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