Why Would It Be Impossible To Draw G With 3 Connected Components If G Had 66 Edges?
Tutorial¶
This guide can help yous starting time working with NetworkX.
Creating a graph¶
Create an empty graph with no nodes and no edges.
>>> import networkx as nx >>> G = nx . Graph () By definition, a Graph is a drove of nodes (vertices) along with identified pairs of nodes (called edges, links, etc). In NetworkX, nodes can be whatsoever hashable object due east.yard., a text string, an image, an XML object, another Graph, a customized node object, etc.
Notation
Python's None object is non allowed to exist used every bit a node. It determines whether optional function arguments accept been assigned in many functions. And information technology can exist used as a sentinel object meaning "not a node".
Nodes¶
The graph G tin can be grown in several ways. NetworkX includes many graph generator functions and facilities to read and write graphs in many formats. To go started though we'll look at uncomplicated manipulations. Yous can add one node at a time,
or add together nodes from whatsoever iterable container, such as a list
>>> G . add_nodes_from ([ 2 , 3 ]) You can besides add together nodes forth with node attributes if your container yields two-tuples of the form (node, node_attribute_dict) :
>>> G . add_nodes_from ([ ... ( 4 , { "color" : "red" }), ... ( five , { "color" : "green" }), ... ]) Node attributes are discussed further below.
Nodes from ane graph can be incorporated into some other:
>>> H = nx . path_graph ( 10 ) >>> Yard . add_nodes_from ( H ) G now contains the nodes of H equally nodes of Chiliad . In dissimilarity, yous could use the graph H as a node in G .
The graph K at present contains H every bit a node. This flexibility is very powerful as it allows graphs of graphs, graphs of files, graphs of functions and much more than. It is worth thinking about how to structure your application so that the nodes are useful entities. Of course you tin can ever use a unique identifier in Yard and have a split dictionary keyed past identifier to the node data if you lot prefer.
Note
You lot should non alter the node object if the hash depends on its contents.
Edges¶
G tin can also be grown by calculation 1 border at a fourth dimension,
>>> G . add_edge ( ane , ii ) >>> e = ( 2 , 3 ) >>> G . add_edge ( * e ) # unpack edge tuple* by adding a listing of edges,
>>> Grand . add_edges_from ([( ane , 2 ), ( i , 3 )]) or by adding any ebunch of edges. An ebunch is whatever iterable container of edge-tuples. An edge-tuple can be a 2-tuple of nodes or a three-tuple with ii nodes followed by an edge aspect dictionary, e.thou., (two, 3, {'weight': 3.1415}) . Edge attributes are discussed further below.
>>> K . add_edges_from ( H . edges ) There are no complaints when adding existing nodes or edges. For case, later removing all nodes and edges,
we add new nodes/edges and NetworkX quietly ignores any that are already present.
>>> Thousand . add_edges_from ([( 1 , 2 ), ( ane , 3 )]) >>> G . add_node ( 1 ) >>> 1000 . add_edge ( i , 2 ) >>> 1000 . add_node ( "spam" ) # adds node "spam" >>> G . add_nodes_from ( "spam" ) # adds 4 nodes: 'south', 'p', 'a', 'm' >>> Thou . add_edge ( 3 , 'one thousand' ) At this stage the graph G consists of eight nodes and 3 edges, equally can be seen by:
>>> G . number_of_nodes () 8 >>> G . number_of_edges () 3 Note
The order of adjacency reporting (e.g., G.adj, One thousand.successors, Yard.predecessors) is the order of edge addition. Yet, the order of Chiliad.edges is the order of the adjacencies which includes both the social club of the nodes and each node's adjacencies. Meet instance below:
>>> DG = nx . DiGraph () >>> DG . add_edge ( two , i ) # adds the nodes in order 2, 1 >>> DG . add_edge ( 1 , 3 ) >>> DG . add_edge ( 2 , 4 ) >>> DG . add_edge ( ane , 2 ) >>> assert list ( DG . successors ( 2 )) == [ 1 , four ] >>> affirm list ( DG . edges ) == [( 2 , i ), ( 2 , 4 ), ( 1 , 3 ), ( ane , two )] Examining elements of a graph¶
We can examine the nodes and edges. 4 bones graph properties facilitate reporting: G.nodes , Yard.edges , 1000.adj and M.degree . These are set-similar views of the nodes, edges, neighbors (adjacencies), and degrees of nodes in a graph. They offer a continually updated read-just view into the graph structure. They are also dict-like in that you can look up node and edge data attributes via the views and iterate with data attributes using methods .items() , .data('bridge') . If you desire a specific container type instead of a view, you lot can specify one. Here we use lists, though sets, dicts, tuples and other containers may be ameliorate in other contexts.
