import networkx as nx
__all__ = ["degree_centrality", "betweenness_centrality", "closeness_centrality"]
[docs]def degree_centrality(G, nodes):
r"""Compute the degree centrality for nodes in a bipartite network.
The degree centrality for a node `v` is the fraction of nodes
connected to it.
Parameters
----------
G : graph
A bipartite network
nodes : list or container
Container with all nodes in one bipartite node set.
Returns
-------
centrality : dictionary
Dictionary keyed by node with bipartite degree centrality as the value.
See Also
--------
betweenness_centrality,
closeness_centrality,
sets,
is_bipartite
Notes
-----
The nodes input parameter must contain all nodes in one bipartite node set,
but the dictionary returned contains all nodes from both bipartite node
sets. See :mod:`bipartite documentation <networkx.algorithms.bipartite>`
for further details on how bipartite graphs are handled in NetworkX.
For unipartite networks, the degree centrality values are
normalized by dividing by the maximum possible degree (which is
`n-1` where `n` is the number of nodes in G).
In the bipartite case, the maximum possible degree of a node in a
bipartite node set is the number of nodes in the opposite node set
[1]_. The degree centrality for a node `v` in the bipartite
sets `U` with `n` nodes and `V` with `m` nodes is
.. math::
d_{v} = \frac{deg(v)}{m}, \mbox{for} v \in U ,
d_{v} = \frac{deg(v)}{n}, \mbox{for} v \in V ,
where `deg(v)` is the degree of node `v`.
References
----------
.. [1] Borgatti, S.P. and Halgin, D. In press. "Analyzing Affiliation
Networks". In Carrington, P. and Scott, J. (eds) The Sage Handbook
of Social Network Analysis. Sage Publications.
http://www.steveborgatti.com/research/publications/bhaffiliations.pdf
"""
top = set(nodes)
bottom = set(G) - top
s = 1.0 / len(bottom)
centrality = {n: d * s for n, d in G.degree(top)}
s = 1.0 / len(top)
centrality.update({n: d * s for n, d in G.degree(bottom)})
return centrality
[docs]def betweenness_centrality(G, nodes):
r"""Compute betweenness centrality for nodes in a bipartite network.
Betweenness centrality of a node `v` is the sum of the
fraction of all-pairs shortest paths that pass through `v`.
Values of betweenness are normalized by the maximum possible
value which for bipartite graphs is limited by the relative size
of the two node sets [1]_.
Let `n` be the number of nodes in the node set `U` and
`m` be the number of nodes in the node set `V`, then
nodes in `U` are normalized by dividing by
.. math::
\frac{1}{2} [m^2 (s + 1)^2 + m (s + 1)(2t - s - 1) - t (2s - t + 3)] ,
where
.. math::
s = (n - 1) \div m , t = (n - 1) \mod m ,
and nodes in `V` are normalized by dividing by
.. math::
\frac{1}{2} [n^2 (p + 1)^2 + n (p + 1)(2r - p - 1) - r (2p - r + 3)] ,
where,
.. math::
p = (m - 1) \div n , r = (m - 1) \mod n .
Parameters
----------
G : graph
A bipartite graph
nodes : list or container
Container with all nodes in one bipartite node set.
Returns
-------
betweenness : dictionary
Dictionary keyed by node with bipartite betweenness centrality
as the value.
See Also
--------
degree_centrality,
closeness_centrality,
sets,
is_bipartite
Notes
-----
The nodes input parameter must contain all nodes in one bipartite node set,
but the dictionary returned contains all nodes from both node sets.
See :mod:`bipartite documentation <networkx.algorithms.bipartite>`
for further details on how bipartite graphs are handled in NetworkX.
