Quick start using graphtool¶
The graph_tool
module provides a Graph
class
and several algorithms that operate on it. The internals of this class,
and of most algorithms, are written in C++ for performance, using the
Boost Graph Library.
The module must be of course imported before it can be used. The package is subdivided into several submodules. To import everything from all of them, one can do:
>>> from graph_tool.all import *
In the following, it will always be assumed that the previous line was run.
Creating and manipulating graphs¶
An empty graph can be created by instantiating a Graph
class:
>>> g = Graph()
By default, newly created graphs are always directed. To construct undirected
graphs, one must pass a value to the directed
parameter:
>>> ug = Graph(directed=False)
A graph can always be switched onthefly from directed to undirected
(and vice versa), with the set_directed()
method. The “directedness” of the graph can be queried with the
is_directed()
method,
>>> ug = Graph()
>>> ug.set_directed(False)
>>> assert ug.is_directed() == False
A graph can also be created by providing another graph, in which case the entire graph (and its internal property maps, see Property maps) is copied:
>>> g1 = Graph()
>>> # ... construct g1 ...
>>> g2 = Graph(g1) # g1 and g2 are copies
Above, g2
is a “deep” copy of g1
, i.e. any modification of
g2
will not affect g1
.
Once a graph is created, it can be populated with vertices and edges. A
vertex can be added with the add_vertex()
method, which returns an instance of a Vertex
class, also called a vertex descriptor. For instance, the following
code creates two vertices, and returns vertex descriptors stored in the
variables v1
and v2
.
>>> v1 = g.add_vertex()
>>> v2 = g.add_vertex()
Edges can be added in an analogous manner, by calling the
add_edge()
method, which returns an edge
descriptor (an instance of the Edge
class):
>>> e = g.add_edge(v1, v2)
The above code creates a directed edge from v1
to v2
. We can
visualize the graph we created so far with the
graph_draw()
function.
>>> graph_draw(g, vertex_text=g.vertex_index, output="twonodes.pdf")
<...>
With vertex and edge descriptors, one can examine and manipulate the
graph in an arbitrary manner. For instance, in order to obtain the
outdegree of a vertex, we can simply call the
out_degree()
method:
>>> print(v1.out_degree())
1
Analogously, we could have used the in_degree()
method to query the indegree.
Note
For undirected graphs, the “outdegree” is synonym for degree, and in this case the indegree of a vertex is always zero.
Edge descriptors have two useful methods, source()
and target()
, which return the source and target
vertex of an edge, respectively.
>>> print(e.source(), e.target())
0 1
The add_vertex()
method also accepts an optional
parameter which specifies the number of vertices to create. If this
value is greater than 1, it returns an iterator on the added vertex
descriptors:
>>> vlist = g.add_vertex(10)
>>> print(len(list(vlist)))
10
Each vertex in a graph has an unique index, which is always between
\(0\) and \(N1\), where \(N\) is the number of
vertices. This index can be obtained by using the
vertex_index
attribute of the graph (which is
a property map, see Property maps), or by converting the
vertex descriptor to an int
.
>>> v = g.add_vertex()
>>> print(g.vertex_index[v])
12
>>> print(int(v))
12
Edges and vertices can also be removed at any time with the
remove_vertex()
and remove_edge()
methods,
>>> g.remove_edge(e) # e no longer exists
>>> g.remove_vertex(v2) # the second vertex is also gone
Note
Removing a vertex is typically an \(O(N)\) operation. The
vertices are internally stored in a STL vector, so removing an
element somewhere in the middle of the list requires the shifting of
the rest of the list. Thus, fast \(O(1)\) removals are only
possible either if one can guarantee that only vertices in the end of
the list are removed (the ones last added to the graph), or if the
relative vertex ordering is invalidated. The latter behavior can be
achieved by passing the option fast == True
, to
remove_vertex()
, which causes the vertex
being deleted to be ‘swapped’ with the last vertex (i.e. with the
largest index), which will in turn inherit the index of the vertex
being deleted.
Warning
Because of the above, removing a vertex with an index smaller than
\(N1\) will invalidate either the last (fast = True
)
or all (fast = False
) descriptors pointing to vertices with
higher index.
