# Weakly Connected Components

## 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.

## Introduction

The Weakly Connected Components (WCC) algorithm finds sets of connected nodes in directed and undirected graphs. Two nodes are connected, if there exists a path between them. The set of all nodes that are connected with each other form a component. In contrast to Strongly Connected Components (SCC), the direction of relationships on the path between two nodes is not considered. For example, in a directed graph `(a)→(b)`, `a` and `b` will be in the same component, even if there is no directed relationship `(b)→(a)`.

WCC is often used early in an analysis to understand the structure of a graph. Using WCC to understand the graph structure enables running other algorithms independently on an identified cluster.

The implementation of the algorithm is based on the following papers:

## Syntax

This section covers the syntax used to execute the Weakly Connected Components 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.

WCC syntax per mode
Run WCC in stream mode on a named graph.
``````CALL gds.wcc.stream(
graphName: String,
configuration: Map
)
YIELD
nodeId: Integer,
componentId: Integer``````
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. Nodes with any of the given labels will be included.

relationshipTypes

List of String

`['*']`

yes

Filter the named graph using the given relationship types. Relationships with any of the given types will be included.

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.

relationshipWeightProperty

String

`null`

yes

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

seedProperty

String

`n/a`

yes

Used to set the initial component for a node. The property value needs to be a number.

threshold

Float

`null`

yes

The value of the weight above which the relationship is considered in the computation.

consecutiveIds

Boolean

`false`

yes

Flag to decide whether component identifiers are mapped into a consecutive id space (requires additional memory).

minComponentSize

Integer

`0`

yes

Only nodes inside communities larger or equal the given value are returned.

Table 3. Results
Name Type Description

nodeId

Integer

Node ID.

componentId

Integer

Component ID.

Run WCC in stats mode on a named graph.
``````CALL gds.wcc.stats(
graphName: String,
configuration: Map
)
YIELD
componentCount: Integer,
preProcessingMillis: Integer,
computeMillis: Integer,
postProcessingMillis: Integer,
componentDistribution: Map,
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. Nodes with any of the given labels will be included.

relationshipTypes

List of String

`['*']`

yes

Filter the named graph using the given relationship types. Relationships with any of the given types will be included.

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.

relationshipWeightProperty

String

`null`

yes

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

seedProperty

String

`n/a`

yes

Used to set the initial component for a node. The property value needs to be a number.

threshold

Float

`null`

yes

The value of the weight above which the relationship is considered in the computation.

consecutiveIds

Boolean

`false`

yes

Flag to decide whether component identifiers are mapped into a consecutive id space (requires additional memory).

Table 6. Results
Name Type Description

componentCount

Integer

The number of computed components.

preProcessingMillis

Integer

Milliseconds for preprocessing the data.

computeMillis

Integer

Milliseconds for running the algorithm.

postProcessingMillis

Integer

Milliseconds for computing component count and distribution statistics.

componentDistribution

Map

Map containing min, max, mean as well as p1, p5, p10, p25, p50, p75, p90, p95, p99 and p999 percentile values of component sizes.

configuration

Map

The configuration used for running the algorithm.

Run WCC in mutate mode on a named graph.
``````CALL gds.wcc.mutate(
graphName: String,
configuration: Map
)
YIELD
componentCount: Integer,
nodePropertiesWritten: Integer,
preProcessingMillis: Integer,
computeMillis: Integer,
mutateMillis: Integer,
postProcessingMillis: Integer,
componentDistribution: Map,
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 component ID 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.

relationshipWeightProperty

String

`null`

yes

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

seedProperty

String

`n/a`

yes

Used to set the initial component for a node. The property value needs to be a number.

threshold

Float

`null`

yes

The value of the weight above which the relationship is considered in the computation.

consecutiveIds

Boolean

`false`

yes

Flag to decide whether component identifiers are mapped into a consecutive id space (requires additional memory).

Table 9. Results
Name Type Description

componentCount

Integer

The number of computed components.

nodePropertiesWritten

Integer

The number of node properties written.

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 component count and distribution statistics.

componentDistribution

Map

Map containing min, max, mean as well as p1, p5, p10, p25, p50, p75, p90, p95, p99 and p999 percentile values of component sizes.

configuration

Map

The configuration used for running the algorithm.

Run WCC in write mode on a named graph.
``````CALL gds.wcc.write(
graphName: String,
configuration: Map
)
YIELD
componentCount: Integer,
nodePropertiesWritten: Integer,
preProcessingMillis: Integer,
computeMillis: Integer,
writeMillis: Integer,
postProcessingMillis: Integer,
componentDistribution: Map,
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. Nodes with any of the given labels will be included.

