path: root/networkx/algorithms/centrality/betweenness.py

"""Betweenness centrality measures."""
from heapq import heappush, heappop
from itertools import count
import warnings

from networkx.utils import py_random_state
from networkx.utils.decorators import not_implemented_for

__all__ = ["betweenness_centrality", "edge_betweenness_centrality", "edge_betweenness"]


@py_random_state(5)
@not_implemented_for("multigraph")
def betweenness_centrality(
    G, k=None, normalized=True, weight=None, endpoints=False, seed=None
):
    r"""Compute the shortest-path betweenness centrality for nodes.

    Betweenness centrality of a node $v$ is the sum of the
    fraction of all-pairs shortest paths that pass through $v$

    .. math::

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

    where $V$ is the set of nodes, $\sigma(s, t)$ 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.

    k : int, optional (default=None)
      If k is not None use k node samples to estimate betweenness.
      The value of k <= n where n is the number of nodes in the graph.
      Higher values give better approximation.

    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.

    endpoints : bool, optional
      If True include the endpoints in the shortest path counts.

    seed : integer, random_state, or None (default)
        Indicator of random number generation state.
        See :ref:`Randomness<randomness>`.
        Note that this is only used if k is not None.

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

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

    Notes
    -----
    The algorithm is from Ulrik Brandes [1]_.
    See [4]_ for the original first published version and [2]_ for details on
    algorithms for variations and related metrics.

    For approximate betweenness calculations set k=#samples to use
    k nodes ("pivots") to estimate the betweenness values. For an estimate
    of the number of pivots needed see [3]_.

    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 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
    .. [3] Ulrik Brandes and Christian Pich:
       Centrality Estimation in Large Networks.
       International Journal of Bifurcation and Chaos 17(7):2303-2318, 2007.
       https://dx.doi.org/10.1142/S0218127407018403
    .. [4] Linton C. Freeman:
       A set of measures of centrality based on betweenness.
       Sociometry 40: 35–41, 1977
       https://doi.org/10.2307/3033543
    """
    betweenness = dict.fromkeys(G, 0.0)  # b[v]=0 for v in G
    if k is None:
        nodes = G
    else:
        nodes = seed.sample(list(G.nodes()), k)
    for s in nodes:
        # single source shortest paths
        if weight is None:  # use BFS
            S, P, sigma, _ = _single_source_shortest_path_basic(G, s)
        else:  # use Dijkstra's algorithm
            S, P, sigma, _ = _single_source_dijkstra_path_basic(G, s, weight)
        # accumulation
        if endpoints:
            betweenness, delta = _accumulate_endpoints(betweenness, S, P, sigma, s)
        else:
            betweenness, delta = _accumulate_basic(betweenness, S, P, sigma, s)
    # rescaling
    betweenness = _rescale(
        betweenness,
        len(G),
        normalized=normalized,
        directed=G.is_directed(),
        k=k,
        endpoints=endpoints,
    )
    return betweenness


@py_random_state(4)
def edge_betweenness_centrality(G, k=None, normalized=True, weight=None, seed=None):
    r"""Compute betweenness centrality for edges.

    Betweenness centrality of an edge $e$ is the sum of the
    fraction of all-pairs shortest paths that pass through $e$

    .. math::

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

    where $V$ is the set of nodes, $\sigma(s, t)$ 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.

    k : int, optional (default=None)
      If k is not None use k node samples to estimate betweenness.
      The value of k <= n where n is the number of nodes in the graph.
      Higher values give better approximation.

    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.

    seed : integer, random_state, or None (default)
        Indicator of random number generation state.
        See :ref:`Randomness<randomness>`.
        Note that this is only used if k is not None.

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

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

    Notes
    -----
    The algorithm is from Ulrik Brandes [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.

