KCore Decomposition
Glossary
 Directed

Directed trait. The algorithm is welldefined 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 welldefined 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. Introduction
The Kcore decomposition constitutes a process of separates the nodes in a graph into groups based on the degree sequence and topology of the graph.
The term icore
refers to a maximal subgraph of the original graph such that each node in this subgraph has degree at least i
.
The maximality ensures that it is not possible to find another subgraph with more nodes where this degree property holds.
The nodes in the subgraph denoted by icore
also belong to the subgraph denoted by jcore
for any j<i
.
The converse however is not true.
Each node u is associated with a core value which denotes the largest value i such that u belongs to the icore
.
The largest core value is called the degeneracy of the graph.
Standard algorithms for KCore Decomposition iteratively remove the node of lowest degree until the graph becomes empty. When a node is removed from the graph, all of its relationships are removed, and the degree of its neighbors is reduced by one. With this approach, the different core groups are discovered onebyone.
The Neo4j GDS Library offers a parallel implementation based on two recent approaches for the problem:
Kcore Decomposition can have applications in several fields ranging from social network analysis to bioinformatics. Some of the possible usecases are presented here.
2. Syntax
This section covers the syntax used to execute the KCore Decomposition 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.kcore.stream(
graphName: String,
configuration: Map
) YIELD
nodeId: Integer,
coreValue: Float
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. 

Boolean 

yes 
If disabled the progress percentage will not be logged. 
Name  Type  Description 

nodeId 
Integer 
Node ID. 
coreValue 
Float 
Core value. 
CALL gds.kcore.stats(
graphName: String,
configuration: Map
) YIELD
degeneracy: 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. 

Boolean 

yes 
If disabled the progress percentage will not be logged. 
Name  Type  Description 

degeneracy 
Integer 
the maximum core value in the graph. 
preProcessingMillis 
Integer 
Milliseconds for preprocessing the graph. 
computeMillis 
Integer 
Milliseconds for running the algorithm. 
postProcessingMillis 
Integer 
Milliseconds for computing the statistics. 
configuration 
Map 
Configuration used for running the algorithm. 
CALL gds.kcore.mutate(
graphName: String,
configuration: Map
) YIELD
degeneracy: Integer,
preProcessingMillis: Integer,
computeMillis: Integer,
postProcessingMillis: Integer,
mutateMillis: Integer,
nodePropertiesWritten: 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 core value 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. 
Name  Type  Description 

degeneracy 
Integer 
the maximum core value in the graph. 
preProcessingMillis 
Integer 
Milliseconds for preprocessing the graph. 
computeMillis 
Integer 
Milliseconds for running the algorithm. 
postProcessingMillis 
Integer 
Milliseconds for computing the statistics. 
mutateMillis 
Integer 
Milliseconds for adding properties to the projected graph. 
nodePropertiesWritten 
Integer 
Number of properties added to the projected graph. 
configuration 
Map 
Configuration used for running the algorithm. 
CALL gds.kcore.write(
graphName: String,
configuration: Map
) YIELD
degeneracy: Integer,
preProcessingMillis: Integer,
computeMillis: Integer,
postProcessingMillis: Integer,
writeMillis: Integer,
nodePropertiesWritten: 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. 

Boolean 

yes 
If disabled the progress percentage will not be logged. 

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 core value is written. 
Name  Type  Description 

