Closeness Centrality
This section describes the Closeness Centrality algorithm in the Neo4j Graph Data Science library.
Closeness centrality is a way of detecting nodes that are able to spread information very efficiently through a graph.
The closeness centrality of a node measures its average farness (inverse distance) to all other nodes. Nodes with a high closeness score have the shortest distances to all other nodes.
This algorithm is in the alpha tier. For more information on algorithm tiers, see Algorithms.
1. History and explanation
For each node, the Closeness Centrality algorithm calculates the sum of its distances to all other nodes, based on calculating the shortest paths between all pairs of nodes. The resulting sum is then inverted to determine the closeness centrality score for that node.
The raw closeness centrality of a node is calculated using the following formula:
raw closeness centrality(node) = 1 / sum(distance from node to all other nodes)
It is more common to normalize this score so that it represents the average length of the shortest paths rather than their sum. This adjustment allow comparisons of the closeness centrality of nodes of graphs of different sizes
The formula for normalized closeness centrality is as follows:
normalized closeness centrality(node) = (number of nodes  1) / sum(distance from node to all other nodes)
2. Usecases  when to use the Closeness Centrality algorithm

Closeness centrality is used to research organizational networks, where individuals with high closeness centrality are in a favourable position to control and acquire vital information and resources within the organization. One such study is "Mapping Networks of Terrorist Cells" by Valdis E. Krebs.

Closeness centrality can be interpreted as an estimated time of arrival of information flowing through telecommunications or package delivery networks where information flows through shortest paths to a predefined target. It can also be used in networks where information spreads through all shortest paths simultaneously, such as infection spreading through a social network. Find more details in "Centrality and network flow" by Stephen P. Borgatti.

Closeness centrality has been used to estimate the importance of words in a document, based on a graphbased keyphrase extraction process. This process is described by Florian Boudin in "A Comparison of Centrality Measures for GraphBased Keyphrase Extraction".
3. Constraints  when not to use the Closeness Centrality algorithm

Academically, closeness centrality works best on connected graphs. If we use the original formula on an unconnected graph, we can end up with an infinite distance between two nodes in separate connected components. This means that we’ll end up with an infinite closeness centrality score when we sum up all the distances from that node.
In practice, a variation on the original formula is used so that we don’t run into these issues.
4. Syntax
CALL gds.alpha.closeness.write(configuration: Map)
YIELD nodes, createMillis, computeMillis, writeMillis, centralityDistribution
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. 
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. 
CALL gds.alpha.closeness.stream(configuration: Map)
YIELD nodeId, centrality
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. 
Name  Type  Description 

node 
long 
Node ID 
centrality 
float 
Closeness centrality score 
5. Closeness Centrality algorithm sample
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]>(a),
(b)[:LINK]>(c),
(c)[:LINK]>(b),
(c)[:LINK]>(d),
(d)[:LINK]>(c),
(d)[:LINK]>(e),
(e)[:LINK]>(d);
CALL gds.alpha.closeness.stream({
nodeProjection: 'Node',
relationshipProjection: 'LINK'
})
YIELD nodeId, centrality
RETURN gds.util.asNode(nodeId).name AS user, centrality
ORDER BY centrality DESC
Name  Centrality weight 

C 
0.6666666666666666 
B 
0.5714285714285714 
D 
0.5714285714285714 
A 
0.4 
E 
0.4 
C is the best connected node in this graph, although B and D aren’t far behind. A and E don’t have close ties to many other nodes, so their scores are lower. Any node that has a direct connection to all other nodes would score 1.
CALL gds.alpha.closeness.write({
nodeProjection: 'Node',
relationshipProjection: 'LINK',
writeProperty: 'centrality'
}) YIELD nodes, writeProperty
nodes  writeProperty 

5 
"centrality" 
6. Cypher projection
CALL gds.alpha.closeness.write({
nodeQuery: 'MATCH (p:Node) RETURN id(p) AS id',
relationshipQuery: 'MATCH (p1:Node)[:LINK]>(p2:Node) RETURN id(p1) AS source, id(p2) AS target'
}) YIELD nodes, writeProperty
nodes  writeProperty 

5 
"centrality" 
Calculation:

count farness in each msbfscallback

divide by N1
N = 5
// number of nodes
k = N1 = 4
// used for normalization
A B C D E  A  0 1 2 3 4 // farness between each pair of nodes B  1 0 1 2 3 C  2 1 0 1 2 D  3 2 1 0 1 E  4 3 2 1 0  S  10 7 6 7 10 // raw closeness centrality =============================== k/S 0.4 0.57 0.67 0.57 0.4 // normalized closeness centrality
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