# 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);

    template<typename Graph, typename CentralityMap, typename EdgeCentralityMap>
    void
    brandes_betweenness_centrality(const Graph& g, CentralityMap centrality,
                                   EdgeCentralityMap edge_centrality_map);

    // non-named parameter versions
    template<typename Graph, typename CentralityMap, typename EdgeCentralityMap,
             typename IncomingMap, typename DistanceMap, typename DependencyMap,
             typename PathCountMap, typename VertexIndexMap, typename Buffer>
    void
    brandes_betweenness_centrality(const Graph& g,
                                   CentralityMap centrality,
                                   EdgeCentralityMap edge_centrality_map,
                                   IncomingMap incoming,
                                   DistanceMap distance,
                                   DependencyMap dependency,
                                   PathCountMap path_count,
                                   VertexIndexMap vertex_index,
                                   Buffer sources,
                                   typename property_traits<DistanceMap>::value_type delta);

    template<typename Graph, typename CentralityMap, typename EdgeCentralityMap,
             typename IncomingMap, typename DistanceMap, typename DependencyMap,
             typename PathCountMap, typename VertexIndexMap, typename WeightMap,
             typename Buffer>
    void
    brandes_betweenness_centrality(const Graph& g,
                                   CentralityMap centrality,
                                   EdgeCentralityMap edge_centrality_map,
                                   IncomingMap incoming,
                                   DistanceMap distance,
                                   DependencyMap dependency,
                                   PathCountMap path_count,
                                   VertexIndexMap vertex_index,
                                   Buffer sources,
                                   typename property_traits<WeightMap>::value_type delta,
                                   WeightMap weight_map);

    // helper functions
    template<typename Graph, typename CentralityMap>
    typename property_traits<CentralityMap>::value_type
    central_point_dominance(const Graph& g, CentralityMap centrality);

The `brandes_betweenness_centrality()` function computes the betweenness centrality of the vertices and edges in a graph. The method of calculating betweenness centrality in O(V) space is due to Brandes [Brandes01]. The algorithm itself is a modification of Brandes algorithm by Edmonds [Edmonds09].

Contents

- Where Defined
- Parameters
- Complexity
- Algorithm Description
- Bibliography

# Where Defined

<`boost/graph/distributed/betweenness_centrality.hpp`>

also accessible from

<`boost/graph/betweenness_centrality.hpp`>

# Parameters

IN: `const Graph& g`
  The graph type must be a model of Distributed Graph. The graph type must also model the Incidence Graph concept. 0-weighted edges in `g` will result in undefined behavior.
IN: `CentralityMap centrality`

  A centrality map may be supplied to the algorithm, if not supplied a `dummy_property_map` will be used and no vertex centrality information will be recorded. The `CentralityMap` type must be a Distributed Property Map. The key type must be the graph's vertex descriptor type.

  Default: A `dummy_property_map`.

IN: `EdgeCentralityMap edge_centrality_map`

  An edge centrality map may be supplied to the algorithm, if not supplied a `dummy_property_map` will be used and no edge centrality information will be recorded. The `EdgeCentralityMap` type must be a Distributed Property Map. The key type must be the graph's vertex descriptor type.

  Default: A `dummy_property_map`.

IN: `IncomingMap incoming`

  The incoming map contains the incoming edges to a vertex that are part of shortest paths to that vertex. The `IncomingMap` type must be a Distributed Property Map. Its key type and value type must both be the graph's vertex descriptor type.

  Default: An `iterator_property_map` created from a
    `std::vector` of `std::vector` of the graph's vertex descriptor type.
IN: `DistanceMap distance`

  The distance map records the distance to vertices during the shortest paths portion of the algorithm. The `DistanceMap` type must be a Distributed Property Map. Its key type must be the graph's vertex descriptor type.

  Default: An `iterator_property_map` created from a
    `std::vector` of the value type of the `CentralityMap`.
IN: `DependencyMap dependency`

  The dependency map records the dependency of each vertex during the centrality calculation portion of the algorithm. The `DependencyMap` type must be a Distributed Property Map. Its key type must be the graph's vertex descriptor type.

  Default: An `iterator_property_map` created from a
    `std::vector` of the value type of the `CentralityMap`.

IN: `PathCountMap path_count`

> The path count map records the number of shortest paths each vertex is on during the centrality calculation portion of the algorithm. The `PathCountMap` type must be a Distributed Property Map. Its key type must be the graph's vertex descriptor type.

> Default: An `iterator_property_map` created from a
>   `std::vector` of the graph's degree size type.

IN: `VertexIndexMap vertex_index`

  A model of Readable Property Map whose key type is the vertex descriptor type of the graph `g` and whose value type is an integral type. The property map should map from vertices to their (local) indices in the range [0, num_vertices(g)).

  Default: `get(vertex_index, g)`

IN: `WeightMap weight_map`
  A model of Readable Property Map whose key type is the edge descriptor type of the graph `g`. If not supplied the betweenness centrality calculation will be unweighted.
IN: `Buffer sources`

  A model of Buffer containing the starting vertices for the algorithm. If `sources` is empty a complete betweenness centrality calculation using all vertices in `g` will be performed. The value type of the Buffer must be the graph's vertex descriptor type.

  Default: An empty `boost::queue` of int.

# Complexity

Computing the shortest paths, counting them, and computing the contribution to the centrality map is O(V log V). Calculating exact betweenness centrality requires counting the shortest paths from all vertices in `g`, thus exact betweenness centrality is O(V^2 log V).

# Algorithm Description

For the vertices in `sources` (or all vertices in `g` when `sources` is empty) the algorithm first calls a customized implementation of delta_stepping_shortest_paths which maintains a shortest path tree using an `IncomingMap`. The `IncomingMap` contains the source of all incoming edges on shortest paths.

The `IncomingMap` defines the shortest path DAG at the target of the edges in the shortest paths tree. In the bidirectional case edge flags could be used to translate the shortest paths information to the source of the edges. Setting edge flags during the shortest path computation rather than using an `IncomingMap` would result in adding an O(V) factor to the inner loop of the shortest paths computation to account for having to clear edge flags when a new shortest path is found. This would increase the complexity of the algorithm. Asymptotically, the current implementation is better, however using edge flags in the bidirectional case would reduce the number of supersteps required by the depth of the shortest paths DAG for each vertex. Currently an `outgoing` map is explicitly constructed by simply reversing the edges in the incoming map. Once the `outgoing` map is constructed it is traversed in dependency order from the source of the shortest paths calculation in order to compute path counts. Once path counts are computed the shortest paths DAG is again traversed in dependency order from the source to calculate the dependency and centrality of each vertex.

The algorithm is complete when the centrality has been computed from all vertices in `g`.

# Bibliography

[Brandes01] | Ulrik Brandes. A Faster Algorithm for Betweenness Centrality. In the Journal of Mathematical Sociology, volume 25 number 2, pages 163--177, 2001.

[Edmonds09] | Nick Edmonds, Torsten Hoefler, and Andrew Lumsdaine. A Space-Efficient Parallel Algorithm for Computing Betweenness Centrality in Sparse Networks. Indiana University tech report. 2009.

Copyright (C) 2009 The Trustees of Indiana University.

Authors: Nick Edmonds and Andrew Lumsdaine