# 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

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

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

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