As a consequence, if more than one vertex is to be removed at a given time, they should always be removed in decreasing index order:
# 'del_list' is a list of vertex descriptors
for v in reversed(sorted(del_list)):
g.remove_vertex(v)
Alternatively (and preferably), a list (or any iterable) may be
passed directly as the vertex
parameter of the
remove_vertex()
function, and the above is
performed internally (in C++).
Note that property map values (see Property maps) are unaffected by the index changes due to vertex removal, as they are modified accordingly by the library.
Note
Removing an edge is an \(O(k_{s} + k_{t})\) operation, where
\(k_{s}\) is the outdegree of the source vertex, and
\(k_{t}\) is the indegree of the target vertex. This can be made
faster by setting set_fast_edge_removal()
to
True, in which case it becomes \(O(1)\), at the expense of
additional data of size \(O(E)\).
No edge descriptors are ever invalidated after edge removal, with the exception of the edge being removed.
Since vertices are uniquely identifiable by their indexes, there is no
need to keep the vertex descriptor lying around to access them at a
later point. If we know its index, we can obtain the descriptor of a
vertex with a given index using the vertex()
method,
>>> v = g.vertex(8)
which takes an index, and returns a vertex descriptor. Edges cannot be
directly obtained by its index, but if the source and target vertices of
a given edge are known, it can be retrieved with the
edge()
method
>>> g.add_edge(g.vertex(2), g.vertex(3))
<...>
>>> e = g.edge(2, 3)
Another way to obtain edge or vertex descriptors is to iterate through them, as described in section Iterating over vertices and edges. This is in fact the most useful way of obtaining vertex and edge descriptors.
Like vertices, edges also have unique indexes, which are given by the
edge_index
property:
>>> e = g.add_edge(g.vertex(0), g.vertex(1))
>>> print(g.edge_index[e])
1
Differently from vertices, edge indexes do not necessarily conform to any specific range. If no edges are ever removed, the indexes will be in the range \([0, E1]\), where \(E\) is the number of edges, and edges added earlier have lower indexes. However if an edge is removed, its index will be “vacant”, and the remaining indexes will be left unmodified, and thus will not all lie in the range \([0, E1]\). If a new edge is added, it will reuse old indexes, in an increasing order.
Iterating over vertices and edges¶
Algorithms must often iterate through vertices, edges, outedges of a
vertex, etc. The Graph
and
Vertex
classes provide different types of iterators
for doing so. The iterators always point to edge or vertex descriptors.
Iterating over all vertices or edges¶
In order to iterate through all the vertices or edges of a graph, the
vertices()
and edges()
methods should be used:
for v in g.vertices():
print(v)
for e in g.edges():
print(e)
The code above will print the vertices and edges of the graph in the order they are found.
Iterating over the neighborhood of a vertex¶
The out and inedges of a vertex, as well as the out and inneighbors can be
iterated through with the out_edges()
,
in_edges()
, out_neighbors()
and in_neighbors()
methods, respectively.
for v in g.vertices():
for e in v.out_edges():
print(e)
for w in v.out_neighbors():
print(w)
# the edge and neighbors order always match
for e, w in zip(v.out_edges(), v.out_neighbors()):
assert e.target() == w
The code above will print the outedges and outneighbors of all vertices in the graph.
Warning
You should never remove vertex or edge descriptors when iterating over them, since this invalidates the iterators. If you plan to remove vertices or edges during iteration, you must first store them somewhere (such as in a list) and remove them only after no iterator is being used. Removal during iteration will cause bad things to happen.
Faster iteration over vertices and edges without descriptors¶
The mode of iteration considered above is convenient, but requires the
creation of vertex and edge descriptor objects, which incurs a
performance overhead. A faster approach involves the use of the methods
iter_vertices()
,
iter_edges()
,
iter_out_edges()
,
iter_in_edges()
,
iter_all_edges()
,
iter_out_neighbors()
,
iter_in_neighbors()
,
iter_all_neighbors()
, which return vertex
indexes and pairs thereof, instead of descriptors objects, to specify
vertex and edges, respectively.
