neurospin.graph.forest

Module: neurospin.graph.forest

Inheritance diagram for nipy.neurospin.graph.forest:

This module implements the DAG class of fff2 Forest(WeightedGraph): This is a special case of a Weighted Graph (i.e. a set of trees)

Main author: Bertrand thirion, 2007-2009

Forest

class nipy.neurospin.graph.forest.Forest(V, parents=None)

Bases: nipy.neurospin.graph.graph.WeightedGraph

This is a Forest structure, i.e. a set of trees - the nodes can be segmented into trees - within each tree a node has one parent and children (hierarchical structure) - some of the nodes can be viewed as leaves, other as roots - the edges within a tree are associated with a weight: +1 from child to parent -1 from parent to child

fields: - V : (int,>0) the number of vertices - E : (int) the number of edges - parents: array of shape (self.V) the parent array - edges: array of shape (self.E,2) reprensenting pairwise neighbors - weights, array of shape (self.E), +1/-1 for scending/descending links - children: list of arrays that represents the childs of any node

Methods

Kruskal
Kruskal_dev
Voronoi_Labelling
Voronoi_diagram
WeightedDegree
adjacency
all_distances
anti_symmeterize
cc
check
cliques
complete
compute_children
converse_edge
copy
cut_redundancies
define_graph_attributes
degrees
depth_from_leaves
dijkstra
eps
floyd
from_3d_grid
from_adjacency
get_E
get_V
get_children
get_descendents
get_edges
get_vertices
get_weights
is_connected
isleaf
isroot
knn
leaves_of_a_subtree
left_incidence
list_of_neighbors
main_cc
merge_simple_branches
mst
normalize
propagate_upward
propagate_upward_and
remove_edges
remove_trivial_edges
reorder
reorder_from_leaves_to_roots
right_incidence
rooted_subtree
set_edges
set_euclidian
set_gaussian
set_weights
show
skeleton
subforest
subgraph
symmeterize
to_neighb
tree_depth

__init__(V, parents=None)

Parameters:

V (int), the number of edges of the graph :

parents = None: array of shape (V) :

the parents of zach vertex if Parents==None , the parents are set to range(V), i.e. each node is its own parent, and each node is a tree

Kruskal()

Creates the Minimum Spanning Tree self using Kruskal’s algo. efficient is self is sparse

Returns:

K: WeightedGraph instance :

the resulting MST

Kruskal_dev()

Creates the Minimum Spanning Tree self using Kruskal’s algo. efficient is self is sparse

Returns:

K: WeightedGraph instance :

the resulting MST

Voronoi_Labelling(seed)

label = self.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

fixme: how is dealt the case of diconnected graph ? :

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) :

WeightedDegree(c)

returns the sum of weighted degree of graph self

Parameters:

c (int): side selection :

if c==0 considering left side if c==1 considering right side of the edges

Returns:

wd : array of shape (self.V),

the resulting weighted degree

Note: slow implementation

adjacency()

Create the adjacency matrix of self

Returns:

A : an ((self.V*self.V),np.double) array

adjacency matrix of the graph

all_distances(seed=None)

returns all the distances of the graph as a tree

Parameters:

seed=None array of shape(nbseed) with valuesin [0..self.V-1] :

set of vertices from which tehe distances are computed

Returns:

dg: array of shape(nseed, self.V): the resulting distance :

anti_symmeterize()

self.anti_symmeterize() anti-symmeterize the self , ie produces the graph whose adjacency matrix would be the antisymmetric part of its current adjacency matrix

cc()

Returns an array of labels corresponding to the different connex components of the graph.

Returns:label: array of shape(self.V), labelling of the vertices :

check()

Check that the proposed is indeed a graph, i.e. contains no loop

Returns:a boolean b=0 iff there are loops, 1 otherwise :

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

complete() makes self a complete graph (i.e. each pair of vertices is an edge)

compute_children()

self.compute_children() define the children list

Returns:

children: a list of self.V lists, :

that yields the children of each node

converse_edge()

Returns the index of the edge (j,i) for each edge (i,j) Note: a C implementation might be necessary

copy()

returns a copy of self

cut_redundancies()

self.cut_redudancies() Remove possibly redundant edges: if an edge (ab) is present twice in the edge matrix, only the first instance in kept. The weights are processed accordingly

Returns:- E(int): the number of edges, self.E :

define_graph_attributes()

define the edge and weights array

degrees()

returns the degree of the graph vertices

Returns:

rdegree: array of shape self.V, the right degree :

ldegree: array of shape self.V, the left degree :

depth_from_leaves()

compute a labelling of the nodes which is 0 for the leaves, 1 for their parents etc and maximal for the roots

Returns:depth: array of shape (self.V): the depth values of the vertices :

dijkstra(seed=0)

returns all the [graph] geodesic distances starting from seed it is mandatory that the graph weights are non-negative

Parameters:

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

eps(X, eps=1.0)

set the graph to be the eps-nearest-neighbours graph of the data

Parameters:

