Fixed database typo and removed unnecessary class identifier.
This commit is contained in:
parent
00ad49a143
commit
45fb349a7d
5098 changed files with 952558 additions and 85 deletions
|
|
@ -0,0 +1,268 @@
|
|||
import networkx as nx
|
||||
|
||||
__all__ = ["degree_centrality", "betweenness_centrality", "closeness_centrality"]
|
||||
|
||||
|
||||
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
|
||||
|
||||
|
||||
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
|
||||
|
||||
|
||||
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
|
||||
Loading…
Add table
Add a link
Reference in a new issue