relationshipTypes

List of String

`['*']`

yes

Filter the named graph using the given relationship types. Relationships with any of the given types will be included.

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 component ID is written.

relationshipWeightProperty

String

`null`

yes

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

seedProperty

String

`n/a`

yes

Used to set the initial component for a node. The property value needs to be a number.

threshold

Float

`null`

yes

The value of the weight above which the relationship is considered in the computation.

consecutiveIds

Boolean

`false`

yes

Flag to decide whether component identifiers are mapped into a consecutive id space (requires additional memory).

minComponentSize

Integer

`0`

yes

Only nodes inside communities larger or equal the given value will be written to the underlying Neo4j database.

Table 12. Results
Name Type Description

componentCount

Integer

The number of computed components.

nodePropertiesWritten

Integer

The number of node properties written.

preProcessingMillis

Integer

Milliseconds for preprocessing the data.

computeMillis

Integer

Milliseconds for running the algorithm.

writeMillis

Integer

Milliseconds for writing result back to Neo4j.

postProcessingMillis

Integer

Milliseconds for computing component count and distribution statistics.

componentDistribution

Map

Map containing min, max, mean as well as p1, p5, p10, p25, p50, p75, p90, p95, p99 and p999 percentile values of component sizes.

configuration

Map

The configuration used for running the algorithm.

## Examples

 All the examples below should be run in an empty database. The examples use native projections as the norm, although Cypher projections can be used as well.

In this section we will show examples of running the Weakly Connected Components 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 user 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
(nAlice:User {name: 'Alice'}),
(nBridget:User {name: 'Bridget'}),
(nCharles:User {name: 'Charles'}),
(nDoug:User {name: 'Doug'}),
(nMark:User {name: 'Mark'}),
(nMichael:User {name: 'Michael'}),

This graph has two connected components, each with three nodes. The relationships that connect the nodes in each component have a property `weight` which determines the strength of the relationship.

The following statement will project a graph using a native projection and store it in the graph catalog under the name 'myGraph'.
``````CALL gds.graph.project(
'myGraph',
'User',
{
relationshipProperties: 'weight'
}
)``````

In the following examples we will demonstrate using the Weakly Connected Components algorithm on this graph.

### 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 in write mode:
``````CALL gds.wcc.write.estimate('myGraph', { writeProperty: 'component' })
YIELD nodeCount, relationshipCount, bytesMin, bytesMax, requiredMemory``````
Table 13. Results
nodeCount relationshipCount bytesMin bytesMax requiredMemory

6

4

112

112

"112 Bytes"

### Stream

In the `stream` execution mode, the algorithm returns the component ID 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 order the results to see the nodes that belong to the same component displayed next to each other.

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

The following will run the algorithm and stream results:
``````CALL gds.wcc.stream('myGraph')
YIELD nodeId, componentId
RETURN gds.util.asNode(nodeId).name AS name, componentId
ORDER BY componentId, name``````
Table 14. Results
name componentId

"Alice"

0

"Bridget"

0

"Charles"

0

"Doug"

3

"Mark"

3

"Michael"

3

The result shows that the algorithm identifies two components. This can be verified in the example graph.

The default behaviour of the algorithm is to run `unweighted`, e.g. without using `relationship` weights. The `weighted` option will be demonstrated in Weighted

### 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 in `stats` mode:
``````CALL gds.wcc.stats('myGraph')
YIELD componentCount``````
Table 15. Results
componentCount

2

The result shows that `myGraph` has two components and this can be verified by looking at the example graph.

