K-Means Clustering

This feature is in the alpha tier. For more information on feature tiers, see Operations reference.

1. Introduction

K-Means clustering is an unsupervised learning algorithm that is used to solve clustering problems. It follows a simple procedure of classifying a given data set into a number of clusters, defined by the parameter k. The clusters are then positioned as points and all observations or data points are associated with the nearest cluster, computed, adjusted and then the process starts over using the new adjustments until a desired result is reached.

For more information on this algorithm, see:

2. Initial Centroid Sampling

The algorithm starts by picking k centroids by randomly sampling from the set of available nodes. There are two different sampling strategies.

Uniform

With uniform sampling, each node has the same probability to be picked as one of the k initial centroids. This is the default sampler for K-Means denoted with the uniform parameter.

K-Means++

This sampling strategy adapts the well-known K-means++ initialization algorithm for K-Means. The sampling begins by choosing the first centroid uniformly at random. Then, the remaining k-1 centroids are picked one-by-one based on weighted random sampling. That is, the probability a node is chosen as the next centroid is proportional to its minimum distance from the already picked centroids. Nodes with larger distance hence have higher chance to be picked as a centroid. This sampling strategy tries to spread the initial clusters more evenly so as to obtain a better final clustering. This option can be enabled by choosing kmeans++ as the initial sampler in the configuration.

It is also possible to explicitly give the list of initial centroids to the algorithm via the seedCentroids parameter. In this case, the value of the initialSampler parameter is ignored, even if changed in the configuration.

3. Considerations

In order for K-Means to work properly, the property arrays for all nodes must have the same number of elements. Also, they should contain exclusively numbers and not contain any NaN values.

4. Syntax

K-Means syntax per mode
Run K-Means in stream mode on a named graph.
CALL gds.alpha.kmeans.stream(
  graphName: String,
  configuration: Map
)
YIELD
  nodeId: Integer,
  communityId: Integer,
  distanceFromCentroid: Float,
  silhouette: 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.

nodeProperty

String

n/a

no

A node property to be used by the algorithm.

k

Integer

10

yes

Number of desired clusters.

maxIterations

Integer

10

yes

The maximum number of iterations of K-Means to run.

deltaThreshold

Float

0.05

yes

Value as a percentage to determine when to stop early. If fewer than 'deltaThreshold * |nodes|' nodes change their cluster , the algorithm stops. Value must be between 0 (exclusive) and 1 (inclusive).

numberOfRestarts

Integer

1

yes

Number of times to execute K-Means with different initial centers. The communities returned are those minimizing the average node-center distances.

randomSeed

Integer

n/a

yes

The seed value to control the initial centroid assignment.

initialSampler

String

"uniform"

yes

The method used to sample the first k centroids. "uniform" and "kmeans++", both case-insensitive, are valid inputs.

seedCentroids

List of List of Float

[]

yes

Parameter to explicitly give the initial centroids. It cannot be enabled together with a non-default value of the numberOfRestarts parameter.

computeSilhouette

Boolean

false

yes

If set to true, the silhouette scores are computed once the clustering has been determined. Silhouette is a metric on how well the nodes have been clustered.

Table 3. Results
Name Type Description

nodeId

Integer

Node ID.

communityId

Integer

The community ID.

distanceFromCentroid

Float

Distance of the node from the centroid of its community.

silhouette

Float

Silhouette score of the node.

