# algorithms.graph.forest¶

## Module: `algorithms.graph.forest`¶

Inheritance diagram for `nipy.algorithms.graph.forest`:

Module implements the Forest class

A Forest is a graph with a hierarchical structure. Each connected component of a forest is a tree. The main characteristic is that each node has a single parent, so that a Forest is fully characterized by a “parent” array, that defines the unique parent of each node. The directed relationships are encoded by the weight sign.

Note that some methods of WeightedGraph class (e.g. dijkstra’s algorithm) require positive weights, so that they cannot work on forests in the current implementation. Specific methods (e.g. all_sidtance()) have been set instead.

Main author: Bertrand thirion, 2007-2011

## `Forest`¶

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

Bases: `WeightedGraph`

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 that describe the associated 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

Attributes:
Vint

int > 0, the number of vertices

Eint

the number of edges

parents(self.V,) array

the parent array

edges(self.E, 2) array

representing pairwise neighbors

weights(self.E,) array

children: list

list of arrays that represents the children any node

__init__(V, parents=None)

Constructor

Parameters:
Vint

the number of edges of the graph

parentsNone or (V,) array

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

returns the adjacency matrix of the graph as a sparse coo matrix

Returns:

that encodes the adjacency matrix of self

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 distances

Notes

By convention infinite distances are given the distance np.inf

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
check()

Check that self is indeed a forest, i.e. contains no loop

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

Notes

Slow implementation, might be rewritten in C or cython

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
compute_children()

Define the children of each node (stored in self.children)

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
define_graph_attributes()

define the edge and weights array

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
depth_from_leaves()

compute an index for each node: 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 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_children(v=-1)

Get the children of a node/each node

Parameters:
v: int, optional

a node index

Returns:
children: list of int the list of children of node v (if v is provided)

a list of lists of int, the children of all nodes otherwise

get_descendants(v, exclude_self=False)

returns the nodes that are children of v as a list

Parameters:
v: int, a node index
Returns:
desc: list of int, the list of all descendant of the input node
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

isleaf()

Identification of the leaves of the forest

Returns:
leaves: bool array of shape(self.V), indicator of the forest’s leaves
isroot()

Returns an indicator of nodes being roots

Returns:
roots, array of shape(self.V, bool), indicator of the forest’s roots
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

leaves_of_a_subtree(ids, custom=False)

tests whether the given nodes are the leaves of a certain subtree

Parameters:
ids: 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()

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)
merge_simple_branches()

Return a subforest, where chained branches are collapsed

Returns:
sf, Forest instance, same as self, without any chain
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

propagate_upward(label)

Propagation of a certain labelling from leaves to roots Assuming that label is a certain positive integer field 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 from leaves to roots some binary property of the nodes 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(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
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.

Returns:
order: array of shape(self.V)

the order of the old vertices in the reordered graph

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.

subforest(valid)

Creates a subforest with the vertices for which valid > 0

Parameters:
valid: array of shape (self.V): indicator of the selected nodes
Returns:
subforest: a new forest instance, with a reduced set of nodes

Notes

The children of deleted vertices become their own parent

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

tree_depth()

Returns the number of hierarchical levels in the tree

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