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# Source code for networkx.algorithms.centrality.betweenness_subset

    """Betweenness centrality measures for subsets of nodes."""

    import networkx as nx
    from networkx.algorithms.centrality.betweenness import (
        _add_edge_keys,
    )
    from networkx.algorithms.centrality.betweenness import (
        _single_source_dijkstra_path_basic as dijkstra,
    )
    from networkx.algorithms.centrality.betweenness import (
        _single_source_shortest_path_basic as shortest_path,
    )

    __all__ = [
        "betweenness_centrality_subset",
        "edge_betweenness_centrality_subset",
    ]

    @nx._dispatchable(edge_attrs="weight")
    def betweenness_centrality_subset(G, sources, targets, normalized=False, weight=None):
        r"""Compute betweenness centrality for a subset of nodes.

        .. math::

           c_B(v) =\sum_{s\in S, t \in T} \frac{\sigma(s, t|v)}{\sigma(s, t)}

        where $S$ is the set of sources, $T$ is the set of targets,
        $\sigma(s, t)$ is the number of shortest $(s, t)$-paths,
        and $\sigma(s, t|v)$ is the number of those paths
        passing through some  node $v$ other than $s, t$.
        If $s = t$, $\sigma(s, t) = 1$,
        and if $v \in {s, t}$, $\sigma(s, t|v) = 0$ [2]_.

        Parameters
        ----------
        G : graph
          A NetworkX graph.

        sources: list of nodes
          Nodes to use as sources for shortest paths in betweenness

        targets: list of nodes
          Nodes to use as targets for shortest paths in betweenness

        normalized : bool, optional
          If True the betweenness values are normalized by $2/((n-1)(n-2))$
          for graphs, and $1/((n-1)(n-2))$ for directed graphs where $n$
          is the number of nodes in G.

        weight : None or string, optional (default=None)
          If None, all edge weights are considered equal.
          Otherwise holds the name of the edge attribute used as weight.
          Weights are used to calculate weighted shortest paths, so they are
          interpreted as distances.

        Returns
        -------
        nodes : dictionary
           Dictionary of nodes with betweenness centrality as the value.

        See Also
        --------
        edge_betweenness_centrality
        load_centrality

        Notes
        -----
        The basic algorithm is from [1]_.

        For weighted graphs the edge weights must be greater than zero.
        Zero edge weights can produce an infinite number of equal length
        paths between pairs of nodes.

        The normalization might seem a little strange but it is
        designed to make betweenness_centrality(G) be the same as
        betweenness_centrality_subset(G,sources=G.nodes(),targets=G.nodes()).

        The total number of paths between source and target is counted
        differently for directed and undirected graphs. Directed paths
        are easy to count. Undirected paths are tricky: should a path
        from "u" to "v" count as 1 undirected path or as 2 directed paths?

        For betweenness_centrality we report the number of undirected
        paths when G is undirected.

        For betweenness_centrality_subset the reporting is different.
        If the source and target subsets are the same, then we want
        to count undirected paths. But if the source and target subsets
        differ -- for example, if sources is {0} and targets is {1},
        then we are only counting the paths in one direction. They are
        undirected paths but we are counting them in a directed way.
        To count them as undirected paths, each should count as half a path.

        References
        ----------
        .. [1] Ulrik Brandes, A Faster Algorithm for Betweenness Centrality.
           Journal of Mathematical Sociology 25(2):163-177, 2001.
           https://doi.org/10.1080/0022250X.2001.9990249
        .. [2] Ulrik Brandes: On Variants of Shortest-Path Betweenness
           Centrality and their Generic Computation.
           Social Networks 30(2):136-145, 2008.
           https://doi.org/10.1016/j.socnet.2007.11.001
        """
        b = dict.fromkeys(G, 0.0)  # b[v]=0 for v in G
        for s in sources:
            # single source shortest paths
            if weight is None:  # use BFS
                S, P, sigma, _ = shortest_path(G, s)
            else:  # use Dijkstra's algorithm
                S, P, sigma, _ = dijkstra(G, s, weight)
            b = _accumulate_subset(b, S, P, sigma, s, targets)
        b = _rescale(b, len(G), normalized=normalized, directed=G.is_directed())
        return b

    @nx._dispatchable(edge_attrs="weight")
    def edge_betweenness_centrality_subset(
        G, sources, targets, normalized=False, weight=None
    ):
        r"""Compute betweenness centrality for edges for a subset of nodes.

        .. math::

           c_B(v) =\sum_{s\in S,t \in T} \frac{\sigma(s, t|e)}{\sigma(s, t)}

        where $S$ is the set of sources, $T$ is the set of targets,
        $\sigma(s, t)$ is the number of shortest $(s, t)$-paths,
        and $\sigma(s, t|e)$ is the number of those paths
        passing through edge $e$ [2]_.