Run K-Means in stats mode on a named graph.
CALL gds.alpha.kmeans.stats(
  graphName: String,
  configuration: Map
)
YIELD
  preProcessingMillis: Integer,
  computeMillis: Integer,
  postProcessingMillis: Integer,
  communityDistribution: Map,
  centroids: List of List of Float,
  averageDistanceToCentroid: Float,
  averageSilhouette: Float,
  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.

nodeProperty

String

n/a

no

A node property to be used by the algorithm.

k

Integer

10

yes

Number of desired clusters.

maxIterations

Integer

10

yes

The maximum number of iterations of K-Means to run.

deltaThreshold

Float

0.05

yes

Value as a percentage to determine when to stop early. If fewer than 'deltaThreshold * |nodes|' nodes change their cluster , the algorithm stops. Value must be between 0 (exclusive) and 1 (inclusive).

numberOfRestarts

Integer

1

yes

Number of times to execute K-Means with different initial centers. The communities returned are those minimizing the average node-center distances.

randomSeed

Integer

n/a

yes

The seed value to control the initial centroid assignment.

initialSampler

String

"uniform"

yes

The method used to sample the first k centroids. "uniform" and "kmeans++", both case-insensitive, are valid inputs.

seedCentroids

List of List of Float

[]

yes

Parameter to explicitly give the initial centroids. It cannot be enabled together with a non-default value of the numberOfRestarts parameter.

computeSilhouette

Boolean

false

yes

If set to true, the silhouette scores are computed once the clustering has been determined. Silhouette is a metric on how well the nodes have been clustered.

Table 6. Results
Name Type Description

preProcessingMillis

Integer

Milliseconds for preprocessing the data.

computeMillis

Integer

Milliseconds for running the algorithm.

postProcessingMillis

Integer

Milliseconds for computing percentiles and community count.

communityDistribution

Map

Map containing min, max, mean as well as p50, p75, p90, p95, p99 and p999 percentile values of community size for the last level.

centroids

List of List of Float

List of centroid coordinates. Each item is a list containing the coordinates of one centroid.

averageDistanceToCentroid

Float

Average distance between node and centroid.

averageSilhouette

Float

Average silhouette score over all nodes.

configuration

Map

The configuration used for running the algorithm.

Run K-Means in mutate mode on a named graph.
CALL gds.alpha.kmeans.mutate(
  graphName: String,
  configuration: Map
)
YIELD
  preProcessingMillis: Integer,
  computeMillis: Integer,
  mutateMillis: Integer,
  postProcessingMillis: Integer,
  nodePropertiesWritten: Integer,
  communityDistribution: Map,
  centroids: List of List of Float,
  averageDistanceToCentroid: Float,
  averageSilhouette: Float,
  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 cluster 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.

nodeProperty

String

n/a

no

A node property to be used by the algorithm.

k

Integer

10

yes

Number of desired clusters.

maxIterations

Integer

10

yes

The maximum number of iterations of K-Means to run.

deltaThreshold

Float

0.05

yes

Value as a percentage to determine when to stop early. If fewer than 'deltaThreshold * |nodes|' nodes change their cluster , the algorithm stops. Value must be between 0 (exclusive) and 1 (inclusive).

numberOfRestarts

Integer

1

yes

Number of times to execute K-Means with different initial centers. The communities returned are those minimizing the average node-center distances.

randomSeed

Integer

n/a

yes

The seed value to control the initial centroid assignment.

initialSampler

String

"uniform"

yes

The method used to sample the first k centroids. "uniform" and "kmeans++", both case-insensitive, are valid inputs.

seedCentroids

List of List of Float

[]

yes

Parameter to explicitly give the initial centroids. It cannot be enabled together with a non-default value of the numberOfRestarts parameter.

computeSilhouette

Boolean

false

yes

If set to true, the silhouette scores are computed once the clustering has been determined. Silhouette is a metric on how well the nodes have been clustered.

Table 9. Results
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 and community count.

nodePropertiesWritten

Integer

Number of properties added to the projected graph.

communityDistribution

Map

Map containing min, max, mean as well as p50, p75, p90, p95, p99 and p999 percentile values of community size for the last level.

centroids

List of List of Float

List of centroid coordinates. Each item is a list containing the coordinates of one centroid.

averageDistanceToCentroid

Float

Average distance between node and centroid.

averageSilhouette

Float

Average silhouette score over all nodes.

configuration

Map

The configuration used for running the algorithm.

