Harmonic Centrality

This section describes the Harmonic Centrality algorithm

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 algorithm is in the alpha tier. For more information on algorithm tiers, see Algorithms.

1. 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.

2. 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.

3. Syntax

The following will run the algorithm and write back results:
CALL gds.alpha.closeness.harmonic.write(configuration: Map)
YIELD nodes, createMillis, computeMillis, writeMillis, centralityDistribution
Table 1. 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 2. Results
Name Type Description

nodes

int

The number of nodes considered.

createMillis

int

Milliseconds for loading 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 3. 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 4. Results
Name Type Description

node

long

Node ID

centrality

float

Harmonic centrality score

4. Harmonic Centrality algorithm sample

The following will create a sample graph:
CREATE (a:Node{id:"A"}),
       (b:Node{id:"B"}),
       (c:Node{id:"C"}),
       (d:Node{id:"D"}),
       (e:Node{id:"E"}),
       (a)-[:LINK]->(b),
       (b)-[:LINK]->(c),
       (d)-[:LINK]->(e)
The following will run the algorithm and stream results:
CALL gds.alpha.closeness.harmonic.stream({
  nodeProjection: 'Node',
  relationshipProjection: 'LINK'
})
YIELD nodeId, centrality
RETURN gds.util.asNode(nodeId).name AS user, centrality
ORDER BY centrality DESC
Table 5. Results
Name Centrality weight

B

0.5

A

0.375

c

0.375

D

0.25

E

0.25

The following will run the algorithm and write back results:
CALL gds.alpha.closeness.harmonic.write({
  nodeProjection: 'Node',
  relationshipProjection: 'LINK',
  writeProperty: 'centrality'
}) YIELD nodes, writeProperty
Table 6. Results
nodes writeProperty

5

"centrality"