This section describes the Closeness Centrality algorithm in the Neo4j Graph Algorithms 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.
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)
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.
The following will create a sample graph:
MERGE (a:Node{id:"A"})
MERGE (b:Node{id:"B"})
MERGE (c:Node{id:"C"})
MERGE (d:Node{id:"D"})
MERGE (e:Node{id:"E"})
MERGE (a)-[:LINK]->(b)
MERGE (b)-[:LINK]->(a)
MERGE (b)-[:LINK]->(c)
MERGE (c)-[:LINK]->(b)
MERGE (c)-[:LINK]->(d)
MERGE (d)-[:LINK]->(c)
MERGE (d)-[:LINK]->(e)
MERGE (e)-[:LINK]->(d);
The following will run the algorithm and stream results:
CALL algo.closeness.stream('Node', 'LINK')
YIELD nodeId, centrality
RETURN algo.getNodeById(nodeId).id AS node, centrality
ORDER BY centrality DESC
LIMIT 20;
The following will run the algorithm and write back results:
CALL algo.closeness('Node', 'LINK', {write:true, writeProperty:'centrality'})
YIELD nodes,loadMillis, computeMillis, writeMillis;
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.
Calculation:
N = 5
// number of nodes
k = N-1 = 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
The following will run the algorithm and write back results:
CALL algo.closeness(label:String, relationship:String,
{write:true, writeProperty:'centrality',graph:'heavy', concurrency:4})
YIELD nodes, loadMillis, computeMillis, writeMillis
Name | Type | Default | Optional | Description |
---|---|---|---|---|
label |
string |
null |
yes |
The label to load from the graph. If null, load all nodes |
relationship |
string |
null |
yes |
The relationship-type to load from the graph. If null, load all relationships |
write |
boolean |
true |
yes |
Specifies if the result should be written back as a node property |
concurrency |
int |
available CPUs |
yes |
The number of concurrent threads |
writeProperty |
string |
'centrality' |
yes |
The property name written back to |
graph |
string |
'heavy' |
yes |
Use 'heavy' when describing the subset of the graph with label and relationship-type parameter,. Use 'cypher' for describing the subset with cypher node-statement and relationship-statement |
Name | Type | Description |
---|---|---|
nodes |
int |
The number of nodes considered |
loadMillis |
int |
Milliseconds for loading data |
evalMillis |
int |
Milliseconds for running the algorithm |
writeMillis |
int |
Milliseconds for writing result data back |
The following will run the algorithm and stream results:
CALL algo.closeness.stream(label:String, relationship:String, {concurrency:4})
YIELD nodeId, centrality
Name | Type | Default | Optional | Description |
---|---|---|---|---|
label |
string |
null |
yes |
The label to load from the graph. If null, load all nodes |
relationship |
string |
null |
yes |
The relationship-type to load from the graph. If null, load all relationships |
concurrency |
int |
available CPUs |
yes |
The number of concurrent threads |
graph |
string |
'heavy' |
yes |
Use 'heavy' when describing the subset of the graph with label and relationship-type parameter. Use 'cypher' for describing the subset with cypher node-statement and relationship-statement |
Name | Type | Description |
---|---|---|
node |
long |
Node ID |
centrality |
float |
Closeness centrality weight |
If our projected graph contains more than 2 billion nodes or relationships, we need to use huge graph projection, as the default label and relationship-type projection has a limitation of 2 billion nodes and 2 billion relationships.
Set graph:'huge'
in the config:
CALL algo.closeness('Node', 'LINK', {graph:'huge'})
YIELD nodes,loadMillis, computeMillis, writeMillis;
If label and relationship-type are not selective enough to describe your subgraph to run the algorithm on, you can use Cypher statements to load or project subsets of your graph. This can also be used to run algorithms on a virtual graph.
Set graph:'cypher'
in the config:
CALL algo.closeness(
'MATCH (p:Node) RETURN id(p) as id',
'MATCH (p1:Node)-[:LINK]->(p2:Node) RETURN id(p1) as source, id(p2) as target',
{graph:'cypher', write: true}
);
The Closeness Centrality algorithm supports the following graph types:
✓ undirected, unweighted