>>> list ( G . nodes ) [ane, ii, 3, 'spam', 'due south', 'p', 'a', 'thou'] >>> listing ( G . edges ) [(1, 2), (ane, 3), (3, 'm')] >>> listing ( G . adj [ i ]) # or list(G.neighbors(1)) [2, 3] >>> Thou . caste [ 1 ] # the number of edges incident to 1 two One tin can specify to report the edges and caste from a subset of all nodes using an nbunch. An nbunch is any of: None (meaning all nodes), a node, or an iterable container of nodes that is not itself a node in the graph.
>>> 1000 . edges ([ 2 , 'thou' ]) EdgeDataView([(two, 1), ('m', 3)]) >>> Thousand . degree ([ ii , iii ]) DegreeView({2: 1, three: 2}) Using the graph constructors¶
Graph objects practice not have to exist built upwardly incrementally - data specifying graph construction can be passed directly to the constructors of the various graph classes. When creating a graph structure by instantiating one of the graph classes you can specify data in several formats.
>>> G . add_edge ( i , ii ) >>> H = nx . DiGraph ( G ) # create a DiGraph using the connections from Thou >>> listing ( H . edges ()) [(1, 2), (2, 1)] >>> edgelist = [( 0 , 1 ), ( 1 , 2 ), ( two , 3 )] >>> H = nx . Graph ( edgelist ) What to apply as nodes and edges¶
You lot might discover that nodes and edges are not specified as NetworkX objects. This leaves yous gratuitous to use meaningful items as nodes and edges. The most mutual choices are numbers or strings, but a node can exist any hashable object (except None ), and an edge can be associated with whatever object 10 using G.add_edge(n1, n2, object=x) .
As an example, n1 and n2 could be poly peptide objects from the RCSB Protein Data Banking concern, and x could refer to an XML record of publications detailing experimental observations of their interaction.
Nosotros take found this ability quite useful, but its abuse tin lead to surprising beliefs unless i is familiar with Python. If in uncertainty, consider using convert_node_labels_to_integers() to obtain a more traditional graph with integer labels.
Accessing edges and neighbors¶
In addition to the views Graph.edges , and Graph.adj , admission to edges and neighbors is possible using subscript annotation.
>>> G = nx . Graph ([( 1 , two , { "color" : "yellow" })]) >>> G [ 1 ] # same as G.adj[i] AtlasView({2: {'color': 'yellow'}}) >>> Grand [ 1 ][ ii ] {'color': 'yellow'} >>> G . edges [ 1 , ii ] {'color': 'yellow'} You tin can become/ready the attributes of an border using subscript annotation if the border already exists.
>>> G . add_edge ( ane , 3 ) >>> G [ 1 ][ three ][ 'color' ] = "bluish" >>> G . edges [ ane , ii ][ 'color' ] = "red" >>> 1000 . edges [ 1 , 2 ] {'color': 'red'} Fast examination of all (node, adjacency) pairs is achieved using G.adjacency() , or G.adj.items() . Note that for undirected graphs, adjacency iteration sees each edge twice.
>>> FG = nx . Graph () >>> FG . add_weighted_edges_from ([( 1 , 2 , 0.125 ), ( one , 3 , 0.75 ), ( 2 , four , 1.two ), ( three , 4 , 0.375 )]) >>> for due north , nbrs in FG . adj . items (): ... for nbr , eattr in nbrs . items (): ... wt = eattr [ 'weight' ] ... if wt < 0.five : print ( f "( { north } , { nbr } , { wt : .3 } )" ) (1, 2, 0.125) (two, 1, 0.125) (iii, 4, 0.375) (four, 3, 0.375) User-friendly admission to all edges is accomplished with the edges belongings.
>>> for ( u , 5 , wt ) in FG . edges . information ( 'weight' ): ... if wt < 0.five : ... print ( f "( { u } , { v } , { wt : .3 } )" ) (1, two, 0.125) (3, 4, 0.375) Adding attributes to graphs, nodes, and edges¶
Attributes such as weights, labels, colors, or whatsoever Python object yous similar, tin exist attached to graphs, nodes, or edges.
Each graph, node, and border can concur key/value attribute pairs in an associated attribute dictionary (the keys must be hashable). By default these are empty, but attributes can be added or changed using add_edge , add_node or direct manipulation of the attribute dictionaries named G.graph , Thou.nodes , and G.edges for a graph G .