### 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 component ID 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.wcc.mutate('myGraph', { mutateProperty: 'componentId' })
YIELD nodePropertiesWritten, componentCount;``````
Table 16. Results
nodePropertiesWritten componentCount

6

2

### Write

The `write` execution mode extends the `stats` mode with an important side effect: writing the component ID 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.wcc.write('myGraph', { writeProperty: 'componentId' })
YIELD nodePropertiesWritten, componentCount;``````
Table 17. Results
nodePropertiesWritten componentCount

6

2

As we can see from the results, the nodes connected to one another are calculated by the algorithm as belonging to the same connected component.

### Weighted

By configuring the algorithm to use a weight we can increase granularity in the way the algorithm calculates component assignment. We do this by specifying the property key with the `relationshipWeightProperty` configuration parameter. Additionally, we can specify a threshold for the weight value. Then, only weights greater than the threshold value will be considered by the algorithm. We do this by specifying the threshold value with the `threshold` configuration parameter.

If a relationship does not have the specified weight property, the algorithm falls back to using a default value of zero.

The following will run the algorithm using relationship weight and stream results:
``````CALL gds.wcc.stream('myGraph', {
relationshipWeightProperty: 'weight',
threshold: 1.0
}) YIELD nodeId, componentId
RETURN gds.util.asNode(nodeId).name AS Name, componentId AS ComponentId
ORDER BY ComponentId, Name``````
Table 18. Results
Name ComponentId

`"Alice"`

`0`

`"Charles"`

`0`

`"Bridget"`

`1`

`"Doug"`

`3`

`"Mark"`

`3`

`"Michael"`

`3`

As we can see from the results, the node named 'Bridget' is now in its own component, due to its relationship weight being less than the configured threshold and thus ignored.

 We are using stream mode to illustrate running the algorithm as weighted or unweighted, all the other algorithm modes also support this configuration parameter.

### Seeded components

It is possible to define preliminary component IDs for nodes using the `seedProperty` configuration parameter. This is helpful if we want to retain components from a previous run and it is known that no components have been split by removing relationships. The property value needs to be a number.

The algorithm first checks if there is a seeded component ID assigned to the node. If there is one, that component ID is used. Otherwise, a new unique component ID is assigned to the node.

Once every node belongs to a component, the algorithm merges components of connected nodes. When components are merged, the resulting component is always the one with the lower component ID. Note that the `consecutiveIds` configuration option cannot be used in combination with seeding in order to retain the seeding values.

 The algorithm assumes that nodes with the same seed value do in fact belong to the same component. If any two nodes in different components have the same seed, behavior is undefined. It is then recommended running WCC without seeds.

To demonstrate this in practice, we will go through a few steps:

1. We will run the algorithm and write the results to Neo4j.

2. Then we will add another node to our graph, this node will not have the property computed in Step 1.

3. We will project a new graph that has the result from Step 1 as `nodeProperty`

4. And then we will run the algorithm again, this time in `stream` mode, and we will use the `seedProperty` configuration parameter.

We will use the weighted variant of WCC.

Step 1

The following will run the algorithm in `write` mode:
``````CALL gds.wcc.write('myGraph', {
writeProperty: 'componentId',
relationshipWeightProperty: 'weight',
threshold: 1.0
})
YIELD nodePropertiesWritten, componentCount;``````
Table 19. Results
nodePropertiesWritten componentCount

6

3

Step 2

After the algorithm has finished writing to Neo4j we want to create a new node in the database.