The equivalent of the above examples can be obtained as:
for v in g.iter_vertices():
print(v)
for e in g.iter_edges():
print(e)
and likewise for the iteration over the neighborhood of a vertex:
for v in g.iter_vertices():
for e in g.iter_out_edges(v):
print(e)
for w in g.iter_out_neighbors(v):
print(w)
Even faster iteration over vertices and edges using arrays¶
While more convenient, looping over the graph as described in the
previous sections are not the most efficient approaches. This is because
the loops are performed in pure Python, and hence it undermines the main
feature of the library, which is the offloading of loops from Python to
C++. Following the numpy
philosophy, graph_tool
also
provides an arraybased interface that avoids loops in Python. This is
done with the get_vertices()
,
get_edges()
,
get_out_edges()
,
get_in_edges()
,
get_all_edges()
,
get_out_neighbors()
,
get_in_neighbors()
,
get_all_neighbors()
,
get_out_degrees()
,
get_in_degrees()
and
get_total_degrees()
methods, which return
numpy.ndarray
instances instead of iterators.
For example, using this interface we can get the outdegree of each node via:
print(g.get_out_degrees(g.get_vertices()))
[1 0 1 0 0 0 0 0 0 0 0 0]
or the sum of the product of the in and outdegrees of the endpoints of each edge with:
edges = g.get_edges()
in_degs = g.get_in_degrees(g.get_vertices())
out_degs = g.get_out_degrees(g.get_vertices())
print((out_degs[edges[:,0]] * in_degs[edges[:,1]]).sum())
2
Property maps¶
Property maps are a way of associating additional information to the
vertices, edges or to the graph itself. There are thus three types of
property maps: vertex, edge and graph. They are handled by the
classes VertexPropertyMap
,
EdgePropertyMap
, and
GraphPropertyMap
. Each created property map has an
associated value type, which must be chosen from the predefined set:
Type name 
Alias 































New property maps can be created for a given graph by calling one of the
methods new_vertex_property()
(alias
new_vp()
),
new_edge_property()
(alias
new_ep()
), or
new_graph_property()
(alias
new_gp()
), for each map type. The values are
then accessed by vertex or edge descriptors, or the graph itself, as
such:
from numpy.random import randint
g = Graph()
g.add_vertex(100)
# insert some random links
for s,t in zip(randint(0, 100, 100), randint(0, 100, 100)):
g.add_edge(g.vertex(s), g.vertex(t))
vprop_double = g.new_vertex_property("double") # Doubleprecision floating point
v = g.vertex(10)
vprop_double[v] = 3.1416
vprop_vint = g.new_vertex_property("vector<int>") # Vector of ints
v = g.vertex(40)
vprop_vint[v] = [1, 3, 42, 54]
eprop_dict = g.new_edge_property("object") # Arbitrary Python object.
e = g.edges().next()
eprop_dict[e] = {"foo": "bar", "gnu": 42} # In this case, a dict.
gprop_bool = g.new_graph_property("bool") # Boolean
gprop_bool[g] = True
Property maps with scalar value types can also be accessed as a
numpy.ndarray
, with the
get_array()
method, or the
a
attribute, e.g.,
from numpy.random import random
# this assigns random values to the vertex properties
vprop_double.get_array()[:] = random(g.num_vertices())
# or more conveniently (this is equivalent to the above)
vprop_double.a = random(g.num_vertices())
Internal property maps¶
Any created property map can be made “internal” to the corresponding
graph. This means that it will be copied and saved to a file together
with the graph. Properties are internalized by including them in the
graph’s dictionarylike attributes
vertex_properties
,
edge_properties
or
graph_properties
(or their aliases,
vp
, ep
or
gp
, respectively). When inserted in the graph,
the property maps must have an unique name (between those of the same
type):
>>> eprop = g.new_edge_property("string")
>>> g.edge_properties["some name"] = eprop
>>> g.list_properties()
some name (edge) (type: string)
Internal graph property maps behave slightly differently. Instead of returning the property map object, the value itself is returned from the dictionaries:
>>> gprop = g.new_graph_property("int")
>>> g.graph_properties["foo"] = gprop # this sets the actual property map
>>> g.graph_properties["foo"] = 42 # this sets its value
>>> print(g.graph_properties["foo"])
42
>>> del g.graph_properties["foo"] # the property map entry is deleted from the dictionary
For convenience, the internal property maps can also be accessed via attributes:
>>> vprop = g.new_vertex_property("double")
>>> g.vp.foo = vprop # equivalent to g.vertex_properties["foo"] = vprop
>>> v = g.vertex(0)
>>> g.vp.foo[v] = 3.14
>>> print(g.vp.foo[v])
3.14
Graph I/O¶
Graphs can be saved and loaded in four formats: graphml, dot, gml
and a custom binary format gt
(see The gt file format).
