egttools.utils.Graph

class Graph(incoming_graph_data=None, **attr)

Bases: object

Base class for undirected graphs.

A Graph stores nodes and edges with optional data, or attributes.

Graphs hold undirected edges. Self loops are allowed but multiple (parallel) edges are not.

Nodes can be arbitrary (hashable) Python objects with optional key/value attributes, except that None is not allowed as a node.

Edges are represented as links between nodes with optional key/value attributes.

Parameters:

• incoming_graph_data (input graph (optional, default: None)) – Data to initialize graph. If None (default) an empty graph is created. The data can be any format that is supported by the to_networkx_graph() function, currently including edge list, dict of dicts, dict of lists, NetworkX graph, 2D NumPy array, SciPy sparse matrix, or PyGraphviz graph.

• attr (keyword arguments, optional (default= no attributes)) – Attributes to add to graph as key=value pairs.

See also

DiGraph, MultiGraph, MultiDiGraph

Examples

Create an empty graph structure (a “null graph”) with no nodes and no edges.

>>> G = nx.Graph()

G can be grown in several ways.

Nodes:

Add one node at a time:

>>> G.add_node(1)

Add the nodes from any container (a list, dict, set or even the lines from a file or the nodes from another graph).

>>> G.add_nodes_from([2, 3])
>>> G.add_nodes_from(range(100, 110))
>>> H = nx.path_graph(10)
>>> G.add_nodes_from(H)

In addition to strings and integers any hashable Python object (except None) can represent a node, e.g. a customized node object, or even another Graph.

>>> G.add_node(H)

Edges:

G can also be grown by adding edges.

Add one edge,

>>> G.add_edge(1, 2)

a list of edges,

>>> G.add_edges_from([(1, 2), (1, 3)])

or a collection of edges,

>>> G.add_edges_from(H.edges)

If some edges connect nodes not yet in the graph, the nodes are added automatically. There are no errors when adding nodes or edges that already exist.

Attributes:

Each graph, node, and edge can hold key/value attribute pairs in an associated attribute dictionary (the keys must be hashable). By default these are empty, but can be added or changed using add_edge, add_node or direct manipulation of the attribute dictionaries named graph, node and edge respectively.

>>> G = nx.Graph(day="Friday")
>>> G.graph
{'day': 'Friday'}

Add node attributes using add_node(), add_nodes_from() or G.nodes

>>> G.add_node(1, time="5pm")
>>> G.add_nodes_from([3], time="2pm")
>>> G.nodes[1]
{'time': '5pm'}
>>> G.nodes[1]["room"] = 714  # node must exist already to use G.nodes
>>> del G.nodes[1]["room"]  # remove attribute
>>> list(G.nodes(data=True))
[(1, {'time': '5pm'}), (3, {'time': '2pm'})]

Add edge attributes using add_edge(), add_edges_from(), subscript notation, or G.edges.

>>> G.add_edge(1, 2, weight=4.7)
>>> G.add_edges_from([(3, 4), (4, 5)], color="red")
>>> G.add_edges_from([(1, 2, {"color": "blue"}), (2, 3, {"weight": 8})])
>>> G[1][2]["weight"] = 4.7
>>> G.edges[1, 2]["weight"] = 4

Warning: we protect the graph data structure by making G.edges a read-only dict-like structure. However, you can assign to attributes in e.g. G.edges[1, 2]. Thus, use 2 sets of brackets to add/change data attributes: G.edges[1, 2][‘weight’] = 4 (For multigraphs: MG.edges[u, v, key][name] = value).

Shortcuts:

Many common graph features allow python syntax to speed reporting.

>>> 1 in G  # check if node in graph
True
>>> [n for n in G if n < 3]  # iterate through nodes
[1, 2]
>>> len(G)  # number of nodes in graph
5

Often the best way to traverse all edges of a graph is via the neighbors. The neighbors are reported as an adjacency-dict G.adj or G.adjacency()

>>> for n, nbrsdict in G.adjacency():
...     for nbr, eattr in nbrsdict.items():
...         if "weight" in eattr:
...             # Do something useful with the edges
...             pass

But the edges() method is often more convenient:

>>> for u, v, weight in G.edges.data("weight"):
...     if weight is not None:
...         # Do something useful with the edges
...         pass

Reporting:

Simple graph information is obtained using object-attributes and methods. Reporting typically provides views instead of containers to reduce memory usage. The views update as the graph is updated similarly to dict-views. The objects nodes, edges and adj provide access to data attributes via lookup (e.g. nodes[n], edges[u, v], adj[u][v]) and iteration (e.g. nodes.items(), nodes.data(‘color’), nodes.data(‘color’, default=’blue’) and similarly for edges) Views exist for nodes, edges, neighbors()/adj and degree.