X array of shape (self.V) or (self.V,p) :

where p = dimension of the features data used for eps-neighbours computation

eps=1. (float), the neighborhood width :

Returns:

self.E the number of edges of the resulting graph :

floyd(seed=None)

Compute all the geodesic distances starting from seeds it is mandatory that the graph weights are non-negative

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

from_3d_grid(xyz, k=18)

set the graph to be the topological neighbours graph of the thre-dimensional coordinate 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 :

from_adjacency(A)

sets the edges of self according to the adjacency matrix M

Parameters:M: array of shape(sef.V,self.V) :

get_E()

get_V()

get_children(v=-1)

returns the list list of children arrays in all the forest if v==-1 or the children of v otherwise

get_descendents(v)

returns the nodes that are children of v

get_edges()

get_vertices()

get_weights()

is_connected()

States whether self is connected or not

isleaf()

returns a bool array of shape(self.V) so that isleaf==1 iff the node is a leaf in the forest (has no kids)

isroot()

returns a bool array of shape(self.V) so that isleaf==1 iff the node is a root in the forest i.e. : is its own parent

knn(X, k=1)

E = knn(X,k) set the graph to be the k-nearest-neighbours graph of the data

Parameters:

X array of shape (self.V) or (self.V,p) :

where p = dimension of the features data used for eps-neighbours computation

k=1 : is the number of neighbours considered

Returns:

- self.E (int): the number of edges of the resulting graph :

leaves_of_a_subtree(ids, custom=False)

tests whether the given nodes within ids represent all the leaves of a certain subtree of self

Parameters:

idds: array of shape (n) that takes values in [0..self.V-1] :

custom == False, boolean :

if custom==true the behavior of the function is more specific - the different connected components are considered as being in a same greater tree - when a node has more than two subbranches, any subset of these children is considered as a subtree

left_incidence()

Returns:

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) :

merge_simple_branches()

merge the branches of the forest that are the only child of the parent branch into their child

mst(X)

makes self the MST of the array X

Parameters:

X: an array of shape (self.V,dim) :

p is the feature dimension of X

Returns:

tl (float) the total length of the mst :

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

propagate_upward(label)

label = self.propagate_upward(label) Assuming that label is a certain positive integer field (i.e. labels) that is defined at the leaves of the tree and can be compared, this propagates these labels to the parents whenever the children nodes have coherent properties otherwise the parent value is unchanged

Parameters:label: array of shape(self.V) : Returns:label: array of shape(self.V) :

propagate_upward_and(prop)

propagates some binary property in the forest that is defined in the leaves so that prop[parents] = logical_and(prop[children])

Parameters:prop, array of shape(self.V), the input property : Returns:prop, array of shape(self.V), the output property field :

remove_edges(valid)

Removes all the edges for which valid==0

Parameters:valid, an array of shape (self.E) :

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 :

reorder(c=0)

Reorder the graph according to the index c

Parameters:

c=0 in {0,1,2}, index that designates the array :

according to which the vectors are jointly reordered c == 0 => reordering makes edges[:,0] increasing,

and edges[:,1] increasing for edges[:,0] fixed

c == 1 => reordering makes edges[:,1] increasing,

and edges[:,0] increasing for edges[:,1] fixed

c == 2 => reordering makes weights increasing

reorder_from_leaves_to_roots()

reorder the tree so that the leaves come first then their parents and so on, and the roots are last the permutation necessary to apply to all vertex-based information

Returns:

order: array of shape(self.V) :

the order of the old vertices in the reordered graph

right_incidence()

Returns:

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

rooted_subtree(k)

l = self.subtree(k) returns an array of the nodes included in the subtree rooted in k

Parameters:k (int): the vertex from which the subtree is searched : Returns:idx : array of shape>=1 the index of the nodes beneath k

set_edges(edges)

sets self.edges=edges if 1. edges has a correct size 2. 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 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 dinstance 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

set_weights(weights)

Parameters:weights : an array of shape(self.V), edges weights

show(X=None, ax=None)

a = self.show(X=None) plots the current graph in 2D

Parameters:

X=None, 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: ax handle, optional :

Returns:

ax: axis handle :

skeleton()

returns a MST that based on self.weights Note: self must be connected

subforest(valid)

creates a subforest with the vertices for which valid>0 and with the correponding set of edges the children of deleted vertices become their own parent

Parameters:valid: array of shape (self.V) : Returns:a new forest instance :

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 :

symmeterize()

symmeterize the graphself , ie produces the graph whose adjacency matrix would be the symmetric part of its current adjacency matrix

to_neighb()

converts the graph to a neighboring system The neighboring system is nothing but a (sparser) representation of the edge matrix

Returns:

ci, ne, we: arrays of shape (self.V+1), (self.E), (self.E) :

such that self.edges, self.weights is coded such that: for j in [ci[a] ci[a+1][, there exists en edge e so that (edge[e,0]=a,edge[e,1]=ne[j],self.weights[e] = we[j])

tree_depth()

return the maximal depth of any node in the tree