"""Functions for finding and evaluating cuts in a graph.
"""
from itertools import chain
import networkx as nx
__all__ = [
"boundary_expansion",
"conductance",
"cut_size",
"edge_expansion",
"mixing_expansion",
"node_expansion",
"normalized_cut_size",
"volume",
]
# TODO STILL NEED TO UPDATE ALL THE DOCUMENTATION!
[docs]def cut_size(G, S, T=None, weight=None):
"""Returns the size of the cut between two sets of nodes.
A *cut* is a partition of the nodes of a graph into two sets. The
*cut size* is the sum of the weights of the edges "between" the two
sets of nodes.
Parameters
----------
G : NetworkX graph
S : collection
A collection of nodes in `G`.
T : collection
A collection of nodes in `G`. If not specified, this is taken to
be the set complement of `S`.
weight : object
Edge attribute key to use as weight. If not specified, edges
have weight one.
Returns
-------
number
Total weight of all edges from nodes in set `S` to nodes in
set `T` (and, in the case of directed graphs, all edges from
nodes in `T` to nodes in `S`).
Examples
--------
In the graph with two cliques joined by a single edges, the natural
bipartition of the graph into two blocks, one for each clique,
yields a cut of weight one::
>>> G = nx.barbell_graph(3, 0)
>>> S = {0, 1, 2}
>>> T = {3, 4, 5}
>>> nx.cut_size(G, S, T)
1
Each parallel edge in a multigraph is counted when determining the
cut size::
>>> G = nx.MultiGraph(["ab", "ab"])
>>> S = {"a"}
>>> T = {"b"}
>>> nx.cut_size(G, S, T)
2
Notes
-----
In a multigraph, the cut size is the total weight of edges including
multiplicity.
"""
edges = nx.edge_boundary(G, S, T, data=weight, default=1)
if G.is_directed():
edges = chain(edges, nx.edge_boundary(G, T, S, data=weight, default=1))
return sum(weight for u, v, weight in edges)
[docs]def volume(G, S, weight=None):
"""Returns the volume of a set of nodes.
The *volume* of a set *S* is the sum of the (out-)degrees of nodes
in *S* (taking into account parallel edges in multigraphs). [1]
Parameters
----------
G : NetworkX graph
S : collection
A collection of nodes in `G`.
weight : object
Edge attribute key to use as weight. If not specified, edges
have weight one.
Returns
-------
number
The volume of the set of nodes represented by `S` in the graph
`G`.
See also
--------
conductance
cut_size
edge_expansion
edge_boundary
normalized_cut_size
References
----------
.. [1] David Gleich.
*Hierarchical Directed Spectral Graph Partitioning*.
<https://www.cs.purdue.edu/homes/dgleich/publications/Gleich%202005%20-%20hierarchical%20directed%20spectral.pdf>
"""
degree = G.out_degree if G.is_directed() else G.degree
return sum(d for v, d in degree(S, weight=weight))
[docs]def normalized_cut_size(G, S, T=None, weight=None):
"""Returns the normalized size of the cut between two sets of nodes.
The *normalized cut size* is the cut size times the sum of the
reciprocal sizes of the volumes of the two sets. [1]
Parameters
----------
G : NetworkX graph
S : collection
A collection of nodes in `G`.
T : collection
A collection of nodes in `G`.
weight : object
Edge attribute key to use as weight. If not specified, edges
have weight one.
Returns
-------
number
The normalized cut size between the two sets `S` and `T`.
Notes
-----
In a multigraph, the cut size is the total weight of edges including
multiplicity.
See also
--------
conductance
cut_size
edge_expansion
volume
References
----------
.. [1] David Gleich.
*Hierarchical Directed Spectral Graph Partitioning*.
<https://www.cs.purdue.edu/homes/dgleich/publications/Gleich%202005%20-%20hierarchical%20directed%20spectral.pdf>
"""
if T is None:
T = set(G) - set(S)
num_cut_edges = cut_size(G, S, T=T, weight=weight)
volume_S = volume(G, S, weight=weight)
volume_T = volume(G, T, weight=weight)
return num_cut_edges * ((1 / volume_S) + (1 / volume_T))
[docs]def conductance(G, S, T=None, weight=None):
"""Returns the conductance of two sets of nodes.
The *conductance* is the quotient of the cut size and the smaller of
the volumes of the two sets. [1]
Parameters
----------
G : NetworkX graph
S : collection
A collection of nodes in `G`.