    References
    ----------
    .. [1]  A Faster Algorithm for Betweenness Centrality. Ulrik Brandes,
       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
    """
    betweenness = dict.fromkeys(G, 0.0)  # b[v]=0 for v in G
    # b[e]=0 for e in G.edges()
    betweenness.update(dict.fromkeys(G.edges(), 0.0))
    if k is None:
        nodes = G
    else:
        nodes = seed.sample(G.nodes(), k)
    for s in nodes:
        # single source shortest paths
        if weight is None:  # use BFS
            S, P, sigma, _ = _single_source_shortest_path_basic(G, s)
        else:  # use Dijkstra's algorithm
            S, P, sigma, _ = _single_source_dijkstra_path_basic(G, s, weight)
        # accumulation
        betweenness = _accumulate_edges(betweenness, S, P, sigma, s)
    # rescaling
    for n in G:  # remove nodes to only return edges
        del betweenness[n]
    betweenness = _rescale_e(
        betweenness, len(G), normalized=normalized, directed=G.is_directed()
    )
    return betweenness


# obsolete name
def edge_betweenness(G, k=None, normalized=True, weight=None, seed=None):
    warnings.warn(
        "edge_betweeness is replaced by edge_betweenness_centrality", DeprecationWarning
    )
    return edge_betweenness_centrality(G, k, normalized, weight, seed)


# helpers for betweenness centrality


def _single_source_shortest_path_basic(G, s):
    S = []
    P = {}
    for v in G:
        P[v] = []
    sigma = dict.fromkeys(G, 0.0)  # sigma[v]=0 for v in G
    D = {}
    sigma[s] = 1.0
    D[s] = 0
    Q = [s]
    while Q:  # use BFS to find shortest paths
        v = Q.pop(0)
        S.append(v)
        Dv = D[v]
        sigmav = sigma[v]
        for w in G[v]:
            if w not in D:
                Q.append(w)
                D[w] = Dv + 1
            if D[w] == Dv + 1:  # this is a shortest path, count paths
                sigma[w] += sigmav
                P[w].append(v)  # predecessors
    return S, P, sigma, D


def _single_source_dijkstra_path_basic(G, s, weight):
    # modified from Eppstein
    S = []
    P = {}
    for v in G:
        P[v] = []
    sigma = dict.fromkeys(G, 0.0)  # sigma[v]=0 for v in G
    D = {}
    sigma[s] = 1.0
    push = heappush
    pop = heappop
    seen = {s: 0}
    c = count()
    Q = []  # use Q as heap with (distance,node id) tuples
    push(Q, (0, next(c), s, s))
    while Q:
        (dist, _, pred, v) = pop(Q)
        if v in D:
            continue  # already searched this node.
        sigma[v] += sigma[pred]  # count paths
        S.append(v)
        D[v] = dist
        for w, edgedata in G[v].items():
            vw_dist = dist + edgedata.get(weight, 1)
            if w not in D and (w not in seen or vw_dist < seen[w]):
                seen[w] = vw_dist
                push(Q, (vw_dist, next(c), v, w))
                sigma[w] = 0.0
                P[w] = [v]
            elif vw_dist == seen[w]:  # handle equal paths
                sigma[w] += sigma[v]
                P[w].append(v)
    return S, P, sigma, D


def _accumulate_basic(betweenness, S, P, sigma, s):
    delta = dict.fromkeys(S, 0)
    while S:
        w = S.pop()
        coeff = (1 + delta[w]) / sigma[w]
        for v in P[w]:
            delta[v] += sigma[v] * coeff
        if w != s:
            betweenness[w] += delta[w]
    return betweenness, delta


def _accumulate_endpoints(betweenness, S, P, sigma, s):
    betweenness[s] += len(S) - 1
    delta = dict.fromkeys(S, 0)
    while S:
        w = S.pop()
        coeff = (1 + delta[w]) / sigma[w]
        for v in P[w]:
            delta[v] += sigma[v] * coeff
        if w != s:
            betweenness[w] += delta[w] + 1
    return betweenness, delta


def _accumulate_edges(betweenness, S, P, sigma, s):
    delta = dict.fromkeys(S, 0)
    while S:
        w = S.pop()
        coeff = (1 + delta[w]) / sigma[w]
        for v in P[w]:
            c = sigma[v] * coeff
            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, k=None, endpoints=False):
    if normalized:
        if endpoints:
            if n < 2:
                scale = None  # no normalization
            else:
                # Scale factor should include endpoint nodes
                scale = 1 / (n * (n - 1))
        elif n <= 2:
            scale = None  # no normalization b=0 for all nodes
        else:
            scale = 1 / ((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:
        if k is not None:
            scale = scale * n / k
        for v in betweenness:
            betweenness[v] *= scale
    return betweenness


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