For details on these and other miscellaneous methods, see below.

Subclasses (Advanced):

The Graph class uses a dict-of-dict-of-dict data structure. The outer dict (node_dict) holds adjacency information keyed by node. The next dict (adjlist_dict) represents the adjacency information and holds edge data keyed by neighbor. The inner dict (edge_attr_dict) represents the edge data and holds edge attribute values keyed by attribute names.

Each of these three dicts can be replaced in a subclass by a user defined dict-like object. In general, the dict-like features should be maintained but extra features can be added. To replace one of the dicts create a new graph class by changing the class(!) variable holding the factory for that dict-like structure.

node_dict_factoryfunction, (default: dict)

Factory function to be used to create the dict containing node attributes, keyed by node id. It should require no arguments and return a dict-like object

node_attr_dict_factory: function, (default: dict)

Factory function to be used to create the node attribute dict which holds attribute values keyed by attribute name. It should require no arguments and return a dict-like object

adjlist_outer_dict_factoryfunction, (default: dict)

Factory function to be used to create the outer-most dict in the data structure that holds adjacency info keyed by node. It should require no arguments and return a dict-like object.

adjlist_inner_dict_factoryfunction, (default: dict)

Factory function to be used to create the adjacency list dict which holds edge data keyed by neighbor. It should require no arguments and return a dict-like object

edge_attr_dict_factoryfunction, (default: dict)

Factory function to be used to create the edge attribute dict which holds attribute values keyed by attribute name. It should require no arguments and return a dict-like object.

graph_attr_dict_factoryfunction, (default: dict)

Factory function to be used to create the graph attribute dict which holds attribute values keyed by attribute name. It should require no arguments and return a dict-like object.

Typically, if your extension doesn’t impact the data structure all methods will inherit without issue except: to_directed/to_undirected. By default these methods create a DiGraph/Graph class and you probably want them to create your extension of a DiGraph/Graph. To facilitate this we define two class variables that you can set in your subclass.

to_directed_classcallable, (default: DiGraph or MultiDiGraph)

Class to create a new graph structure in the to_directed method. If None, a NetworkX class (DiGraph or MultiDiGraph) is used.

to_undirected_classcallable, (default: Graph or MultiGraph)

Class to create a new graph structure in the to_undirected method. If None, a NetworkX class (Graph or MultiGraph) is used.

Subclassing Example

Create a low memory graph class that effectively disallows edge attributes by using a single attribute dict for all edges. This reduces the memory used, but you lose edge attributes.

>>> class ThinGraph(nx.Graph):
...     all_edge_dict = {"weight": 1}
...
...     def single_edge_dict(self):
...         return self.all_edge_dict
...
...     edge_attr_dict_factory = single_edge_dict
>>> G = ThinGraph()
>>> G.add_edge(2, 1)
>>> G[2][1]
{'weight': 1}
>>> G.add_edge(2, 2)
>>> G[2][1] is G[2][2]
True

Initialize a graph with edges, name, or graph attributes.

Parameters:

• incoming_graph_data (input graph (optional, default: None)) – Data to initialize graph. If None (default) an empty graph is created. The data can be an edge list, or any NetworkX graph object. If the corresponding optional Python packages are installed the data can also be a 2D NumPy array, a SciPy sparse array, or a PyGraphviz graph.

• attr (keyword arguments, optional (default= no attributes)) – Attributes to add to graph as key=value pairs.

See also

convert

Examples

>>> G = nx.Graph()  # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G = nx.Graph(name="my graph")
>>> e = [(1, 2), (2, 3), (3, 4)]  # list of edges
>>> G = nx.Graph(e)

