393 lines
9.5 KiB
Python
393 lines
9.5 KiB
Python
"""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!
|
||
|
||
|
||
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)
|
||
|
||
|
||
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))
|
||
|
||
|
||
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))
|
||
|
||
|
||
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)
|
||
|
||
|
||
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))
|
||
|
||
|
||
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))`?
|
||
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))`?
|
||
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)
|