Warning
The binary format gt
and the textbased graphml
are the
preferred formats, since they are by far the most complete. Both
these formats are equally complete, but the gt
format is faster
and requires less storage.
The dot
and gml
formats are fully supported, but since they
contain no precise type information, all properties are read as
strings (or also as double, in the case of gml
), and must be
converted by hand to the desired type. Therefore you should always
use either gt
or graphml
, since they implement an exact
bitforbit representation of all supported Property maps
types, except when interfacing with other software, or existing
data, which uses dot
or gml
.
A graph can be saved or loaded to a file with the save
and load
methods, which take either a file name or a
filelike object. A graph can also be loaded from disc with the
load_graph()
function, as such:
g = Graph()
# ... fill the graph ...
g.save("my_graph.xml.gz")
g2 = load_graph("my_graph.xml.gz")
# g and g2 should be copies of each other
Graph classes can also be pickled with the pickle
module.
An Example: Building a Price Network¶
A Price network is the first known model of a “scalefree” graph,
invented in 1976 by de Solla Price. It is defined
dynamically, where at each time step a new vertex is added to the graph,
and connected to an old vertex, with probability proportional to its
indegree. The following program implements this construction using
graphtool
.
Note
Note that it would be much faster just to use the
price_network()
function, which is
implemented in C++, as opposed to the script below which is in pure
Python. The code below is merely a demonstration on how to use the
library.
1#! /usr/bin/env python
2
3# We will need some things from several places
4from __future__ import division, absolute_import, print_function
5import sys
6if sys.version_info < (3,):
7 range = xrange
8import os
9from pylab import * # for plotting
10from numpy.random import * # for random sampling
11seed(42)
12
13# We need to import the graph_tool module itself
14from graph_tool.all import *
15
16# let's construct a Price network (the one that existed before Barabasi). It is
17# a directed network, with preferential attachment. The algorithm below is
18# very naive, and a bit slow, but quite simple.
19
20# We start with an empty, directed graph
21g = Graph()
22
23# We want also to keep the age information for each vertex and edge. For that
24# let's create some property maps
25v_age = g.new_vertex_property("int")
26e_age = g.new_edge_property("int")
27
28# The final size of the network
29N = 100000
30
31# We have to start with one vertex
32v = g.add_vertex()
33v_age[v] = 0
34
35# we will keep a list of the vertices. The number of times a vertex is in this
36# list will give the probability of it being selected.
37vlist = [v]
38
39# let's now add the new edges and vertices
40for i in range(1, N):
41 # create our new vertex
42 v = g.add_vertex()
43 v_age[v] = i
44
45 # we need to sample a new vertex to be the target, based on its indegree +
46 # 1. For that, we simply randomly sample it from vlist.
47 i = randint(0, len(vlist))
48 target = vlist[i]
49
50 # add edge
51 e = g.add_edge(v, target)
52 e_age[e] = i
53
54 # put v and target in the list
55 vlist.append(target)
56 vlist.append(v)
57
58# now we have a graph!
59
60# let's do a random walk on the graph and print the age of the vertices we find,
61# just for fun.
62
63v = g.vertex(randint(0, g.num_vertices()))
64while True:
65 print("vertex:", int(v), "indegree:", v.in_degree(), "outdegree:",
66 v.out_degree(), "age:", v_age[v])
67
68 if v.out_degree() == 0:
69 print("Nowhere else to go... We found the main hub!")
