| """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", |
| ] |
|
|
|
|
| |
|
|
|
|
| @nx._dispatchable(edge_attrs="weight") |
| 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) |
|
|
|
|
| @nx._dispatchable(edge_attrs="weight") |
| 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)) |
|
|
|
|
| @nx._dispatchable(edge_attrs="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)) |
|
|
|
|
| @nx._dispatchable(edge_attrs="weight") |
| 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) |
|
|
|
|
| @nx._dispatchable(edge_attrs="weight") |
| 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)) |
|
|
|
|
| @nx._dispatchable(edge_attrs="weight") |
| 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) |
|
|
|
|
| |
| |
| @nx._dispatchable |
| 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) |
|
|
|
|
| @nx._dispatchable |
| def boundary_expansion(G, S): |
| """Returns the boundary expansion of the set `S`. |
| |
| The *boundary expansion* of a set `S` is the ratio between the size of its |
| node boundary and the cardinality of the set itself [1]_ . |
| |
| Parameters |
| ---------- |
| G : NetworkX graph |
| The input graph. |
| |
| S : collection |
| A collection of nodes in `G`. |
| |
| Returns |
| ------- |
| number |
| The boundary expansion ratio: size of node boundary / size of `S`. |
| |
| Examples |
| -------- |
| The node boundary is {2, 3} (size 2), divided by ``|S|=2``: |
| |
| >>> G = nx.cycle_graph(4) |
| >>> S = {0, 1} |
| >>> nx.boundary_expansion(G, S) |
| 1.0 |
| |
| For disconnected sets, e.g. here where the node boundary is ``{1, 3, 5}``: |
| |
| >>> G = nx.cycle_graph(6) |
| >>> S = {0, 2, 4} |
| >>> nx.boundary_expansion(G, S) |
| 1.0 |
| |
| See also |
| -------- |
| :func:`~networkx.algorithms.boundary.node_boundary` |
| edge_expansion |
| mixing_expansion |
| node_expansion |
| |
| Notes |
| ----- |
| The node boundary is defined as all nodes not in `S` that are adjacent to |
| nodes in `S`. |
| |
| 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) |
|
|
