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
+1/-1 for ascending/descending links
- 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
- 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
- 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