Boost Graph Library: Brandes' Betweenness Centrality

# `brandes_betweenness_centrality`

    // named parameter versions
    template<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 versions
    template<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 functions
    template<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.

### Where Defined

`boost/graph/betweenness_centrality.hpp`

### Parameters

IN: `const Graph& g`

> The graph object on which the algorithm will be applied. The type `Graph` must 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 named `graph`.

UTIL: `IncomingMap incoming`

> 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.The `IncomingMap` type 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., an `std::vector`) containing edge descriptors.
> Default: `iterator_property_map` created from a `std::vector` of `std::vector<Edge>`, where `Edge` is the edge descriptor type of the graph.
> Python: Unsupported parameter.

UTIL: `DistanceMap distance_map`

> The shortest path weight from each source vertex `s` to each vertex in the graph `g` is recorded in this property map, but the result is only used internally. The type `DistanceMap` must 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_map` created from a `std::vector` of the `WeightMap`'s value type (or the `vertices_size_type` of the graph when no weight map exists) of size `num_vertices(g)` and using the `vertex_index` for the index map.
> Python: Unsupported parameter.

UTIL: `DependencyMap dependency`

> Property map used internally to accumulate partial betweenness centrality results. The type `DependencyMap` must 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_map` created from a `std::vector` of the `CentralityMap`'s value type of size `num_vertices(g)` and using the `vertex_index` for the index map.
> Python: Unsupported parameter.

UTIL: `PathCountMap path_count`

> Property map used internally to accumulate the number of paths that pass through each particular vertex. The type `PathCountMap` must 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_map` created from a `std::vector` of the `degree_size_type` of the graph of size `num_vertices(g)` and using the `vertex_index` for the index map.
> Python: Unsupported parameter.

### Named parameters

OUT/UTIL: `CentralityMap centrality_map`

> This property map is used to accumulate the betweenness centrality of each vertex, and is the primary output of the algorithm. The type `CentralityMap` must 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: a `dummy_property_map`, which requires no work to compute and returns no answer.
> Python: The color map must be a `vertex_double_map` for the graph.
> Python default: `graph.get_vertex_double_map("centrality")`

OUT/UTIL: `EdgeCentralityMap edge_centrality_map`

> This property map is used to accumulate the betweenness centrality of each edge, and is a secondary form of output for the algorithm. The type `EdgeCentralityMap` must 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 the `CentralityMap` property map.
> Default: a `dummy_property_map`, which requires no work to compute and returns no answer.
> Python: The color map must be a `edge_double_map` for the graph.
> Python default: `graph.get_edge_double_map("centrality")`

IN: `vertex_index_map(VertexIndexMap vertex_index)`

> This maps each vertex to an integer in the range `[0, num_vertices(g))`. This is necessary for efficient updates of the heap data structure when an edge is relaxed. The type `VertexIndexMap` must 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 internal `vertex_index` property. For example, `adjacency_list` with `VertexList=listS` does not have an internal `vertex_index` property.
> Python: Unsupported parameter.

IN: `weight_map(WeightMap w_map)`

> The weight or ``length'' of each edge in the graph. The weights must all be non-negative, and the algorithm will throw a `negative_edge` exception is one of the edges is negative. The type `WeightMap` must 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 an `edge_double_map` for the graph.

### Complexity

The time complexity is O(VE) for unweighted graphs and O(VE + V(V+E) log V) for weighted graphs. The space complexity is O(VE).

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

Last modified: Tue Mar 1 14:14:51 EST 2005