Run K-Means in write mode on a named graph.
CALL gds.alpha.kmeans.write(
  graphName: String,
  configuration: Map
)
YIELD
  preProcessingMillis: Integer,
  computeMillis: Integer,
  writeMillis: Integer,
  postProcessingMillis: Integer,
  nodePropertiesWritten: Integer,
  communityDistribution: Map,
  centroids: List of List of Float,
  averageDistanceToCentroid: Float,
  averageSilhouette: Float,
  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.

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 cluster is written.

nodeProperty

String

n/a

no

A node property to be used by the algorithm.

k

Integer

10

yes

Number of desired clusters.

maxIterations

Integer

10

yes

The maximum number of iterations of K-Means to run.

deltaThreshold

Float

0.05

yes

Value as a percentage to determine when to stop early. If fewer than 'deltaThreshold * |nodes|' nodes change their cluster , the algorithm stops. Value must be between 0 (exclusive) and 1 (inclusive).

numberOfRestarts

Integer

1

yes

Number of times to execute K-Means with different initial centers. The communities returned are those minimizing the average node-center distances.

randomSeed

Integer

n/a

yes

The seed value to control the initial centroid assignment.

initialSampler

String

"uniform"

yes

The method used to sample the first k centroids. "uniform" and "kmeans++", both case-insensitive, are valid inputs.

seedCentroids

List of List of Float

[]

yes

Parameter to explicitly give the initial centroids. It cannot be enabled together with a non-default value of the numberOfRestarts parameter.

computeSilhouette

Boolean

false

yes

If set to true, the silhouette scores are computed once the clustering has been determined. Silhouette is a metric on how well the nodes have been clustered.

Table 12. Results
Name Type Description

preProcessingMillis

Integer

Milliseconds for preprocessing the data.

computeMillis

Integer

Milliseconds for running the algorithm.

writeMillis

Integer

Milliseconds for adding properties to the Neo4j database.

postProcessingMillis

Integer

Milliseconds for computing percentiles and community count.

nodePropertiesWritten

Integer

Number of properties added to the projected graph.

communityDistribution

Map

Map containing min, max, mean as well as p50, p75, p90, p95, p99 and p999 percentile values of community size for the last level.

centroids

List of List of Float

List of centroid coordinates. Each item is a list containing the coordinates of one centroid.

averageDistanceToCentroid

Float

Average distance between node and centroid.

averageSilhouette

Float

Average silhouette score over all nodes.

configuration

Map

The configuration used for running the algorithm.

5. Examples

In this section we will show examples of running the K-Means 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 cities graph of a handful nodes connected in a particular pattern. The example graph looks like this:

Visualization of the example graph
The following Cypher statement will create the example graph in the Neo4j database:
CREATE
  (:City {name: 'Surbiton', coordinates: [51.39148, -0.29825]}),
  (:City {name: 'Liverpool', coordinates: [53.41058, -2.97794]}),
  (:City {name: 'Kingston upon Thames', coordinates: [51.41259, -0.2974]}),
  (:City {name: 'Sliven', coordinates: [42.68583, 26.32917]}),
  (:City {name: 'Solna', coordinates: [59.36004, 18.00086]}),
  (:City {name: 'Örkelljunga', coordinates: [56.28338, 13.27773]}),
  (:City {name: 'Malmö', coordinates: [55.60587, 13.00073]}),
  (:City {name: 'Xánthi', coordinates: [41.13488, 24.888]});

This graph is composed of various City nodes, in three global locations - United Kingdom, Sweden and the Balkan region in Europe.

We can now project the graph and store it in the graph catalog. We load the City node label with coordinates node property.

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 the graph and store it in the graph catalog.
CALL gds.graph.project(
    'cities',
    {
      City: {
        properties: 'coordinates'
      }
    },
    '*'
)

In the following examples we will demonstrate using the K-Means algorithm on this graph to find communities of cities that are close to each other geographically.