Arbitrary graph attribute pairs (key=value) may be assigned

>>> G = nx.Graph(e, day="Friday")
>>> G.graph
{'day': 'Friday'}

Methods

add_edge	Add an edge between u and v.
add_edges_from	Add all the edges in ebunch_to_add.
add_node	Add a single node node_for_adding and update node attributes.
add_nodes_from	Add multiple nodes.
add_weighted_edges_from	Add weighted edges in ebunch_to_add with specified weight attr
adjacency	Returns an iterator over (node, adjacency dict) tuples for all nodes.
clear	Remove all nodes and edges from the graph.
clear_edges	Remove all edges from the graph without altering nodes.
copy	Returns a copy of the graph.
edge_subgraph	Returns the subgraph induced by the specified edges.
get_edge_data	Returns the attribute dictionary associated with edge (u, v).
has_edge	Returns True if the edge (u, v) is in the graph.
has_node	Returns True if the graph contains the node n.
is_directed	Returns True if graph is directed, False otherwise.
is_multigraph	Returns True if graph is a multigraph, False otherwise.
nbunch_iter	Returns an iterator over nodes contained in nbunch that are also in the graph.
neighbors	Returns an iterator over all neighbors of node n.
number_of_edges	Returns the number of edges between two nodes.
number_of_nodes	Returns the number of nodes in the graph.
order	Returns the number of nodes in the graph.
remove_edge	Remove the edge between u and v.
remove_edges_from	Remove all edges specified in ebunch.
remove_node	Remove node n.
remove_nodes_from	Remove multiple nodes.
size	Returns the number of edges or total of all edge weights.
subgraph	Returns a SubGraph view of the subgraph induced on nodes.
to_directed	Returns a directed representation of the graph.
to_directed_class	Returns the class to use for empty directed copies.
to_undirected	Returns an undirected copy of the graph.
to_undirected_class	Returns the class to use for empty undirected copies.
update	Update the graph using nodes/edges/graphs as input.

Attributes

adj	Graph adjacency object holding the neighbors of each node.
degree	A DegreeView for the Graph as G.degree or G.degree().
edges	An EdgeView of the Graph as G.edges or G.edges().
name	String identifier of the graph.
nodes	A NodeView of the Graph as G.nodes or G.nodes().

adjlist_inner_dict_factory

alias of dict

adjlist_outer_dict_factory

alias of dict

edge_attr_dict_factory

alias of dict

graph_attr_dict_factory

alias of dict

node_attr_dict_factory

alias of dict

node_dict_factory

alias of dict

__contains__(n)

Returns True if n is a node, False otherwise. Use: ‘n in G’.

Examples

>>> G = nx.path_graph(4)  # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> 1 in G
True

__getitem__(n)

Returns a dict of neighbors of node n. Use: ‘G[n]’.

Parameters:

n (node) – A node in the graph.

Returns:

adj_dict – The adjacency dictionary for nodes connected to n.

Return type:

dictionary

Notes

G[n] is the same as G.adj[n] and similar to G.neighbors(n) (which is an iterator over G.adj[n])

Examples

>>> G = nx.path_graph(4)  # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G[0]
AtlasView({1: {}})

__init__(incoming_graph_data=None, **attr)

Initialize a graph with edges, name, or graph attributes.

Parameters:

• incoming_graph_data (input graph (optional, default: None)) – Data to initialize graph. If None (default) an empty graph is created. The data can be an edge list, or any NetworkX graph object. If the corresponding optional Python packages are installed the data can also be a 2D NumPy array, a SciPy sparse array, or a PyGraphviz graph.

• attr (keyword arguments, optional (default= no attributes)) – Attributes to add to graph as key=value pairs.

See also

convert

Examples

>>> G = nx.Graph()  # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G = nx.Graph(name="my graph")
>>> e = [(1, 2), (2, 3), (3, 4)]  # list of edges
>>> G = nx.Graph(e)

Arbitrary graph attribute pairs (key=value) may be assigned

>>> G = nx.Graph(e, day="Friday")
>>> G.graph
{'day': 'Friday'}

__iter__()

Iterate over the nodes. Use: ‘for n in G’.

Returns:

niter – An iterator over all nodes in the graph.

Return type:

iterator

Examples

>>> G = nx.path_graph(4)  # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> [n for n in G]
[0, 1, 2, 3]
>>> list(G)
[0, 1, 2, 3]

__len__()

Returns the number of nodes in the graph. Use: ‘len(G)’.

Returns:

nnodes – The number of nodes in the graph.

Return type:

int

See also

number_of_nodes

identical method

order

identical method

Examples

>>> G = nx.path_graph(4)  # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> len(G)
4

__str__()

Returns a short summary of the graph.

Returns:

info – Graph information including the graph name (if any), graph type, and the number of nodes and edges.

Return type:

string

Examples

>>> G = nx.Graph(name="foo")
>>> str(G)
"Graph named 'foo' with 0 nodes and 0 edges"

>>> G = nx.path_graph(3)
>>> str(G)
'Graph with 3 nodes and 2 edges'

add_edge(u_of_edge, v_of_edge, **attr)

Add an edge between u and v.

The nodes u and v will be automatically added if they are not already in the graph.

Edge attributes can be specified with keywords or by directly accessing the edge’s attribute dictionary. See examples below.

