Harmonic Centrality

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.

Harmonic centrality (also known as valued centrality) is a variant of closeness centrality, that was invented to solve the problem the original formula had when dealing with unconnected graphs. As with many of the centrality algorithms, it originates from the field of social network analysis.

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

History and explanation

Harmonic centrality was proposed by Marchiori and Latora in Harmony in the Small World while trying to come up with a sensible notion of "average shortest path".

They suggested a different way of calculating the average distance to that used in the Closeness Centrality algorithm. Rather than summing the distances of a node to all other nodes, the harmonic centrality algorithm sums the inverse of those distances. This enables it deal with infinite values.

The raw harmonic centrality for a node is calculated using the following formula:

raw harmonic centrality(node) = sum(1 / distance from node to every other node excluding itself)

As with closeness centrality, we can also calculate a normalized harmonic centrality with the following formula:

normalized harmonic centrality(node) = sum(1 / distance from node to every other node excluding itself) / (number of nodes - 1)

In this formula, ∞ values are handled cleanly.

Use-cases - when to use the Harmonic Centrality algorithm

Harmonic centrality was proposed as an alternative to closeness centrality, and therefore has similar use cases.

For example, we might use it if we’re trying to identify where in the city to place a new public service so that it’s easily accessible for residents. If we’re trying to spread a message on social media we could use the algorithm to find the key influencers that can help us achieve our goal.

Syntax

This section covers the syntax used to execute the Harmonic Centrality 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.

Harmonic Centrality syntax per mode

The following will run the algorithm and write back results:

CALL gds.closeness.harmonic.write(
  graphName: String,
  configuration: Map
)
YIELD
  centralityDistribution: Map,
  preProcessingMillis: Integer,
  computeMillis: Integer,
  postProcessingMillis: Integer,
  writeMillis: Integer,
  nodePropertiesWritten: Integer,
  configuration: Map

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
concurrency	int	4	yes	The number of concurrent threads used for running the algorithm. Also provides the default value for 'readConcurrency' and 'writeConcurrency'.
readConcurrency	int	value of 'concurrency'	yes	The number of concurrent threads used for reading the graph.
writeConcurrency	int	value of 'concurrency'	yes	The number of concurrent threads used for writing the result.
writeProperty	string	N/A	no	The node property in the Neo4j database to which the centrality score is written.

Table 3. Results
Name	Type	Description
centralityDistribution	Map	Map containing min, max, mean as well as p50, p75, p90, p95, p99 and p999 percentile values of centrality values.
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.

The following will run the algorithm and stream results:

CALL gds.closeness.harmonic.stream(
  graphName: String,
  configuration: Map
)
YIELD
  nodeId: Integer,
  score: Float

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.

Table 6. Results
Name	Type	Description
nodeId	Integer	Node ID.
score	Float	Harmonic centrality score.

Alpha syntax

The following will run the algorithm and write back results:

CALL gds.alpha.closeness.harmonic.write(configuration: Map)
YIELD nodes, preProcessingMillis, computeMillis, writeMillis, centralityDistribution

Table 7. Parameters
Name	Type	Default	Optional	Description
concurrency	int	4	yes	The number of concurrent threads used for running the algorithm. Also provides the default value for 'readConcurrency' and 'writeConcurrency'.
readConcurrency	int	value of 'concurrency'	yes	The number of concurrent threads used for reading the graph.
writeConcurrency	int	value of 'concurrency'	yes	The number of concurrent threads used for writing the result.
writeProperty	string	'centrality'	yes	The property name written back to.

Table 8. Results
Name	Type	Description
nodes	int	The number of nodes considered.
preProcessingMillis	int	Milliseconds for preprocessing the data.
computeMillis	int	Milliseconds for running the algorithm.
writeMillis	int	Milliseconds for writing result data back.
writeProperty	string	The property name written back to.
centralityDistribution	Map	Map containing min, max, mean as well as p50, p75, p90, p95, p99 and p999 percentile values of centrality values.

The following will run the algorithm and stream results:

CALL gds.alpha.closeness.harmonic.stream(configuration: Map)
YIELD nodeId, centrality

Table 9. Parameters
Name	Type	Default	Optional	Description
concurrency	int	4	yes	The number of concurrent threads used for running the algorithm. Also provides the default value for 'readConcurrency' and 'writeConcurrency'.
readConcurrency	int	value of 'concurrency'	yes	The number of concurrent threads used for reading the graph.

Table 10. Results
Name	Type	Description
node	long	Node ID
centrality	float	Harmonic centrality score

Harmonic Centrality algorithm example

The following will create a sample graph:

CREATE (a:User {name: "Alice"}),
       (b:User {name: "Bob"}),
       (c:User {name: "Charles"}),
       (d:User {name: "Doug"}),
       (e:User {name: "Ethan"}),
       (a)-[:LINK]->(b),
       (b)-[:LINK]->(c),
       (d)-[:LINK]->(e)

The following will project and store a named graph:

MATCH (source:User)-[r:LINK]->(target:User)
RETURN gds.graph.project(
  'graph',
  source,
  target,
  {},
  { undirectedRelationshipTypes: ['*'] }
)

The following will run the algorithm and stream results:

CALL gds.closeness.harmonic.stream('graph', {})
YIELD nodeId, score
RETURN gds.util.asNode(nodeId).name AS user, score
ORDER BY score DESC

Table 11. Results
user	score
"Bob"	0.5
"Alice"	0.375
"Charles"	0.375
"Doug"	0.25
"Ethan"	0.25

The following will run the algorithm and write back results:

CALL gds.closeness.harmonic.write('graph', {writeProperty: 'score'})
YIELD nodePropertiesWritten

Table 12. Results
nodePropertiesWritten
5