5.1. 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.

The following will estimate the memory requirements for running the algorithm:
CALL gds.alpha.kmeans.write.estimate('cities', {
  writeProperty: 'kmeans',
  nodeProperty: 'coordinates'
})
YIELD nodeCount, bytesMin, bytesMax, requiredMemory
Table 13. Results
nodeCount bytesMin bytesMax requiredMemory

8

33264

54256

"[32 KiB ... 52 KiB]"

5.2. Stream

In the stream execution mode, the algorithm returns the cluster for each node. This allows us to inspect the results directly or post-process them in Cypher without any side effects.

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

The following will run the algorithm and stream results:
CALL gds.alpha.kmeans.stream('cities', {
  nodeProperty: 'coordinates',
  k: 3,
  randomSeed: 42
})
YIELD nodeId, communityId
RETURN gds.util.asNode(nodeId).name AS name, communityId
ORDER BY communityId, name ASC
Table 14. Results
name communityId

"Kingston upon Thames"

0

"Liverpool"

0

"Surbiton"

0

"Sliven"

1

"Xánthi"

1

"Malmö"

2

"Solna"

2

"Örkelljunga"

2

In the example above we can see that the cities are geographically clustered together.

5.3. 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.alpha.kmeans.stats('cities', {
  nodeProperty: 'coordinates',
  k: 3,
  randomSeed: 42
})
YIELD communityDistribution
Table 15. Results
communityDistribution

{ "p99": 3, "min": 2, "max": 3, "mean": 2.6666666666666665, "p90": 3, "p50": 3, "p999": 3, "p95": 3, "p75": 3 }

5.4. 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 cluster 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 and store the results in cities graph:
CALL gds.alpha.kmeans.mutate('cities', {
  nodeProperty: 'coordinates',
  k: 3,
  randomSeed: 42,
  mutateProperty: 'kmeans'
})
YIELD communityDistribution
Table 16. Results
communityDistribution

{ "p99": 3, "min": 2, "max": 3, "mean": 2.6666666666666665, "p90": 3, "p50": 3, "p999": 3, "p95": 3, "p75": 3 }

In mutate mode, only a single row is returned by the procedure. The result is written to the GDS in-memory graph instead of the Neo4j database.

5.5. Write

The write execution mode extends the stats mode with an important side effect: writing the cluster 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 and write the results back to Neo4j:
CALL gds.alpha.kmeans.write('cities', {
  nodeProperty: 'coordinates',
  k: 3,
  randomSeed: 42,
  writeProperty: 'kmeans'
})
YIELD nodePropertiesWritten
Table 17. Results
nodePropertiesWritten

8

In write mode, only a single row is returned by the procedure. The result is written to the Neo4j database instead of the GDS in-memory graph.

5.6. Seeding initial centroids

We now see the effect that seeding centroids has on K-Means. We run K-Means with initial seeds the coordinates of New York, Amsterdam, and Rome.

The following will run the algorithm and stream results:
CALL gds.alpha.kmeans.stream('cities', {
  nodeProperty: 'coordinates',
  k: 3,
  seedCentroids: [[40.712776,-74.005974], [52.370216,4.895168],[41.902782,12.496365]]
})
YIELD nodeId, communityId
RETURN gds.util.asNode(nodeId).name AS name, communityId
ORDER BY communityId, name ASC
Table 18. Results
name communityId

"Kingston upon Thames"

1

"Liverpool"

1

"Malmö"

1

"Solna"

1

"Surbiton"

1

"Örkelljunga"

1

"Sliven"

2

"Xánthi"

2

Notice that in this case the cities have been geographically clustered into two clusters: one contains cities in Northern Europe whereas the other contains in Southern Europe. On the other hand, the cluster with New York as the initial centroid was not the closest to any city at the first phase.