70 break
71
72 n_list = []
73 for w in v.out_neighbors():
74 n_list.append(w)
75 v = n_list[randint(0, len(n_list))]
76
77# let's save our graph for posterity. We want to save the age properties as
78# well... To do this, they must become "internal" properties:
79
80g.vertex_properties["age"] = v_age
81g.edge_properties["age"] = e_age
82
83# now we can save it
84g.save("price.xml.gz")
85
86
87# Let's plot its indegree distribution
88in_hist = vertex_hist(g, "in")
89
90y = in_hist[0]
91err = sqrt(in_hist[0])
92err[err >= y] = y[err >= y]  1e2
93
94figure(figsize=(6,4))
95errorbar(in_hist[1][:1], in_hist[0], fmt="o", yerr=err,
96 label="in")
97gca().set_yscale("log")
98gca().set_xscale("log")
99gca().set_ylim(1e1, 1e5)
100gca().set_xlim(0.8, 1e3)
101subplots_adjust(left=0.2, bottom=0.2)
102xlabel("$k_{in}$")
103ylabel("$NP(k_{in})$")
104tight_layout()
105savefig("pricedegdist.pdf")
106savefig("pricedegdist.svg")
The following is what should happen when the program is run.
vertex: 36063 indegree: 0 outdegree: 1 age: 36063
vertex: 9075 indegree: 4 outdegree: 1 age: 9075
vertex: 5967 indegree: 3 outdegree: 1 age: 5967
vertex: 1113 indegree: 7 outdegree: 1 age: 1113
vertex: 25 indegree: 84 outdegree: 1 age: 25
vertex: 10 indegree: 541 outdegree: 1 age: 10
vertex: 5 indegree: 140 outdegree: 1 age: 5
vertex: 2 indegree: 459 outdegree: 1 age: 2
vertex: 1 indegree: 520 outdegree: 1 age: 1
vertex: 0 indegree: 210 outdegree: 0 age: 0
Nowhere else to go... We found the main hub!
Below is the degree distribution, with \(10^5\) nodes (in order to the asymptotic behavior to be even clearer, the number of vertices needs to be increased to something like \(10^6\) or \(10^7\)).
We can draw the graph to see some other features of its topology. For that we
use the graph_draw()
function.
g = load_graph("price.xml.gz")
age = g.vertex_properties["age"]
pos = sfdp_layout(g)
graph_draw(g, pos, output_size=(1000, 1000), vertex_color=[1,1,1,0],
vertex_fill_color=age, vertex_size=1, edge_pen_width=1.2,
vcmap=matplotlib.cm.gist_heat_r, output="price.pdf")
Graph filtering¶
One of the very nice features of graphtool
is the “onthefly” filtering of
edges and/or vertices. Filtering means the temporary masking of vertices/edges,
which are in fact not really removed, and can be easily recovered. Vertices or
edges which are to be filtered should be marked with a
PropertyMap
with value type bool
, and then set with
set_vertex_filter()
or
set_edge_filter()
methods. By default, vertex or edges
with value “1” are kept in the graphs, and those with value “0” are filtered
out. This behaviour can be modified with the inverted
parameter of the
respective functions. All manipulation functions and algorithms will work as if
the marked edges or vertices were removed from the graph, with minimum overhead.
Note
It is important to emphasize that the filtering functionality does not add any overhead when the graph is not being filtered. In this case, the algorithms run just as fast as if the filtering functionality didn’t exist.
Here is an example which obtains the minimum spanning tree of a graph,
using the function min_spanning_tree()
and
edge filtering.
g, pos = triangulation(random((500, 2)) * 4, type="delaunay")
tree = min_spanning_tree(g)
graph_draw(g, pos=pos, edge_color=tree, output="min_tree.svg")
The tree
property map has a bool type, with value “1” if the edge belongs to
the tree, and “0” otherwise. Below is an image of the original graph, with the
marked edges.