degeneracy 
Integer 
the maximum core value in the graph. 
preProcessingMillis 
Integer 
Milliseconds for preprocessing the graph. 
computeMillis 
Integer 
Milliseconds for running the algorithm. 
postProcessingMillis 
Integer 
Milliseconds for computing the statistics. 
writeMillis 
Integer 
Milliseconds for writing result data back. 
nodePropertiesWritten 
Integer 
Number of properties written to Neo4j. 
configuration 
Map 
The configuration used for running the algorithm. 
3. Examples
In this section we will show examples of running the KCore Decomposition 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:User {name: 'Alice'}),
(bridget:User {name: 'Bridget'}),
(charles:User {name: 'Charles'}),
(doug:User {name: 'Doug'}),
(eli:User {name: 'Eli'}),
(filip:User {name: 'Filip'}),
(greg:User {name: 'Greg'}),
(harry:User {name: 'Harry'}),
(ian:User {name: 'Ian'}),
(james:User {name: 'James'}),
(alice)[:FRIEND]>(bridget),
(bridget)[:FRIEND]>(charles),
(charles)[:FRIEND]>(doug),
(charles)[:FRIEND]>(harry),
(doug)[:FRIEND]>(eli),
(doug)[:FRIEND]>(filip),
(doug)[:FRIEND]>(greg),
(eli)[:FRIEND]>(filip),
(eli)[:FRIEND]>(greg),
(filip)[:FRIEND]>(greg),
(greg)[:FRIEND]>(harry),
(ian)[:FRIEND]>(james)
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 User
nodes and the FRIEND
relationships.
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(
'graph',
'User',
{
FRIEND: {
orientation: 'UNDIRECTED'
}
}
)
The graph is projected in the UNDIRECTED
orientation as the friendship relationship is associative.
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.kcore.write.estimate('graph', { writeProperty: 'coreValue' })
YIELD nodeCount, relationshipCount, bytesMin, bytesMax, requiredMemory
nodeCount  relationshipCount  bytesMin  bytesMax  requiredMemory 

10 
24 
1456 
1456 
"1456 Bytes" 
3.2. Stream
In the stream
execution mode, the algorithm returns the core value 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 core values.
For more details on the stream
mode in general, see Stream.
stream
mode:CALL gds.kcore.stream('graph')
YIELD nodeId, coreValue
RETURN gds.util.asNode(nodeId).name AS name, coreValue
ORDER BY coreValue ASC, name DESC
name  coreValue 

"James" 
1 
"Ian" 
1 
"Bridget" 
1 
"Alice" 
1 
"Harry" 
2 
"Charles" 
2 
"Greg" 
3 
"Filip" 
3 
"Eli" 
3 
"Doug" 
3 
The algorithm has separated the nodes in the graph in three distinct groups. The first group where all nodes have core value qual to 1 includes James, Ian, Bridget, and Alice. The second group includes Harry and Charles. Here, all the nodes have core value equal to 2. The third group includes Greg, Filip, Eli, and Doug, and all the nodes have core value equal to 3.
As it was explained in introduction, nodes with core value i have degree at least i in the subgraph containing only nodes with core value at least i. For example, although Charles has degree 3, he cannot be part of the 3core subgraph since one of its neighbors is Bridget from the first group of core value 1. Once Bridget is excluded, Charles is left with a degree of 2, which acts as an upper bound on its core value. One of its two remaining neighbors is Doug who belongs to the 3core.
Note that as the results show, the nodes in different connected components might be part of the same core group (for example Ian and Alice).
3.3. 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.
stats
mode:CALL gds.kcore.stats('graph')
YIELD degeneracy
RETURN degeneracy
degeneracy 

3 
As the results from stream example also confirm, the degeneracy, i.e., the largest core value, is equal to three.
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 core value 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.kcore.mutate('graph', { mutateProperty: 'coreValue' })
YIELD degeneracy, nodePropertiesWritten
RETURN degeneracy , nodePropertiesWritten
degeneracy  nodePropertiesWritten 

3 
10 
The returned result is the same as in the stats
example.
Additionally, the inmemory graph now has a node property coreValue
which stores the core value of each node.
To find out how to inspect the new schema of the inmemory graph, see Listing graphs in the catalog.
3.5. Write
The write
execution mode extends the stats
mode with an important side effect: writing the core value 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.kcore.write('graph', { writeProperty: 'coreValue' })
YIELD degeneracy, nodePropertiesWritten
RETURN degeneracy , nodePropertiesWritten
degeneracy  nodePropertiesWritten 

3 
10 
The returned result is the same as in the stats
example.
Additionally, each of the seven nodes now has a new property coreValue
in the Neo4j database, containing the core value for that node.
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