Parameters:

• u_of_edge (nodes) – Nodes can be, for example, strings or numbers. Nodes must be hashable (and not None) Python objects.

• v_of_edge (nodes) – Nodes can be, for example, strings or numbers. Nodes must be hashable (and not None) Python objects.

• attr (keyword arguments, optional) – Edge data (or labels or objects) can be assigned using keyword arguments.

See also

add_edges_from

add a collection of edges

Notes

Adding an edge that already exists updates the edge data.

Many NetworkX algorithms designed for weighted graphs use an edge attribute (by default weight) to hold a numerical value.

Examples

The following all add the edge e=(1, 2) to graph G:

>>> G = nx.Graph()  # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> e = (1, 2)
>>> G.add_edge(1, 2)  # explicit two-node form
>>> G.add_edge(*e)  # single edge as tuple of two nodes
>>> G.add_edges_from([(1, 2)])  # add edges from iterable container

Associate data to edges using keywords:

>>> G.add_edge(1, 2, weight=3)
>>> G.add_edge(1, 3, weight=7, capacity=15, length=342.7)

For non-string attribute keys, use subscript notation.

>>> G.add_edge(1, 2)
>>> G[1][2].update({0: 5})
>>> G.edges[1, 2].update({0: 5})

add_edges_from(ebunch_to_add, **attr)

Add all the edges in ebunch_to_add.

Parameters:

• ebunch_to_add (container of edges) – Each edge given in the container will be added to the graph. The edges must be given as 2-tuples (u, v) or 3-tuples (u, v, d) where d is a dictionary containing edge data.

• attr (keyword arguments, optional) – Edge data (or labels or objects) can be assigned using keyword arguments.

See also

add_edge

add a single edge

add_weighted_edges_from

convenient way to add weighted edges

Notes

Adding the same edge twice has no effect but any edge data will be updated when each duplicate edge is added.

Edge attributes specified in an ebunch take precedence over attributes specified via keyword arguments.

When adding edges from an iterator over the graph you are changing, a RuntimeError can be raised with message: RuntimeError: dictionary changed size during iteration. This happens when the graph’s underlying dictionary is modified during iteration. To avoid this error, evaluate the iterator into a separate object, e.g. by using list(iterator_of_edges), and pass this object to G.add_edges_from.

Examples

>>> G = nx.Graph()  # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_edges_from([(0, 1), (1, 2)])  # using a list of edge tuples
>>> e = zip(range(0, 3), range(1, 4))
>>> G.add_edges_from(e)  # Add the path graph 0-1-2-3

Associate data to edges

>>> G.add_edges_from([(1, 2), (2, 3)], weight=3)
>>> G.add_edges_from([(3, 4), (1, 4)], label="WN2898")

Evaluate an iterator over a graph if using it to modify the same graph

>>> G = nx.Graph([(1, 2), (2, 3), (3, 4)])
>>> # Grow graph by one new node, adding edges to all existing nodes.
>>> # wrong way - will raise RuntimeError
>>> # G.add_edges_from(((5, n) for n in G.nodes))
>>> # correct way - note that there will be no self-edge for node 5
>>> G.add_edges_from(list((5, n) for n in G.nodes))

add_node(node_for_adding, **attr)

Add a single node node_for_adding and update node attributes.

Parameters:

• node_for_adding (node) – A node can be any hashable Python object except None.

• attr (keyword arguments, optional) – Set or change node attributes using key=value.

See also

add_nodes_from

Examples

>>> G = nx.Graph()  # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_node(1)
>>> G.add_node("Hello")
>>> K3 = nx.Graph([(0, 1), (1, 2), (2, 0)])
>>> G.add_node(K3)
>>> G.number_of_nodes()
3

Use keywords set/change node attributes:

>>> G.add_node(1, size=10)
>>> G.add_node(3, weight=0.4, UTM=("13S", 382871, 3972649))

Notes

A hashable object is one that can be used as a key in a Python dictionary. This includes strings, numbers, tuples of strings and numbers, etc.

On many platforms hashable items also include mutables such as NetworkX Graphs, though one should be careful that the hash doesn’t change on mutables.

add_nodes_from(nodes_for_adding, **attr)

Add multiple nodes.

Parameters:

• nodes_for_adding (iterable container) – A container of nodes (list, dict, set, etc.). OR A container of (node, attribute dict) tuples. Node attributes are updated using the attribute dict.

• attr (keyword arguments, optional (default= no attributes)) – Update attributes for all nodes in nodes. Node attributes specified in nodes as a tuple take precedence over attributes specified via keyword arguments.

