# Strongly Connected Components

The Strongly Connected Components (SCC) algorithm finds maximal sets of connected nodes in a directed graph. A set is considered a strongly connected component if there is a directed path between each pair of nodes within the set. It is often used early in a graph analysis process to help us get an idea of how our graph is structured.

This feature is in the alpha tier. For more information on feature tiers, see API Tiers.

Supported algorithm traits:

## 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.

## 1. History and explanation

SCC is one of the earliest graph algorithms, and the first linear-time algorithm was described by Tarjan in 1972. Decomposing a directed graph into its strongly connected components is a classic application of the depth-first search algorithm.

## 2. Use-cases - when to use the Strongly Connected Components algorithm

• In the analysis of powerful transnational corporations, SCC can be used to find the set of firms in which every member owns directly and/or indirectly owns shares in every other member. Although it has benefits, such as reducing transaction costs and increasing trust, this type of structure can weaken market competition. Read more in "The Network of Global Corporate Control".

• SCC can be used to compute the connectivity of different network configurations when measuring routing performance in multihop wireless networks. Read more in "Routing performance in the presence of unidirectional links in multihop wireless networks"

• Strongly Connected Components algorithms can be used as a first step in many graph algorithms that work only on strongly connected graph. In social networks, a group of people are generally strongly connected (For example, students of a class or any other common place). Many people in these groups generally like some common pages, or play common games. The SCC algorithms can be used to find such groups, and suggest the commonly liked pages or games to the people in the group who have not yet liked those pages or games.

## 3. Syntax

Decomposition syntax per mode

The following will run the algorithm and stream results:
``````CALL gds.scc.stream(graphName: String, configuration: Map)
YIELD  nodeId,
componentId``````
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.

consecutiveIds

Boolean

`false`

yes

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

Table 3. Results
Name Type Description

nodeId

Integer

Node ID.

componentId

Integer

Component ID.

The following will run the algorithm in stats mode
``````CALL gds.scc.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.

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 strongly connected 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.

The following will run the algorithm and mutate the in-memory graph:
``````CALL gds.scc.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 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.

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 strongly connected 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 mutating the in-memory 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.

The following will run the algorithm and write back results:
``````CALL gds.scc.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 is written.

consecutiveIds

Boolean

`false`

yes

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

Table 12. Results
Name Type Description

componentCount

Integer

The number of computed strongly connected 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.

## 4. Strongly Connected Components algorithm examples

The following will create a sample graph:
``````CREATE (nAlice:User {name:'Alice'})
CREATE (nBridget:User {name:'Bridget'})
CREATE (nCharles:User {name:'Charles'})
CREATE (nDoug:User {name:'Doug'})
CREATE (nMark:User {name:'Mark'})
CREATE (nMichael:User {name:'Michael'})

CREATE (nAlice)-[:FOLLOW]->(nBridget)
CREATE (nAlice)-[:FOLLOW]->(nCharles)
CREATE (nMark)-[:FOLLOW]->(nDoug)
CREATE (nMark)-[:FOLLOW]->(nMichael)
CREATE (nBridget)-[:FOLLOW]->(nMichael)
CREATE (nDoug)-[:FOLLOW]->(nMark)
CREATE (nMichael)-[:FOLLOW]->(nAlice)
CREATE (nAlice)-[:FOLLOW]->(nMichael)
CREATE (nBridget)-[:FOLLOW]->(nAlice)
CREATE (nMichael)-[:FOLLOW]->(nBridget);``````
The following will project and store a named graph:
``CALL gds.graph.project('graph', 'User', 'FOLLOW')``

### 4.1. 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.scc.write.estimate('graph', { writeProperty: 'componentId' })
YIELD nodeCount, relationshipCount, bytesMin, bytesMax, requiredMemory``````
Table 13. Results
nodeCount relationshipCount bytesMin bytesMax requiredMemory

6

10

33332

33332

"32 KiB"

### 4.2. Stream

In the `stream` execution mode, the algorithm returns the component 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 back results:
``````CALL gds.scc.stream('graph', {})
YIELD nodeId, componentId
RETURN gds.util.asNode(nodeId).name AS Name, componentId AS Component
ORDER BY Component DESC``````
Table 14. Results
Name Component

`"Doug"`

`3`

`"Mark"`

`3`

`"Charles"`

`2`

`"Alice"`

`0`

`"Bridget"`

`0`

`"Michael"`

`0`

We have 3 strongly connected components in our sample graph.

The first, and biggest, component has members Alice, Bridget, and Michael, while the second component has Doug and Mark. Charles ends up in his own component because there isn’t an outgoing relationship from that node to any of the others.

### 4.3. 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.scc.stats('graph')
YIELD componentCount``````
Table 15. Results
componentCount

3

### 4.4. 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 component 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 `graph`:
``````CALL gds.scc.mutate('graph', { mutateProperty: 'componentId'})
YIELD componentCount``````
Table 16. Results
componentCount

3

### 4.5. Write

The `write` execution mode extends the `stats` mode with an important side effect: writing the component 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 back results:
``````CALL gds.scc.write('graph', {
writeProperty: 'componentId'
})
YIELD componentCount, componentDistribution
RETURN componentCount,componentDistribution.max as maxSetSize, componentDistribution.min as minSetSize``````
Table 17. Results
componentCount maxSetSize minSetSize

`3`

`3`

`1`

The following will find the largest partition:
``````MATCH (u:User)
RETURN u.componentId AS Component, count(*) AS ComponentSize
ORDER BY ComponentSize DESC
LIMIT 1``````
Table 18. Results
Component ComponentSize

`0`

`3`