References
----------
.. [1] Borgatti, S.P. and Halgin, D. In press. "Analyzing Affiliation
Networks". In Carrington, P. and Scott, J. (eds) The Sage Handbook
of Social Network Analysis. Sage Publications.
http://www.steveborgatti.com/research/publications/bhaffiliations.pdf
"""
top = set(nodes)
bottom = set(G) - top
n = float(len(top))
m = float(len(bottom))
s = (n - 1) // m
t = (n - 1) % m
bet_max_top = (
((m ** 2) * ((s + 1) ** 2))
+ (m * (s + 1) * (2 * t - s - 1))
- (t * ((2 * s) - t + 3))
) / 2.0
p = (m - 1) // n
r = (m - 1) % n
bet_max_bot = (
((n ** 2) * ((p + 1) ** 2))
+ (n * (p + 1) * (2 * r - p - 1))
- (r * ((2 * p) - r + 3))
) / 2.0
betweenness = nx.betweenness_centrality(G, normalized=False, weight=None)
for node in top:
betweenness[node] /= bet_max_top
for node in bottom:
betweenness[node] /= bet_max_bot
return betweenness
[docs]def closeness_centrality(G, nodes, normalized=True):
r"""Compute the closeness centrality for nodes in a bipartite network.
The closeness of a node is the distance to all other nodes in the
graph or in the case that the graph is not connected to all other nodes
in the connected component containing that node.
Parameters
----------
G : graph
A bipartite network
nodes : list or container
Container with all nodes in one bipartite node set.
normalized : bool, optional
If True (default) normalize by connected component size.
Returns
-------
closeness : dictionary
Dictionary keyed by node with bipartite closeness centrality
as the value.
See Also
--------
betweenness_centrality,
degree_centrality
sets,
is_bipartite
Notes
-----
The nodes input parameter must contain all nodes in one bipartite node set,
but the dictionary returned contains all nodes from both node sets.
See :mod:`bipartite documentation <networkx.algorithms.bipartite>`
for further details on how bipartite graphs are handled in NetworkX.
Closeness centrality is normalized by the minimum distance possible.
In the bipartite case the minimum distance for a node in one bipartite
node set is 1 from all nodes in the other node set and 2 from all
other nodes in its own set [1]_. Thus the closeness centrality
for node `v` in the two bipartite sets `U` with
`n` nodes and `V` with `m` nodes is
.. math::
c_{v} = \frac{m + 2(n - 1)}{d}, \mbox{for} v \in U,
c_{v} = \frac{n + 2(m - 1)}{d}, \mbox{for} v \in V,
where `d` is the sum of the distances from `v` to all
other nodes.
Higher values of closeness indicate higher centrality.
As in the unipartite case, setting normalized=True causes the
values to normalized further to n-1 / size(G)-1 where n is the
number of nodes in the connected part of graph containing the
node. If the graph is not completely connected, this algorithm
computes the closeness centrality for each connected part
separately.
References
----------
.. [1] Borgatti, S.P. and Halgin, D. In press. "Analyzing Affiliation
Networks". In Carrington, P. and Scott, J. (eds) The Sage Handbook
of Social Network Analysis. Sage Publications.
http://www.steveborgatti.com/research/publications/bhaffiliations.pdf
"""
closeness = {}
path_length = nx.single_source_shortest_path_length
top = set(nodes)
bottom = set(G) - top
n = float(len(top))
m = float(len(bottom))
for node in top:
sp = dict(path_length(G, node))
totsp = sum(sp.values())
if totsp > 0.0 and len(G) > 1:
closeness[node] = (m + 2 * (n - 1)) / totsp
if normalized:
s = (len(sp) - 1.0) / (len(G) - 1)
closeness[node] *= s
else:
closeness[n] = 0.0
for node in bottom:
sp = dict(path_length(G, node))
totsp = sum(sp.values())
if totsp > 0.0 and len(G) > 1:
closeness[node] = (n + 2 * (m - 1)) / totsp
if normalized:
s = (len(sp) - 1.0) / (len(G) - 1)
closeness[node] *= s
else:
closeness[n] = 0.0
return closeness