algorithms.graph.graph¶
Module: algorithms.graph.graph
¶
Inheritance diagram for nipy.algorithms.graph.graph
:
This module implements two graph classes:
Graph: basic topological graph, i.e. vertices and edges. This kind of object only has topological properties
WeightedGraph (Graph): also has a value associated with edges, called weights, that are used in some computational procedures (e.g. path length computation). Importantly these objects are equivalent to square sparse matrices, which is used to perform certain computations.
This module also provides several functions to instantiate WeightedGraphs from data:  k nearest neighbours (where samples are rows of a 2Darray)  epsilonneighbors (where sample rows of a 2Darray)  representation of the neighbors on a 3d grid (6, 18 and 26neighbors)  Minimum Spanning Tree (where samples are rows of a 2Darray)
Author: Bertrand Thirion, 2006–2011
Classes¶
Graph
¶

class
nipy.algorithms.graph.graph.
Graph
(V, E=0, edges=None)¶ Bases:
object
Basic topological (nonweighted) directed Graph class
Member variables:
 V (int > 0): the number of vertices
 E (int >= 0): the number of edges
Properties:
 vertices (list, type=int, shape=(V,)) vertices id
 edges (list, type=int, shape=(E,2)): edges as vertices id tuples

__init__
(V, E=0, edges=None)¶ Constructor
Parameters: V : int
the number of vertices
E : int, optional
the number of edges
edges : None or shape (E, 2) array, optional
edges of graph

adjacency
()¶ returns the adjacency matrix of the graph as a sparse coo matrix
Returns: adj: scipy.sparse matrix instance,
that encodes the adjacency matrix of self

cc
()¶ Compte the different connected components of the graph.
Returns: label: array of shape(self.V), labelling of the vertices

degrees
()¶ Returns the degree of the graph vertices.
Returns: rdegree: (array, type=int, shape=(self.V,)), the right degrees
ldegree: (array, type=int, shape=(self.V,)), the left degrees

get_E
()¶ To get the number of edges in the graph

get_V
()¶ To get the number of vertices in the graph

get_edges
()¶ To get the graph’s edges

get_vertices
()¶ To get the graph’s vertices (as id)

main_cc
()¶ Returns the indexes of the vertices within the main cc
Returns: idx: array of shape (sizeof main cc)

set_edges
(edges)¶ Sets the graph’s edges
Preconditions:
 edges has a correct size
 edges take values in [1..V]

show
(ax=None)¶ Shows the graph as a planar one.
Parameters: ax, axis handle Returns: ax, axis handle

to_coo_matrix
()¶ Return adjacency matrix as coo sparse
Returns: sp: scipy.sparse matrix instance,
that encodes the adjacency matrix of self
WeightedGraph
¶

class
nipy.algorithms.graph.graph.
WeightedGraph
(V, edges=None, weights=None)¶ Bases:
nipy.algorithms.graph.graph.Graph
Basic weighted, directed graph class
Member variables:
 V (int): the number of vertices
 E (int): the number of edges
Methods
 vertices (list, type=int, shape=(V,)): vertices id
 edges (list, type=int, shape=(E,2)): edges as vertices id tuples
 weights (list, type=int, shape=(E,)): weights / lengths of the graph’s edges

__init__
(V, edges=None, weights=None)¶ Constructor
Parameters: V : int
(int > 0) the number of vertices
edges : (E, 2) array, type int
edges of the graph
weights : (E, 2) array, type=int
weights/lenghts of the edges

anti_symmeterize
()¶ antisymmeterize self, i.e. produces the graph whose adjacency matrix would be the antisymmetric part of its current adjacency matrix

cliques
()¶ Extraction of the graphe cliques these are defined using replicator dynamics equations
Returns: cliques: array of shape (self.V), type (np.int)
labelling of the vertices according to the clique they belong to

compact_neighb
()¶ returns a compact representation of self
Returns: idx: array of of shape(self.V + 1):
the positions where to find the neighors of each node within neighb and weights
neighb: array of shape(self.E), concatenated list of neighbors
weights: array of shape(self.E), concatenated list of weights

copy
()¶ returns a copy of self

cut_redundancies
()¶ Returns a graph with redundant edges removed: ecah edge (ab) is present ony once in the edge matrix: the correspondng weights are added.
Returns: the resulting WeightedGraph