The following will create a new node in the Neo4j graph, with no component ID:
``````MATCH (b:User {name: 'Bridget'})
CREATE (b)-[:LINK {weight: 2.0}]->(new:User {name: 'Mats'})``````

Step 3

Note, that we cannot use our already projected graph as it does not contain the component id. We will therefore project a second graph that contains the previously computed component id.

The following will project a new graph containing the previously computed component id:
``````CALL gds.graph.project(
'myGraph-seeded',
'User',
{
nodeProperties: 'componentId',
relationshipProperties: 'weight'
}
)``````

Step 4

The following will run the algorithm in `stream` mode using `seedProperty`:
``````CALL gds.wcc.stream('myGraph-seeded', {
seedProperty: 'componentId',
relationshipWeightProperty: 'weight',
threshold: 1.0
}) YIELD nodeId, componentId
RETURN gds.util.asNode(nodeId).name AS name, componentId
ORDER BY componentId, name``````
Table 20. Results
name componentId

"Alice"

0

"Charles"

0

"Bridget"

1

"Mats"

1

"Doug"

3

"Mark"

3

"Michael"

3

The result shows that despite not having the `seedProperty` when it was projected, the node 'Mats' has been assigned to the same component as the node 'Bridget'. This is correct because these two nodes are connected.

### Writing Seeded components

In the previous section we demonstrated the `seedProperty` usage in `stream` mode. It is also available in the other modes of the algorithm. Below is an example on how to use `seedProperty` in `write` mode. Note that the example below relies on Steps 1 - 3 from the previous section.

The following will run the algorithm in `write` mode using `seedProperty`:
``````CALL gds.wcc.write('myGraph-seeded', {
seedProperty: 'componentId',
writeProperty: 'componentId',
relationshipWeightProperty: 'weight',
threshold: 1.0
})
YIELD nodePropertiesWritten, componentCount;``````
Table 21. Results
nodePropertiesWritten componentCount

1

3

 If the `seedProperty` configuration parameter has the same value as `writeProperty`, the algorithm only writes properties for nodes where the component ID has changed. If they differ, the algorithm writes properties for all nodes.

### Graph Sampling optimization

The WCC implementation provides two compute strategies:

While both strategies provide very good performance, the sampled strategy is usually the faster one. The decision, which strategy to use, depends on the input graph. If the relationships of the graph are …​

• …​ undirected, the algorithm picks the sampled strategy.

• …​ directed, the algorithm picks the unsampled strategy.

• …​ directed and inverse indexed, the algorithm picks the sampled strategy.

The direction of a relationship is defined by the `orientation` which can be set during a graph projection. While `NATURAL` and `REVERSE` orientation result in a directed graph, the `UNDIRECTED` orientation leads to undirected relationships. In order to create a directed graph with inverse indexed relationships, one can use the `indexInverse` parameter as part of the relationship projection. An inverse index allows the algorithm to traverse the relationships of a node according to the opposite orientation. If the graph is projected using a `NATURAL` orientation, the inverse index represents the `REVERSE` orientation and vice versa.

The following statement will project the above example graph using a native projection with inverse index and store it in the graph catalog under the name `myIndexedGraph`.
``````CALL gds.graph.project(
'myIndexedGraph',
'User',
{LINK: {orientation: 'NATURAL', indexInverse: true }}
)``````

The following query is identical to the stream example in the previous section. This time, we execute WCC on `myIndexedGraph` which will allow the algorithm to use the sampled strategy.

The following will run the algorithm with sampled strategy and stream results:
``````CALL gds.wcc.stream('myIndexedGraph', {concurrency: 1, consecutiveIds: true})
YIELD nodeId, componentId
RETURN gds.util.asNode(nodeId).name AS name, componentId
ORDER BY componentId, name``````
Table 22. Results
name componentId

"Alice"

0

"Bridget"

0

"Charles"

0

"Doug"

1

"Mark"

1

"Michael"

1

 Because of the randomness in the Graph sampling optimization we are using `concurrency: 1` in the example. Running it with higher concurrency may yield different componentIds but the actual components should stay the same. For our case here it would be equally plausible to get the inverse solution, f.i. when our community `0` nodes are mapped to community `3` instead, and vice versa.