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 2D-array) - epsilon-neighbors (where sample rows of a 2D-array) - representation of the neighbors on a 3d grid (6-, 18- and 26-neighbors) - Minimum Spanning Tree (where samples are rows of a 2D-array)
Author: Bertrand Thirion, 2006–2011
Classes¶
Graph
¶
- class nipy.algorithms.graph.graph.Graph(V, E=0, edges=None)¶
Bases:
object
Basic topological (non-weighted) 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:
- Vint
the number of vertices
- Eint, optional
the number of edges
- edgesNone 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:
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:
- Vint
(int > 0) the number of vertices
- edges(E, 2) array, type int
edges of the graph
- weights(E, 2) array, type=int
weights/lengths of the edges
- 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
- anti_symmeterize()¶
anti-symmeterize self, i.e. produces the graph whose adjacency matrix would be the antisymmetric part of its current adjacency matrix
- cc()¶
Compte the different connected components of the graph.
- Returns:
- label: array of shape(self.V), labelling of the vertices
- 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 neighbors 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 only once in the edge matrix: the correspondng weights are added.
- Returns:
- the resulting WeightedGraph
- 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
- 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 non-negative
- 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 non-negative. 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 three-dimensional coordinates set xyz, in the k-connectivity 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_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)
- 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
- main_cc()¶
Returns the indexes of the vertices within the main cc
- Returns:
- idx: array of shape (sizeof main cc)
- 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_edges(edges)¶
Sets the graph’s edges
Preconditions:
edges has a correct size
edges take values in [1..V]
- set_euclidian(X)¶
Compute the weights of the graph as the distances between the corresponding rows of X, which represents an embedding 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 embedding 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:
- XNone 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 corresponding 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 eps-nearest-neighbours 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 k-nearest-neighbours 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 components 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 non-sparse 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 three-dimensional coordinates set xyz, in the k-connectivity 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