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.

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 u is calculated using the following formula:

Figure 1. Raw harmonic formula for node u

That is, for every node j (excluding u) we compute its minimum distance from u and sum up its inverse. If there is no path from u to j, 1/d(u,v) is treated as 0.

Similar to Closeness centrality, we can also calculate a normalized harmonic centrality value for node with the following formula:

Figure 2. Normalized harmonic formula for node u

Here, we divide the raw value by the number of nodes minus one to normalize the returned value. In this formula, ∞ values are handled cleanly. The Neo4j GDS Library calculates normalized harmonic centrality values.

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 stream results:

CALL gds.closeness.harmonic.stream(
  graphName: String,
  configuration: Map
)
YIELD
  nodeId: Integer,
  score: 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. 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 ^[1]	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.
1. In a GDS Session the default is the number of available processors

Table 3. Results
Name	Type	Description
nodeId	Integer	Node ID.
score	Float	Harmonic centrality score.

The following will run the algorithm and return statistics:

CALL gds.closeness.harmonic.stats(
  graphName: String,
  configuration: Map
)
YIELD
  centralityDistribution: Map,
  preProcessingMillis: Integer,
  computeMillis: Integer,
  postProcessingMillis: Integer,
  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
concurrency	Integer	4 ^[2]	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.
2. In a GDS Session the default is the number of available processors

Table 6. 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.
configuration	Map	The configuration used for running the algorithm.

The following will run the algorithm and update the graph:

CALL gds.closeness.harmonic.mutate(
  graphName: String,
  configuration: Map
)
YIELD
  centralityDistribution: Map,
  preProcessingMillis: Integer,
  computeMillis: Integer,
  mutateMillis: Integer,
  nodePropertiesWritten: Integer,
  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
concurrency	Integer	4 ^[3]	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.
mutateProperty	String	n/a	no	The node property in the GDS graph to which the score 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.
3. In a GDS Session the default is the number of available processors

Table 9. 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.
mutateMillis	Integer	Milliseconds for adding properties to the projected graph.
nodePropertiesWritten	Integer	Number of properties written to the projected graph.
configuration	Map	The configuration used for running the algorithm.

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 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
concurrency	Integer	4 ^[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.
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 ^[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 score is written.
4. In a GDS Session the default is the number of available processors

Table 12. 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.

Examples

All the examples below should be run in an empty database.

The examples use Cypher projections as the norm. Native projections will be deprecated in a future release.

In this section we will show examples of running the Harmonic Centrality 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 (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 statement will project a graph using a Cypher projection and store it in the graph catalog under the name 'graph'.

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

In the following examples we will demonstrate using the Harmonic Centrality 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 stream 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 stream mode:

CALL gds.closeness.harmonic.stream.estimate('graph', {})
YIELD nodeCount, relationshipCount, bytesMin, bytesMax, requiredMemory

Table 13. Results
nodeCount	relationshipCount	bytesMin	bytesMax	requiredMemory
5	6	1368	1368	"1368 Bytes"

Stream

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

Stats

The following will run the algorithm and return statistics:

CALL gds.closeness.harmonic.stats('graph', {})
YIELD centralityDistribution

Table 15. Results
centralityDistribution
{max=0.5000038147, mean=0.3500003815, min=0.25, p50=0.375, p75=0.375, p90=0.5000019073, p95=0.5000019073, p99=0.5000019073, p999=0.5000019073}

Mutate

The following will run the algorithm in mutate mode and update the in-memory graph:

CALL gds.closeness.harmonic.mutate('graph', {mutateProperty: 'harmonicScore'})
YIELD nodePropertiesWritten, centralityDistribution

Table 16. Results
nodePropertiesWritten	centralityDistribution
5	{max=0.5000038147, mean=0.3500003815, min=0.25, p50=0.375, p75=0.375, p90=0.5000019073, p95=0.5000019073, p99=0.5000019073, p999=0.5000019073}

Write

The following will run the algorithm and write the results back to the Neo4j database:

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

Table 17. Results
nodePropertiesWritten
5