...one of the most highly
regarded and expertly designed C++ library projects in the
world.

— Herb Sutter and Andrei
Alexandrescu, C++
Coding Standards

// named parameter versionstemplate<typename Graph, typename Param, typename Tag, typename Rest> void brandes_betweenness_centrality(const Graph& g, const bgl_named_params<Param,Tag,Rest>& params); template<typename Graph, typename CentralityMap> void brandes_betweenness_centrality(const Graph& g, CentralityMap centrality_map); template<typename Graph, typename CentralityMap, typename EdgeCentralityMap> void brandes_betweenness_centrality(const Graph& g, CentralityMap centrality_map, EdgeCentralityMap edge_centrality);// non-named parameter versionstemplate<typename Graph, typename CentralityMap, typename EdgeCentralityMap, typename IncomingMap, typename DistanceMap, typename DependencyMap, typename PathCountMap, typename VertexIndexMap> void brandes_betweenness_centrality(const Graph& g, CentralityMap centrality_map, EdgeCentralityMap edge_centrality, IncomingMap incoming, DistanceMap distance, DependencyMap dependency, PathCountMap path_count, VertexIndexMap vertex_index); template<typename Graph, typename CentralityMap, typename EdgeCentralityMap, typename IncomingMap, typename DistanceMap, typename DependencyMap, typename PathCountMap, typename VertexIndexMap, typename WeightMap> void brandes_betweenness_centrality(const Graph& g, CentralityMap centrality_map, EdgeCentralityMap edge_centrality, IncomingMap incoming, DistanceMap distance, DependencyMap dependency, PathCountMap path_count, VertexIndexMap vertex_index, WeightMap weight_map);// helper functionstemplate<typename Graph, typename CentralityMap> void relative_betweenness_centrality(const Graph& g, CentralityMap centrality_map); template<typename Graph, typename CentralityMap> typename property_traits<CentralityMap>::value_type central_point_dominance(const Graph& g, CentralityMap centrality_map);

This algorithm [54]
computes the *betweenness centrality* [55,56] of each vertex or each
edge in the graph. The betweenness centrality of a vertex *v*
is defined by

,

where is the number of shortest paths from
vertex *s* to vertex *t* and
is the number of shortest paths from vertex *s* to vertex
*t* that pass through vertex *v*.

The edge betweenness centrality indicates for each edge the betweenness centrality that was contributed to the target(s) of the edge (plural for undirected graphs). Similar to (vertex) betweenness centrality, edge betweenness centrality can be used to determine the edges through which most shortest paths must pass. A single invocation of this algorithm can compute either the vertex or edge centrality (or both).

This algorithm can operate either on weighted graphs (if a suitable
edge weight map is supplied) or unweighted graphs (if no edge weight
map is supplied). The result is the absolute betweenness centrality;
to convert to the relative betweenness centrality, which scales each
absolute centrality by
(where *n* is the number of vertices in the graph), use
`relative_betweenness_centrality`. Given the relative
betweenness centrality, one can compute the *central point
dominance* [55], which is a measure of the maximum "betweenness" of any
point in the graph: it will be 0 for complete graphs and
1 for "wheel" graphs (in which there is a central vertex that all
paths include; see Fig. 1). Let be the vertex with the largest relative betweenness centrality; then, the central point dominance is defined as:

Fig. 1: A wheel graph, where every path travels through the central node. The central point dominance of this graph is 1. |
---|

The graph object on which the algorithm will be applied. The typeUTIL:Graphmust be a model of Vertex List Graph and Incidence Graph. When an edge centrality map is supplied, it must also model Edge List Graph.

Python: The parameter is namedgraph.

This property map records the set of edges incoming to each vertex that comprise a shortest path from a particular source vertex through this vertex, and is used internally by the algorithm.TheUTIL:IncomingMaptype must be a Lvalue Property Map whose key type is the same as the vertex descriptor type of the graph and whose value type is a Sequence (e.g., anstd::vector) containing edge descriptors.

Default:iterator_property_mapcreated from astd::vectorofstd::vector<Edge>, whereEdgeis the edge descriptor type of the graph.

Python: Unsupported parameter.

The shortest path weight from each source vertexUTIL:sto each vertex in the graphgis recorded in this property map, but the result is only used internally. The typeDistanceMapmust be a model of Read/Write Property Map. The vertex descriptor type of the graph needs to be usable as the key type of the distance map. The value type of the distance map is the element type of a Monoid.

Default:iterator_property_mapcreated from astd::vectorof theWeightMap's value type (or thevertices_size_typeof the graph when no weight map exists) of sizenum_vertices(g)and using thevertex_indexfor the index map.

Python: Unsupported parameter.

Property map used internally to accumulate partial betweenness centrality results. The typeUTIL:DependencyMapmust be a model of Read/Write Property Map. The vertex descriptor type of the graph needs to be usable as the key type of the dependency map. The value type of the dependency map must be compatible with the value type of the centrality map.

Default:iterator_property_mapcreated from astd::vectorof theCentralityMap's value type of sizenum_vertices(g)and using thevertex_indexfor the index map.

Python: Unsupported parameter.

Property map used internally to accumulate the number of paths that pass through each particular vertex. The typePathCountMapmust be a model of Read/Write Property Map. The vertex descriptor type of the graph needs to be usable as the key type of the dependency map. The value type of the dependency map must be an integral type large enough to store the number of paths in the graph.

Default:iterator_property_mapcreated from astd::vectorof thedegree_size_typeof the graph of sizenum_vertices(g)and using thevertex_indexfor the index map.

Python: Unsupported parameter.

This property map is used to accumulate the betweenness centrality of each vertex, and is the primary output of the algorithm. The typeOUT/UTIL:CentralityMapmust be a model of Read/Write Property Map, with the graph's vertex descriptor type as its key type. The value type of this property map should be a floating-point or rational type.

Default:adummy_property_map, which requires no work to compute and returns no answer.

Python: The color map must be avertex_double_mapfor the graph.

Python default:graph.get_vertex_double_map("centrality")

This property map is used to accumulate the betweenness centrality of each edge, and is a secondary form of output for the algorithm. The typeIN:EdgeCentralityMapmust be a model of Read/Write Property Map, with the graph's edge descriptor type as its key type. The value type of this property map should be the same as the value type of theCentralityMapproperty map.

Default:adummy_property_map, which requires no work to compute and returns no answer.

Python: The color map must be aedge_double_mapfor the graph.

Python default:graph.get_edge_double_map("centrality")

This maps each vertex to an integer in the rangeIN:[0, num_vertices(g)). This is necessary for efficient updates of the heap data structure when an edge is relaxed. The typeVertexIndexMapmust be a model of Readable Property Map. The value type of the map must be an integer type. The vertex descriptor type of the graph needs to be usable as the key type of the map.

Default:get(vertex_index, g). Note: if you use this default, make sure your graph has an internalvertex_indexproperty. For example,adjacency_listwithVertexList=listSdoes not have an internalvertex_indexproperty.

Python: Unsupported parameter.

The weight or ``length'' of each edge in the graph. The weights must all be non-negative, and the algorithm will throw anegative_edgeexception is one of the edges is negative. The typeWeightMapmust be a model of Readable Property Map. The edge descriptor type of the graph needs to be usable as the key type for the weight map. The value type for this map must be the same as the value type of the distance map.

Default:All edge weights are assumed to be equivalent.Python: If supplied, must be anedge_double_mapfor the graph.

Copyright © 2004 |
Douglas Gregor, Indiana University (dgregor@cs.indiana.edu) Andrew Lumsdaine, Indiana University (lums@osl.iu.edu) |