELD nodeA, nodeB, nodeC
RETURN
gds.util.asNode(nodeA).name AS nodeA,
gds.util.asNode(nodeB).name AS nodeB,
gds.util.asNode(nodeC).name AS nodeC
nodeA  nodeB  nodeC 

"Michael" 
"Karin" 
"Chris" 
"Michael" 
"Chris" 
"Will" 
"Michael" 
"Will" 
"Mark" 
We can see that there are three triangles in the graph: "Will, Michael, and Chris", "Will, Mark, and Michael", and "Michael, Karin, and Chris". The node "Alice" is not part of any triangle and thus does not appear in the triangles listing.
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