dijkstra
(seed=0)¶ Returns all the [graph] geodesic distances starting from seed x
 seed (int, >1, <self.V) or array of shape(p)
 edge(s) from which the distances are computed
Returns: dg: array of shape (self.V),
the graph distance dg from ant vertex to the nearest seed
Notes
It is mandatory that the graph weights are nonnegative

floyd
(seed=None)¶ Compute all the geodesic distances starting from seeds
Parameters: seed= None: array of shape (nbseed), type np.int
vertex indexes from which the distances are computed if seed==None, then every edge is a seed point
Returns: dg array of shape (nbseed, self.V)
the graph distance dg from each seed to any vertex
Notes
It is mandatory that the graph weights are nonnegative. The algorithm proceeds by repeating Dijkstra’s algo for each seed. Floyd’s algo is not used (O(self.V)^3 complexity…)

from_3d_grid
(xyz, k=18)¶ Sets the graph to be the topological neighbours graph of the threedimensional coordinates set xyz, in the kconnectivity scheme
Parameters: xyz: array of shape (self.V, 3) and type np.int,
k = 18: the number of neighbours considered. (6, 18 or 26)
Returns: E(int): the number of edges of self

get_weights
()¶

is_connected
()¶ States whether self is connected or not

kruskal
()¶ Creates the Minimum Spanning Tree of self using Kruskal’s algo. efficient is self is sparse
Returns: K, WeightedGraph instance: the resulting MST Notes
If self contains several connected components, will have the same number k of connected components

left_incidence
()¶ Return left incidence matrix
Returns: left_incid: list
the left incidence matrix of self as a list of lists: i.e. the list[[e.0.0, .., e.0.i(0)], .., [e.V.0, E.V.i(V)]] where e.i.j is the set of edge indexes so that e.i.j[0] = i

list_of_neighbors
()¶ returns the set of neighbors of self as a list of arrays

normalize
(c=0)¶ Normalize the graph according to the index c Normalization means that the sum of the edges values that go into or out each vertex must sum to 1
Parameters: c=0 in {0, 1, 2}, optional: index that designates the way
according to which D is normalized c == 0 => for each vertex a, sum{edge[e, 0]=a} D[e]=1 c == 1 => for each vertex b, sum{edge[e, 1]=b} D[e]=1 c == 2 => symmetric (‘l2’) normalization
Notes
Note that when sum_{edge[e, .] == a } D[e] = 0, nothing is performed

remove_edges
(valid)¶ Removes all the edges for which valid==0
Parameters: valid : (self.E,) array

remove_trivial_edges
()¶ Removes trivial edges, i.e. edges that are (vv)like self.weights and self.E are corrected accordingly
Returns: self.E (int): The number of edges

right_incidence
()¶ Return right incidence matrix
Returns: right_incid: list
the right incidence matrix of self as a list of lists: i.e. the list[[e.0.0, .., e.0.i(0)], .., [e.V.0, E.V.i(V)]] where e.i.j is the set of edge indexes so that e.i.j[1] = i

set_euclidian
(X)¶ Compute the weights of the graph as the distances between the corresponding rows of X, which represents an embdedding of self
Parameters: X array of shape (self.V, edim),
the coordinate matrix of the embedding

set_gaussian
(X, sigma=0)¶ Compute the weights of the graph as a gaussian function of the distance between the corresponding rows of X, which represents an embdedding of self
Parameters: X array of shape (self.V, dim)
the coordinate matrix of the embedding
sigma=0, float: the parameter of the gaussian function
Notes
When sigma == 0, the following value is used:
sigma = sqrt(mean(X[self.edges[:, 0], :]X[self.edges[:, 1], :]^2))

set_weights
(weights)¶ Set edge weights
Parameters: weights: array
array shape(self.V): edges weights

show
(X=None, ax=None)¶ Plots the current graph in 2D
Parameters: X : None or array of shape (self.V, 2)
a set of coordinates that can be used to embed the vertices in 2D. If X.shape[1]>2, a svd reduces X for display. By default, the graph is presented on a circle
ax: None or int, optional
ax handle
Returns: ax: axis handle
Notes
This should be used only for small graphs.