        Parameters
        ----------
        G : graph
          A networkx graph.

        sources: list of nodes
          Nodes to use as sources for shortest paths in betweenness

        targets: list of nodes
          Nodes to use as targets for shortest paths in betweenness

        normalized : bool, optional
          If True the betweenness values are normalized by `2/(n(n-1))`
          for graphs, and `1/(n(n-1))` for directed graphs where `n`
          is the number of nodes in G.

        weight : None or string, optional (default=None)
          If None, all edge weights are considered equal.
          Otherwise holds the name of the edge attribute used as weight.
          Weights are used to calculate weighted shortest paths, so they are
          interpreted as distances.

        Returns
        -------
        edges : dictionary
           Dictionary of edges with Betweenness centrality as the value.

        See Also
        --------
        betweenness_centrality
        edge_load

        Notes
        -----
        The basic algorithm is from [1]_.

        For weighted graphs the edge weights must be greater than zero.
        Zero edge weights can produce an infinite number of equal length
        paths between pairs of nodes.

        The normalization might seem a little strange but it is the same
        as in edge_betweenness_centrality() and is designed to make
        edge_betweenness_centrality(G) be the same as
        edge_betweenness_centrality_subset(G,sources=G.nodes(),targets=G.nodes()).

        References
        ----------
        .. [1] Ulrik Brandes, A Faster Algorithm for Betweenness Centrality.
           Journal of Mathematical Sociology 25(2):163-177, 2001.
           https://doi.org/10.1080/0022250X.2001.9990249
        .. [2] Ulrik Brandes: On Variants of Shortest-Path Betweenness
           Centrality and their Generic Computation.
           Social Networks 30(2):136-145, 2008.
           https://doi.org/10.1016/j.socnet.2007.11.001
        """
        b = dict.fromkeys(G, 0.0)  # b[v]=0 for v in G
        b.update(dict.fromkeys(G.edges(), 0.0))  # b[e] for e in G.edges()
        for s in sources:
            # single source shortest paths
            if weight is None:  # use BFS
                S, P, sigma, _ = shortest_path(G, s)
            else:  # use Dijkstra's algorithm
                S, P, sigma, _ = dijkstra(G, s, weight)
            b = _accumulate_edges_subset(b, S, P, sigma, s, targets)
        for n in G:  # remove nodes to only return edges
            del b[n]
        b = _rescale_e(b, len(G), normalized=normalized, directed=G.is_directed())
        if G.is_multigraph():
            b = _add_edge_keys(G, b, weight=weight)
        return b

    def _accumulate_subset(betweenness, S, P, sigma, s, targets):
        delta = dict.fromkeys(S, 0.0)
        target_set = set(targets) - {s}
        while S:
            w = S.pop()
            if w in target_set:
                coeff = (delta[w] + 1.0) / sigma[w]
            else:
                coeff = delta[w] / sigma[w]
            for v in P[w]:
                delta[v] += sigma[v] * coeff
            if w != s:
                betweenness[w] += delta[w]
        return betweenness

    def _accumulate_edges_subset(betweenness, S, P, sigma, s, targets):
        """edge_betweenness_centrality_subset helper."""
        delta = dict.fromkeys(S, 0)
        target_set = set(targets)
        while S:
            w = S.pop()
            for v in P[w]:
                if w in target_set:
                    c = (sigma[v] / sigma[w]) * (1.0 + delta[w])
                else:
                    c = delta[w] / len(P[w])
                if (v, w) not in betweenness:
                    betweenness[(w, v)] += c
                else:
                    betweenness[(v, w)] += c
                delta[v] += c
            if w != s:
                betweenness[w] += delta[w]
        return betweenness

    def _rescale(betweenness, n, normalized, directed=False):
        """betweenness_centrality_subset helper."""
        if normalized:
            if n <= 2:
                scale = None  # no normalization b=0 for all nodes
            else:
                scale = 1.0 / ((n - 1) * (n - 2))
        else:  # rescale by 2 for undirected graphs
            if not directed:
                scale = 0.5
            else:
                scale = None
        if scale is not None:
            for v in betweenness:
                betweenness[v] *= scale
        return betweenness

    def _rescale_e(betweenness, n, normalized, directed=False):
        """edge_betweenness_centrality_subset helper."""
        if normalized:
            if n <= 1:
                scale = None  # no normalization b=0 for all nodes
            else:
                scale = 1.0 / (n * (n - 1))
        else:  # rescale by 2 for undirected graphs
            if not directed:
                scale = 0.5
            else:
                scale = None
        if scale is not None:
            for v in betweenness:
                betweenness[v] *= scale
        return betweenness