path: root/.venv/lib/python3.12/site-packages/networkx/algorithms/centrality/betweenness.py

"""Betweenness centrality measures."""

import math
from collections import deque
from heapq import heappop, heappush
from itertools import count

import networkx as nx
from networkx.algorithms.shortest_paths.weighted import _weight_function
from networkx.utils import py_random_state
from networkx.utils.decorators import not_implemented_for

__all__ = ["betweenness_centrality", "edge_betweenness_centrality"]


@py_random_state(5)
@nx._dispatchable(edge_attrs="weight")
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.

    This algorithm is not guaranteed to be correct if edge weights
    are floating point numbers. As a workaround you can use integer
    numbers by multiplying the relevant edge attributes by a convenient
    constant factor (eg 100) and converting to integers.

    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 == len(G):
        # This is done for performance; the result is the same regardless.
        k = None
    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, _ = _accumulate_endpoints(betweenness, S, P, sigma, s)
        else:
            betweenness, _ = _accumulate_basic(betweenness, S, P, sigma, s)
    # rescaling
    betweenness = _rescale(
        betweenness,
        len(G),
        normalized=normalized,
        directed=G.is_directed(),
        k=k,
        endpoints=endpoints,
        sampled_nodes=nodes,
    )
    return betweenness


@py_random_state(4)
@nx._dispatchable(edge_attrs="weight")
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(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
        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()
    )
    if G.is_multigraph():
        betweenness = _add_edge_keys(G, betweenness, weight=weight)
    return betweenness


# 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 = deque([s])
    while Q:  # use BFS to find shortest paths
        v = Q.popleft()
        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):
    weight = _weight_function(G, 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
    seen = {s: 0}
    c = count()
    Q = []  # use Q as heap with (distance,node id) tuples
    heappush(Q, (0, next(c), s, s))
    while Q:
        (dist, _, pred, v) = heappop(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 + weight(v, w, edgedata)
            if w not in D and (w not in seen or vw_dist < seen[w]):
                seen[w] = vw_dist
                heappush(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, k, endpoints, sampled_nodes):
    # N is used to count the number of valid (s, t) pairs where s != t that
    # could have a path pass through v. If endpoints is False, then v must
    # not be the target t, hence why we subtract by 1.
    N = n if endpoints else n - 1
    if N < 2:
        # No rescaling necessary: b=0 for all nodes
        return betweenness

    K_source = N if k is None else k

    if k is None or endpoints:
        # No sampling adjustment needed
        if normalized:
            # Divide by the number of valid (s, t) node pairs that could have
            # a path through v where s != t.
            scale = 1 / (K_source * (N - 1))
        else:
            # Scale to the full BC
            if not directed:
                # The non-normalized BC values are computed the same way for
                # directed and undirected graphs: shortest paths are computed and
                # counted for each *ordered* (s, t) pair. Undirected graphs should
                # only count valid *unordered* node pairs {s, t}; that is, (s, t)
                # and (t, s) should be counted only once. We correct for this here.
                correction = 2
            else:
                correction = 1
            scale = N / (K_source * correction)

        if scale != 1:
            for v in betweenness:
                betweenness[v] *= scale
        return betweenness

    # Sampling adjustment needed when excluding endpoints when using k. In this
    # case, we need to handle source nodes differently from non-source nodes,
    # because source nodes can't include themselves since endpoints are excluded.
    # Without this, k == n would be a special case that would violate the
    # assumption that node `v` is not one of the (s, t) node pairs.
    if normalized:
        # NaN for undefined 0/0; there is no data for source node when k=1
        scale_source = 1 / ((K_source - 1) * (N - 1)) if K_source > 1 else math.nan
        scale_nonsource = 1 / (K_source * (N - 1))
    else:
        correction = 1 if directed else 2
        scale_source = N / ((K_source - 1) * correction) if K_source > 1 else math.nan
        scale_nonsource = N / (K_source * correction)

    sampled_nodes = set(sampled_nodes)
    for v in betweenness:
        betweenness[v] *= scale_source if v in sampled_nodes else scale_nonsource
    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


@not_implemented_for("graph")
def _add_edge_keys(G, betweenness, weight=None):
    r"""Adds the corrected betweenness centrality (BC) values for multigraphs.

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

    betweenness : dictionary
        Dictionary mapping adjacent node tuples to betweenness centrality values.

    weight : string or function
        See `_weight_function` for details. Defaults to `None`.

    Returns
    -------
    edges : dictionary
        The parameter `betweenness` including edges with keys and their
        betweenness centrality values.

    The BC value is divided among edges of equal weight.
    """
    _weight = _weight_function(G, weight)

    edge_bc = dict.fromkeys(G.edges, 0.0)
    for u, v in betweenness:
        d = G[u][v]
        wt = _weight(u, v, d)
        keys = [k for k in d if _weight(u, v, {k: d[k]}) == wt]
        bc = betweenness[(u, v)] / len(keys)
        for k in keys:
            edge_bc[(u, v, k)] = bc

    return edge_bc