subgraph
(valid)¶ Creates a subgraph with the vertices for which valid>0 and with the correponding set of edges
Parameters: valid, array of shape (self.V): nonzero for vertices to be retained Returns: G, WeightedGraph instance, the desired subgraph of self Notes
The vertices are renumbered as [1..p] where p = sum(valid>0) when sum(valid==0) then None is returned

symmeterize
()¶ Symmeterize self, modify edges and weights so that self.adjacency becomes the symmetric part of the current self.adjacency.

to_coo_matrix
()¶ Return adjacency matrix as coo sparse
Returns: sp: scipy.sparse matrix instance
that encodes the adjacency matrix of self

voronoi_diagram
(seeds, samples)¶ Defines the graph as the Voronoi diagram (VD) that links the seeds. The VD is defined using the sample points.
Parameters: seeds: array of shape (self.V, dim)
samples: array of shape (nsamples, dim)
Notes
By default, the weights are a Gaussian function of the distance The implementation is not optimal

voronoi_labelling
(seed)¶ Performs a voronoi labelling of the graph
Parameters: seed: array of shape (nseeds), type (np.int),
vertices from which the cells are built
Returns: labels: array of shape (self.V) the labelling of the vertices
Functions¶

nipy.algorithms.graph.graph.
complete_graph
(n)¶ returns a complete graph with n vertices

nipy.algorithms.graph.graph.
concatenate_graphs
(G1, G2)¶ Returns the concatenation of the graphs G1 and G2 It is thus assumed that the vertices of G1 and G2 represent disjoint sets
Parameters: G1, G2: the two WeightedGraph instances to be concatenated Returns: G, WeightedGraph, the concatenated graph Notes
This implies that the vertices of G corresponding to G2 are labeled [G1.V .. G1.V+G2.V]

nipy.algorithms.graph.graph.
eps_nn
(X, eps=1.0)¶ Returns the epsnearestneighbours graph of the data
Parameters: X, array of shape (n_samples, n_features), input data
eps, float, optional: the neighborhood width
Returns: the resulting graph instance

nipy.algorithms.graph.graph.
graph_3d_grid
(xyz, k=18)¶ Utility that computes the six neighbors on a 3d grid
Parameters: xyz: array of shape (n_samples, 3); grid coordinates of the points
k: neighboring system, equal to 6, 18, or 26
Returns: i, j, d 3 arrays of shape (E),
where E is the number of edges in the resulting graph (i, j) represent the edges, d their weights

nipy.algorithms.graph.graph.
knn
(X, k=1)¶ returns the knearestneighbours graph of the data
Parameters: X, array of shape (n_samples, n_features): the input data
k, int, optional: is the number of neighbours considered
Returns: the corresponding WeightedGraph instance
Notes
The knn system is symmeterized: if (ab) is one of the edges then (ba) is also included

nipy.algorithms.graph.graph.
lil_cc
(lil)¶ Returns the connected comonents of a graph represented as a list of lists
Parameters: lil: a list of list representing the graph neighbors Returns: label a vector of shape len(lil): connected components labelling Notes
Dramatically slow for nonsparse graphs

nipy.algorithms.graph.graph.
mst
(X)¶ Returns the WeightedGraph that is the minimum Spanning Tree of X
Parameters: X: data array, of shape(n_samples, n_features) Returns: the corresponding WeightedGraph instance

nipy.algorithms.graph.graph.
wgraph_from_3d_grid
(xyz, k=18)¶ Create graph as the set of topological neighbours of the threedimensional coordinates set xyz, in the kconnectivity scheme
Parameters: xyz: array of shape (nsamples, 3) and type np.int,
k = 18: the number of neighbours considered. (6, 18 or 26)
Returns: the WeightedGraph instance

nipy.algorithms.graph.graph.
wgraph_from_adjacency
(x)¶ Instantiates a weighted graph from a square 2D array
Parameters: x: 2D array instance, the input array Returns: wg: WeightedGraph instance

nipy.algorithms.graph.graph.
wgraph_from_coo_matrix
(x)¶ Instantiates a weighted graph from a (sparse) coo_matrix
Parameters: x: scipy.sparse.coo_matrix instance, the input matrix Returns: wg: WeightedGraph instance