Fixed database typo and removed unnecessary class identifier.

venv/Lib/site-packages/networkx/algorithms/centrality/load.py

@@ -0,0 +1,197 @@
+"""Load centrality."""
+from operator import itemgetter
+
+import networkx as nx
+
+__all__ = ["load_centrality", "edge_load_centrality"]
+
+
+def newman_betweenness_centrality(G, v=None, cutoff=None, normalized=True, weight=None):
+    """Compute load centrality for nodes.
+
+    The load centrality of a node is the fraction of all shortest
+    paths that pass through that node.
+
+    Parameters
+    ----------
+    G : graph
+      A networkx graph.
+
+    normalized : bool, optional (default=True)
+      If True the betweenness values are normalized by b=b/(n-1)(n-2) where
+      n is the number of nodes in G.
+
+    weight : None or string, optional (default=None)
+      If None, edge weights are ignored.
+      Otherwise holds the name of the edge attribute used as weight.
+
+    cutoff : bool, optional (default=None)
+      If specified, only consider paths of length <= cutoff.
+
+    Returns
+    -------
+    nodes : dictionary
+       Dictionary of nodes with centrality as the value.
+
+    See Also
+    --------
+    betweenness_centrality()
+
+    Notes
+    -----
+    Load centrality is slightly different than betweenness. It was originally
+    introduced by [2]_. For this load algorithm see [1]_.
+
+    References
+    ----------
+    .. [1] Mark E. J. Newman:
+       Scientific collaboration networks. II.
+       Shortest paths, weighted networks, and centrality.
+       Physical Review E 64, 016132, 2001.
+       http://journals.aps.org/pre/abstract/10.1103/PhysRevE.64.016132
+    .. [2] Kwang-Il Goh, Byungnam Kahng and Doochul Kim
+       Universal behavior of Load Distribution in Scale-Free Networks.
+       Physical Review Letters 87(27):1–4, 2001.
+       http://phya.snu.ac.kr/~dkim/PRL87278701.pdf
+    """
+    if v is not None:  # only one node
+        betweenness = 0.0
+        for source in G:
+            ubetween = _node_betweenness(G, source, cutoff, False, weight)
+            betweenness += ubetween[v] if v in ubetween else 0
+        if normalized:
+            order = G.order()
+            if order <= 2:
+                return betweenness  # no normalization b=0 for all nodes
+            betweenness *= 1.0 / ((order - 1) * (order - 2))
+        return betweenness
+    else:
+        betweenness = {}.fromkeys(G, 0.0)
+        for source in betweenness:
+            ubetween = _node_betweenness(G, source, cutoff, False, weight)
+            for vk in ubetween:
+                betweenness[vk] += ubetween[vk]
+        if normalized:
+            order = G.order()
+            if order <= 2:
+                return betweenness  # no normalization b=0 for all nodes
+            scale = 1.0 / ((order - 1) * (order - 2))
+            for v in betweenness:
+                betweenness[v] *= scale
+        return betweenness  # all nodes
+
+
+def _node_betweenness(G, source, cutoff=False, normalized=True, weight=None):
+    """Node betweenness_centrality helper:
+
+    See betweenness_centrality for what you probably want.
+    This actually computes "load" and not betweenness.
+    See https://networkx.lanl.gov/ticket/103
+
+    This calculates the load of each node for paths from a single source.
+    (The fraction of number of shortests paths from source that go
+    through each node.)
+
+    To get the load for a node you need to do all-pairs shortest paths.
+
+    If weight is not None then use Dijkstra for finding shortest paths.
+    """
+    # get the predecessor and path length data
+    if weight is None:
+        (pred, length) = nx.predecessor(G, source, cutoff=cutoff, return_seen=True)
+    else:
+        (pred, length) = nx.dijkstra_predecessor_and_distance(G, source, cutoff, weight)
+
+    # order the nodes by path length
+    onodes = [(l, vert) for (vert, l) in length.items()]
+    onodes.sort()
+    onodes[:] = [vert for (l, vert) in onodes if l > 0]
+
+    # initialize betweenness
+    between = {}.fromkeys(length, 1.0)
+
+    while onodes:
+        v = onodes.pop()
+        if v in pred:
+            num_paths = len(pred[v])  # Discount betweenness if more than
+            for x in pred[v]:  # one shortest path.
+                if x == source:  # stop if hit source because all remaining v
+                    break  # also have pred[v]==[source]
+                between[x] += between[v] / float(num_paths)
+    #  remove source
+    for v in between:
+        between[v] -= 1
+    # rescale to be between 0 and 1
+    if normalized:
+        l = len(between)
+        if l > 2:
+            # scale by 1/the number of possible paths
+            scale = 1.0 / float((l - 1) * (l - 2))
+            for v in between:
+                between[v] *= scale
+    return between
+
+
+load_centrality = newman_betweenness_centrality
+
+
+def edge_load_centrality(G, cutoff=False):
+    """Compute edge load.
+
+    WARNING: This concept of edge load has not been analysed
+    or discussed outside of NetworkX that we know of.
+    It is based loosely on load_centrality in the sense that
+    it counts the number of shortest paths which cross each edge.
+    This function is for demonstration and testing purposes.
+
+    Parameters
+    ----------
+    G : graph
+        A networkx graph
+
+    cutoff : bool, optional (default=False)
+        If specified, only consider paths of length <= cutoff.
+
+    Returns
+    -------
+    A dict keyed by edge 2-tuple to the number of shortest paths
+    which use that edge. Where more than one path is shortest
+    the count is divided equally among paths.
+    """
+    betweenness = {}
+    for u, v in G.edges():
+        betweenness[(u, v)] = 0.0
+        betweenness[(v, u)] = 0.0
+
+    for source in G:
+        ubetween = _edge_betweenness(G, source, cutoff=cutoff)
+        for e, ubetweenv in ubetween.items():
+            betweenness[e] += ubetweenv  # cumulative total
+    return betweenness
+
+
+def _edge_betweenness(G, source, nodes=None, cutoff=False):
+    """Edge betweenness helper."""
+    # get the predecessor data
+    (pred, length) = nx.predecessor(G, source, cutoff=cutoff, return_seen=True)
+    # order the nodes by path length
+    onodes = [n for n, d in sorted(length.items(), key=itemgetter(1))]
+    # initialize betweenness, doesn't account for any edge weights
+    between = {}
+    for u, v in G.edges(nodes):
+        between[(u, v)] = 1.0
+        between[(v, u)] = 1.0
+
+    while onodes:  # work through all paths
+        v = onodes.pop()
+        if v in pred:
+            # Discount betweenness if more than one shortest path.
+            num_paths = len(pred[v])
+            for w in pred[v]:
+                if w in pred:
+                    # Discount betweenness, mult path
+                    num_paths = len(pred[w])
+                    for x in pred[w]:
+                        between[(w, x)] += between[(v, w)] / num_paths
+                        between[(x, w)] += between[(w, v)] / num_paths
+    return between