T : collection
A collection of nodes in `G`.
weight : object
Edge attribute key to use as weight. If not specified, edges
have weight one.
Returns
-------
number
The conductance between the two sets `S` and `T`.
See also
--------
cut_size
edge_expansion
normalized_cut_size
volume
References
----------
.. [1] David Gleich.
*Hierarchical Directed Spectral Graph Partitioning*.
<https://www.cs.purdue.edu/homes/dgleich/publications/Gleich%202005%20-%20hierarchical%20directed%20spectral.pdf>
"""
if T is None:
T = set(G) - set(S)
num_cut_edges = cut_size(G, S, T, weight=weight)
volume_S = volume(G, S, weight=weight)
volume_T = volume(G, T, weight=weight)
return num_cut_edges / min(volume_S, volume_T)
[docs]def edge_expansion(G, S, T=None, weight=None):
"""Returns the edge expansion between two node sets.
The *edge expansion* is the quotient of the cut size and the smaller
of the cardinalities of the two sets. [1]
Parameters
----------
G : NetworkX graph
S : collection
A collection of nodes in `G`.
T : collection
A collection of nodes in `G`.
weight : object
Edge attribute key to use as weight. If not specified, edges
have weight one.
Returns
-------
number
The edge expansion between the two sets `S` and `T`.
See also
--------
boundary_expansion
mixing_expansion
node_expansion
References
----------
.. [1] Fan Chung.
*Spectral Graph Theory*.
(CBMS Regional Conference Series in Mathematics, No. 92),
American Mathematical Society, 1997, ISBN 0-8218-0315-8
<http://www.math.ucsd.edu/~fan/research/revised.html>
"""
if T is None:
T = set(G) - set(S)
num_cut_edges = cut_size(G, S, T=T, weight=weight)
return num_cut_edges / min(len(S), len(T))
[docs]def mixing_expansion(G, S, T=None, weight=None):
"""Returns the mixing expansion between two node sets.
The *mixing expansion* is the quotient of the cut size and twice the
number of edges in the graph. [1]
Parameters
----------
G : NetworkX graph
S : collection
A collection of nodes in `G`.
T : collection
A collection of nodes in `G`.
weight : object
Edge attribute key to use as weight. If not specified, edges
have weight one.
Returns
-------
number
The mixing expansion between the two sets `S` and `T`.
See also
--------
boundary_expansion
edge_expansion
node_expansion
References
----------
.. [1] Vadhan, Salil P.
"Pseudorandomness."
*Foundations and Trends
in Theoretical Computer Science* 7.1–3 (2011): 1–336.
<https://doi.org/10.1561/0400000010>
"""
num_cut_edges = cut_size(G, S, T=T, weight=weight)
num_total_edges = G.number_of_edges()
return num_cut_edges / (2 * num_total_edges)
# TODO What is the generalization to two arguments, S and T? Does the
# denominator become `min(len(S), len(T))`?
[docs]def node_expansion(G, S):
"""Returns the node expansion of the set `S`.
The *node expansion* is the quotient of the size of the node
boundary of *S* and the cardinality of *S*. [1]
Parameters
----------
G : NetworkX graph
S : collection
A collection of nodes in `G`.
Returns
-------
number
The node expansion of the set `S`.
See also
--------
boundary_expansion
edge_expansion
mixing_expansion
References
----------
.. [1] Vadhan, Salil P.
"Pseudorandomness."
*Foundations and Trends
in Theoretical Computer Science* 7.1–3 (2011): 1–336.
<https://doi.org/10.1561/0400000010>
"""
neighborhood = set(chain.from_iterable(G.neighbors(v) for v in S))
return len(neighborhood) / len(S)
# TODO What is the generalization to two arguments, S and T? Does the
# denominator become `min(len(S), len(T))`?
[docs]def boundary_expansion(G, S):
"""Returns the boundary expansion of the set `S`.
The *boundary expansion* is the quotient of the size
of the node boundary and the cardinality of *S*. [1]
Parameters
----------
G : NetworkX graph
S : collection
A collection of nodes in `G`.
Returns
-------
number
The boundary expansion of the set `S`.
See also
--------
edge_expansion
mixing_expansion
node_expansion
References
----------
.. [1] Vadhan, Salil P.
"Pseudorandomness."
*Foundations and Trends in Theoretical Computer Science*
7.1–3 (2011): 1–336.
<https://doi.org/10.1561/0400000010>
"""
return len(nx.node_boundary(G, S)) / len(S)