Graph attributes¶
Assign graph attributes when creating a new graph
>>> G = nx . Graph ( 24-hour interval = "Friday" ) >>> G . graph {'day': 'Friday'} Or you lot can alter attributes later
>>> G . graph [ 'twenty-four hour period' ] = "Monday" >>> G . graph {'day': 'Mon'} Node attributes¶
Add node attributes using add_node() , add_nodes_from() , or G.nodes
>>> G . add_node ( 1 , time = '5pm' ) >>> K . add_nodes_from ([ three ], time = '2pm' ) >>> G . nodes [ i ] {'time': '5pm'} >>> G . nodes [ one ][ 'room' ] = 714 >>> G . nodes . information () NodeDataView({1: {'time': '5pm', 'room': 714}, iii: {'time': '2pm'}}) Note that adding a node to G.nodes does not add it to the graph, use G.add_node() to add new nodes. Similarly for edges.
Edge Attributes¶
Add/change edge attributes using add_edge() , add_edges_from() , or subscript annotation.
>>> G . add_edge ( 1 , two , weight = 4.7 ) >>> 1000 . add_edges_from ([( 3 , 4 ), ( 4 , v )], color = 'carmine' ) >>> G . add_edges_from ([( 1 , 2 , { 'color' : 'bluish' }), ( 2 , 3 , { 'weight' : 8 })]) >>> Thou [ 1 ][ ii ][ 'weight' ] = four.seven >>> G . edges [ 3 , 4 ][ 'weight' ] = 4.two The special attribute weight should exist numeric as it is used by algorithms requiring weighted edges.
Directed graphs¶
The DiGraph class provides additional methods and properties specific to directed edges, e.yard., DiGraph.out_edges , DiGraph.in_degree , DiGraph.predecessors() , DiGraph.successors() etc. To allow algorithms to work with both classes easily, the directed versions of neighbors() is equivalent to successors() while degree reports the sum of in_degree and out_degree even though that may feel inconsistent at times.
>>> DG = nx . DiGraph () >>> DG . add_weighted_edges_from ([( ane , 2 , 0.v ), ( three , one , 0.75 )]) >>> DG . out_degree ( ane , weight = 'weight' ) 0.five >>> DG . degree ( 1 , weight = 'weight' ) 1.25 >>> list ( DG . successors ( 1 )) [2] >>> list ( DG . neighbors ( ane )) [2] Some algorithms work only for directed graphs and others are not well defined for directed graphs. Indeed the tendency to lump directed and undirected graphs together is dangerous. If you want to treat a directed graph as undirected for some measurement you should probably convert information technology using Graph.to_undirected() or with
>>> H = nx . Graph ( G ) # create an undirected graph H from a directed graph G Multigraphs¶
NetworkX provides classes for graphs which allow multiple edges betwixt any pair of nodes. The MultiGraph and MultiDiGraph classes allow you to add the same edge twice, possibly with different edge data. This tin be powerful for some applications, but many algorithms are not well divers on such graphs. Where results are well defined, eastward.k., MultiGraph.degree() we provide the part. Otherwise you should convert to a standard graph in a way that makes the measurement well defined.
>>> MG = nx . MultiGraph () >>> MG . add_weighted_edges_from ([( 1 , 2 , 0.v ), ( one , two , 0.75 ), ( 2 , three , 0.5 )]) >>> dict ( MG . degree ( weight = 'weight' )) {1: 1.25, 2: 1.75, iii: 0.5} >>> GG = nx . Graph () >>> for n , nbrs in MG . adjacency (): ... for nbr , edict in nbrs . items (): ... minvalue = min ([ d [ 'weight' ] for d in edict . values ()]) ... GG . add_edge ( n , nbr , weight = minvalue ) ... >>> nx . shortest_path ( GG , i , three ) [i, 2, 3] Graph generators and graph operations¶
In addition to amalgam graphs node-by-node or edge-past-edge, they tin also be generated by
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Applying classic graph operations, such as:
| | Returns the subgraph induced on nodes in nbunch. |
| | Return the union of graphs G and H. |
| | Return the disjoint union of graphs G and H. |
| | Returns the Cartesian product of G and H. |
| | Returns a new graph of Yard equanimous with H. |
| | Returns the graph complement of Yard. |
| | Returns a copy of the graph Grand with all of the edges removed. |
| | Returns an undirected view of the graph |
| | Returns a directed view of the graph |
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Using a call to one of the archetype small graphs, e.g.,
| | Returns the Petersen graph. |
| | Returns the Tutte graph. |
| | Render a small maze with a cycle. |
| | Render the iii-regular Platonic Tetrahedral graph. |
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Using a (effective) generator for a archetype graph, e.m.,
| | Return the complete graph |
| | Returns the complete bipartite graph |
| | Returns the Barbell Graph: ii complete graphs continued by a path. |
| | Returns the Lollipop Graph; |
like so:
>>> K_5 = nx . complete_graph ( v ) >>> K_3_5 = nx . complete_bipartite_graph ( 3 , v ) >>> barbell = nx . barbell_graph ( 10 , ten ) >>> lollipop = nx . lollipop_graph ( 10 , 20 ) -
Using a stochastic graph generator, e.chiliad,
| | Returns a \(G_{n,p}\) random graph, also known as an Erdős-Rényi graph or a binomial graph. |
| | Returns a Watts–Strogatz small-earth graph. |
| | Returns a random graph using Barabási–Albert preferential attachment |
| | Returns a random lobster graph. |
like and so:
>>> er = nx . erdos_renyi_graph ( 100 , 0.15 ) >>> ws = nx . watts_strogatz_graph ( 30 , three , 0.1 ) >>> ba = nx . barabasi_albert_graph ( 100 , v ) >>> scarlet = nx . random_lobster ( 100 , 0.ix , 0.ix ) -
Reading a graph stored in a file using common graph formats, such as edge lists, adjacency lists, GML, GraphML, pickle, LEDA and others.
