"""Functions for computing the harmonic centrality of a graph."""
from functools import partial
import networkx as nx
__all__ = ["harmonic_centrality"]
[docs]def harmonic_centrality(G, nbunch=None, distance=None, sources=None):
r"""Compute harmonic centrality for nodes.
Harmonic centrality [1]_ of a node `u` is the sum of the reciprocal
of the shortest path distances from all other nodes to `u`
.. math::
C(u) = \sum_{v \neq u} \frac{1}{d(v, u)}
where `d(v, u)` is the shortest-path distance between `v` and `u`.
If `sources` is given as an argument, the returned harmonic centrality
values are calculated as the sum of the reciprocals of the shortest
path distances from the nodes specified in `sources` to `u` instead
of from all nodes to `u`.
Notice that higher values indicate higher centrality.
Parameters
----------
G : graph
A NetworkX graph
nbunch : container (default: all nodes in G)
Container of nodes for which harmonic centrality values are calculated.
sources : container (default: all nodes in G)
Container of nodes `v` over which reciprocal distances are computed.
Nodes not in `G` are silently ignored.
distance : edge attribute key, optional (default=None)
Use the specified edge attribute as the edge distance in shortest
path calculations. If `None`, then each edge will have distance equal to 1.
Returns
-------
nodes : dictionary
Dictionary of nodes with harmonic centrality as the value.
See Also
--------
betweenness_centrality, load_centrality, eigenvector_centrality,
degree_centrality, closeness_centrality
Notes
-----
If the 'distance' keyword is set to an edge attribute key then the
shortest-path length will be computed using Dijkstra's algorithm with
that edge attribute as the edge weight.
References
----------
.. [1] Boldi, Paolo, and Sebastiano Vigna. "Axioms for centrality."
Internet Mathematics 10.3-4 (2014): 222-262.
"""
nbunch = set(G.nbunch_iter(nbunch)) if nbunch is not None else set(G.nodes)
sources = set(G.nbunch_iter(sources)) if sources is not None else G.nodes
spl = partial(nx.shortest_path_length, G, weight=distance)
centrality = {u: 0 for u in nbunch}
for v in sources:
dist = spl(v)
for u in nbunch.intersection(dist):
d = dist[u]
if d == 0: # handle u == v and edges with 0 weight
continue
centrality[u] += 1 / d
return centrality