Graph Centrality

Overview

Graph centrality of a node is measured by the maximum shortest distance from the node to all other reachable nodes. This measurement, along with other measurements like closeness centrality and graph diameter, can be considered jointly to determine whether a node is literally located at the very center of the graph.

Graph centrality takes on values between 0 to 1, nodes with higher scores are closer to the center.

Concepts

Shortest Distance

The shortest distance of two nodes is the number of edges contained in the shortest path between them. Please refer to Closeness Centrality for more details.

Graph Centrality

Graph centrality score of a node defined by this algorithm is the inverse of the maximum shortest distance from the node to all other reachable nodes. The formula is:

where x is the target node, y is any node that connects with x along edges (x itself is excluded), d(x,y) is the shortest distance between x and y.

In this graph, the green number and red number next to each node is the shortest distance between the node and the green node and red node. Graph centrality scores of the green and red nodes are 1/4 = 0.25 and 1/3 = 0.3333 respectively.

Regarding closeness centrality, the green node has score 8/(1+1+1+1+2+3+4+3) = 0.5, the red node has score 8/(3+3+3+2+1+1+2+1) = 0.5. When two nodes have the same closeness centrality score, graph centrality can be viewed as the subsidiary basis to determine which node is closer to the center.

NOTE

Graph Centrality algorithm consumes considerable computing resources. For a graph with V nodes, it is recommended to perform (uniform) sampling when V > 10,000, and the suggested number of samples is the base-10 logarithm of the number of nodes (log(V)).

For each execution of the algorithm, sampling is performed only once, centrality score of each node is computed based on the shortest distance between the node and all sample nodes.

Considerations

• The graph centrality score of isolated nodes is 0.
• The Graph Centrality algorithm ignores the direction of edges but calculates them as undirected edges.

Syntax

• Command: algo(graph_centrality)
• Parameters:

Examples

The example graph is as follows:

File Writeback

UQL
algo(graph_centrality).params().write({
  file:{ 
    filename: 'res'
  }
})

Results: File res

File
J,0
I,0.25
H,0.2
F,0.333333
G,0.25
D,0.2
E,0.333333
C,0.2
A,0.25
B,0.2

Property Writeback

UQL
algo(graph_centrality).params().write({
  db:{ 
    property: 'gc'
  }
})

Results: Centrality score for each node is written to a new property named gc

Direct Return

UQL
algo(graph_centrality).params({
  ids: ['A', 'B', 'C'],
  order: 'asc'
}) as gc
return gc

Results: gc

Stream Return

UQL
algo(graph_centrality).params().stream() as gc
where gc.centrality > 0.25
return gc

Results: gc