central_point_dominance

graph_tool.centrality.central_point_dominance(g, betweenness)

Calculate the central point dominance of the graph, given the betweenness centrality of each vertex.

Parameters:

gGraph

Graph to be used.

betweennessVertexPropertyMap

Vertex property map with the betweenness centrality values. The values must be normalized.

Returns:

cpfloat

The central point dominance.

See also

betweenness

betweenness centrality

Notes

Let \(v^*\) be the vertex with the largest relative betweenness centrality; then, the central point dominance [freeman-set-1977] is defined as:

\[C'_B = \frac{1}{|V|-1} \sum_{v} C_B(v^*) - C_B(v)\]

where \(C_B(v)\) is the normalized betweenness centrality of vertex v. The value of \(C_B\) lies in the range [0,1].

The algorithm has a complexity of \(O(V)\).

References

[freeman-set-1977]

Linton C. Freeman, “A Set of Measures of Centrality Based on Betweenness”, Sociometry, Vol. 40, No. 1, pp. 35-41, 1977, DOI: 10.2307/3033543 [sci-hub, @tor]

Examples

>>> g = gt.collection.data["polblogs"]
>>> g = gt.GraphView(g, vfilt=gt.label_largest_component(g))
>>> vp, ep = gt.betweenness(g)
>>> print(gt.central_point_dominance(g, vp))
0.218286...