>>> nx . write_gml ( red , "path.to.file" ) >>> mygraph = nx . read_gml ( "path.to.file" ) For details on graph formats run into Reading and writing graphs and for graph generator functions see Graph generators
Analyzing graphs¶
The structure of G can be analyzed using various graph-theoretic functions such as:
>>> G = nx . Graph () >>> G . add_edges_from ([( 1 , ii ), ( ane , 3 )]) >>> G . add_node ( "spam" ) # adds node "spam" >>> list ( nx . connected_components ( G )) [{1, 2, 3}, {'spam'}] >>> sorted ( d for n , d in G . degree ()) [0, ane, 1, 2] >>> nx . clustering ( G ) {1: 0, two: 0, iii: 0, 'spam': 0} Some functions with large output iterate over (node, value) 2-tuples. These are easily stored in a dict structure if yous desire.
>>> sp = dict ( nx . all_pairs_shortest_path ( G )) >>> sp [ iii ] {three: [3], 1: [3, one], 2: [iii, one, two]} Run into Algorithms for details on graph algorithms supported.
Drawing graphs¶
NetworkX is non primarily a graph cartoon package but bones cartoon with Matplotlib also as an interface to use the open source Graphviz software package are included. These are part of the networkx.drawing module and will exist imported if possible.
First import Matplotlib's plot interface (pylab works likewise)
>>> import matplotlib.pyplot every bit plt To examination if the import of networkx.drawing was successful draw M using i of
>>> M = nx . petersen_graph () >>> subax1 = plt . subplot ( 121 ) >>> nx . draw ( G , with_labels = Truthful , font_weight = 'bold' ) >>> subax2 = plt . subplot ( 122 ) >>> nx . draw_shell ( Yard , nlist = [ range ( 5 , 10 ), range ( 5 )], with_labels = Truthful , font_weight = 'assuming' ) (png, hires.png, pdf)
when drawing to an interactive display. Note that you may need to upshot a Matplotlib
command if you are non using matplotlib in interactive fashion (meet this Matplotlib FAQ).
>>> options = { ... 'node_color' : 'black' , ... 'node_size' : 100 , ... 'width' : iii , ... } >>> subax1 = plt . subplot ( 221 ) >>> nx . draw_random ( G , ** options ) >>> subax2 = plt . subplot ( 222 ) >>> nx . draw_circular ( G , ** options ) >>> subax3 = plt . subplot ( 223 ) >>> nx . draw_spectral ( Thou , ** options ) >>> subax4 = plt . subplot ( 224 ) >>> nx . draw_shell ( Thousand , nlist = [ range ( 5 , x ), range ( five )], ** options ) (png, hires.png, pdf)
You lot tin find boosted options via draw_networkx() and layouts via layout . Yous tin can utilize multiple shells with draw_shell() .
>>> G = nx . dodecahedral_graph () >>> shells = [[ 2 , three , four , 5 , half dozen ], [ viii , 1 , 0 , 19 , 18 , 17 , 16 , 15 , 14 , vii ], [ 9 , 10 , 11 , 12 , 13 ]] >>> nx . draw_shell ( M , nlist = shells , ** options ) (png, hires.png, pdf)
To save drawings to a file, employ, for case
>>> nx . draw ( G ) >>> plt . savefig ( "path.png" ) writes to the file path.png in the local directory. If Graphviz and PyGraphviz or pydot, are available on your organisation, y'all can also use nx_agraph.graphviz_layout(G) or nx_pydot.graphviz_layout(One thousand) to go the node positions, or write the graph in dot format for further processing.
>>> from networkx.drawing.nx_pydot import write_dot >>> pos = nx . nx_agraph . graphviz_layout ( G ) >>> nx . draw ( M , pos = pos ) >>> write_dot ( Chiliad , 'file.dot' ) Meet Drawing for additional details.
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Source: https://networkx.org/documentation/stable/tutorial.html
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