We can now filter out the edges which don’t belong to the minimum spanning tree.
g.set_edge_filter(tree)
graph_draw(g, pos=pos, output="min_tree_filtered.svg")
This is how the graph looks when filtered:
Everything should work transparently on the filtered graph, simply as if the
masked edges were removed. For instance, the following code will calculate the
betweenness()
centrality of the edges and vertices,
and draws them as colors and line thickness in the graph.
bv, be = betweenness(g)
be.a /= be.a.max() / 5
graph_draw(g, pos=pos, vertex_fill_color=bv, edge_pen_width=be,
output="filteredbt.svg")
The original graph can be recovered by setting the edge filter to None
.
g.set_edge_filter(None)
bv, be = betweenness(g)
be.a /= be.a.max() / 5
graph_draw(g, pos=pos, vertex_fill_color=bv, edge_pen_width=be,
output="nonfilteredbt.svg")
Everything works in analogous fashion with vertex filtering.
Additionally, the graph can also have its edges reversed with the
set_reversed()
method. This is also an \(O(1)\)
operation, which does not really modify the graph.
As mentioned previously, the directedness of the graph can also be changed
“onthefly” with the set_directed()
method.
Graph views¶
It is often desired to work with filtered and unfiltered graphs
simultaneously, or to temporarily create a filtered version of graph for
some specific task. For these purposes, graphtool provides a
GraphView
class, which represents a filtered “view”
of a graph, and behaves as an independent graph object, which shares the
underlying data with the original graph. Graph views are constructed by
instantiating a GraphView
class, and passing a
graph object which is supposed to be filtered, together with the desired
filter parameters. For example, to create a directed view of the graph
g
constructed above, one should do:
>>> ug = GraphView(g, directed=True)
>>> ug.is_directed()
True
Graph views also provide a much more direct and convenient approach to vertex/edge filtering: To construct a filtered minimum spanning tree like in the example above, one must only pass the filter property as the “efilt” parameter:
>>> tv = GraphView(g, efilt=tree)
Note that this is an \(O(1)\) operation, since it is equivalent (in
speed) to setting the filter in graph g
directly, but in this case
the object g
remains unmodified.
Like above, the result should be the isolated minimum spanning tree:
>>> bv, be = betweenness(tv)
>>> be.a /= be.a.max() / 5
>>> graph_draw(tv, pos=pos, vertex_fill_color=bv,
... edge_pen_width=be, output="mstview.svg")
<...>
Note
GraphView
objects behave exactly like regular
Graph
objects. In fact,
GraphView
is a subclass of
Graph
. The only difference is that a
GraphView
object shares its internal data with
its parent Graph
class. Therefore, if the
original Graph
object is modified, this
modification will be reflected immediately in the
GraphView
object, and vice versa.
For even more convenience, one can supply a function as filter
parameter, which takes a vertex or an edge as single parameter, and
returns True
if the vertex/edge should be kept and False
otherwise. For instance, if we want to keep only the most “central”
edges, we can do:
>>> bv, be = betweenness(g)
>>> u = GraphView(g, efilt=lambda e: be[e] > be.a.max() / 2)
This creates a graph view u
which contains only the edges of g
which have a normalized betweenness centrality larger than half of the
maximum value. Note that, differently from the case above, this is an
\(O(E)\) operation, where \(E\) is the number of edges, since
the supplied function must be called \(E\) times to construct a
filter property map. Thus, supplying a constructed filter map is always
faster, but supplying a function can be more convenient.
The graph view constructed above can be visualized as
>>> be.a /= be.a.max() / 5
>>> graph_draw(u, pos=pos, vertex_fill_color=bv, output="centraledgesview.svg")
<...>
Composing graph views¶
Since graph views are regular graphs, one can just as easily create graph views of graph views. This provides a convenient way of composing filters. For instance, in order to isolate the minimum spanning tree of all vertices of the example above which have a degree larger than four, one can do:
>>> u = GraphView(g, vfilt=lambda v: v.out_degree() > 4)
>>> tree = min_spanning_tree(u)
>>> u = GraphView(u, efilt=tree)
The resulting graph view can be visualized as
>>> graph_draw(u, pos=pos, output="composedfilter.svg")
<...>