See also

add_node

Notes

When adding nodes from an iterator over the graph you are changing, a RuntimeError can be raised with message: RuntimeError: dictionary changed size during iteration. This happens when the graph’s underlying dictionary is modified during iteration. To avoid this error, evaluate the iterator into a separate object, e.g. by using list(iterator_of_nodes), and pass this object to G.add_nodes_from.

Examples

>>> G = nx.Graph()  # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_nodes_from("Hello")
>>> K3 = nx.Graph([(0, 1), (1, 2), (2, 0)])
>>> G.add_nodes_from(K3)
>>> sorted(G.nodes(), key=str)
[0, 1, 2, 'H', 'e', 'l', 'o']

Use keywords to update specific node attributes for every node.

>>> G.add_nodes_from([1, 2], size=10)
>>> G.add_nodes_from([3, 4], weight=0.4)

Use (node, attrdict) tuples to update attributes for specific nodes.

>>> G.add_nodes_from([(1, dict(size=11)), (2, {"color": "blue"})])
>>> G.nodes[1]["size"]
11
>>> H = nx.Graph()
>>> H.add_nodes_from(G.nodes(data=True))
>>> H.nodes[1]["size"]
11

Evaluate an iterator over a graph if using it to modify the same graph

>>> G = nx.Graph([(0, 1), (1, 2), (3, 4)])
>>> # wrong way - will raise RuntimeError
>>> # G.add_nodes_from(n + 1 for n in G.nodes)
>>> # correct way
>>> G.add_nodes_from(list(n + 1 for n in G.nodes))

add_weighted_edges_from(ebunch_to_add, weight='weight', **attr)

Add weighted edges in ebunch_to_add with specified weight attr

Parameters:

• ebunch_to_add (container of edges) – Each edge given in the list or container will be added to the graph. The edges must be given as 3-tuples (u, v, w) where w is a number.

• weight (string, optional (default= 'weight')) – The attribute name for the edge weights to be added.

• attr (keyword arguments, optional (default= no attributes)) – Edge attributes to add/update for all edges.

See also

add_edge

add a single edge

add_edges_from

add multiple edges

Notes

Adding the same edge twice for Graph/DiGraph simply updates the edge data. For MultiGraph/MultiDiGraph, duplicate edges are stored.

When adding edges from an iterator over the graph you are changing, a RuntimeError can be raised with message: RuntimeError: dictionary changed size during iteration. This happens when the graph’s underlying dictionary is modified during iteration. To avoid this error, evaluate the iterator into a separate object, e.g. by using list(iterator_of_edges), and pass this object to G.add_weighted_edges_from.

Examples

>>> G = nx.Graph()  # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_weighted_edges_from([(0, 1, 3.0), (1, 2, 7.5)])

Evaluate an iterator over edges before passing it

>>> G = nx.Graph([(1, 2), (2, 3), (3, 4)])
>>> weight = 0.1
>>> # Grow graph by one new node, adding edges to all existing nodes.
>>> # wrong way - will raise RuntimeError
>>> # G.add_weighted_edges_from(((5, n, weight) for n in G.nodes))
>>> # correct way - note that there will be no self-edge for node 5
>>> G.add_weighted_edges_from(list((5, n, weight) for n in G.nodes))

adjacency()

Returns an iterator over (node, adjacency dict) tuples for all nodes.

For directed graphs, only outgoing neighbors/adjacencies are included.

Returns:

adj_iter – An iterator over (node, adjacency dictionary) for all nodes in the graph.

Return type:

iterator

Examples

>>> G = nx.path_graph(4)  # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> [(n, nbrdict) for n, nbrdict in G.adjacency()]
[(0, {1: {}}), (1, {0: {}, 2: {}}), (2, {1: {}, 3: {}}), (3, {2: {}})]

clear()

Remove all nodes and edges from the graph.

This also removes the name, and all graph, node, and edge attributes.

Examples

>>> G = nx.path_graph(4)  # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.clear()
>>> list(G.nodes)
[]
>>> list(G.edges)
[]

clear_edges()

Remove all edges from the graph without altering nodes.

Examples

>>> G = nx.path_graph(4)  # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.clear_edges()
>>> list(G.nodes)
[0, 1, 2, 3]
>>> list(G.edges)
[]

copy(as_view=False)

Returns a copy of the graph.

The copy method by default returns an independent shallow copy of the graph and attributes. That is, if an attribute is a container, that container is shared by the original an the copy. Use Python’s copy.deepcopy for new containers.

If as_view is True then a view is returned instead of a copy.

Notes

All copies reproduce the graph structure, but data attributes may be handled in different ways. There are four types of copies of a graph that people might want.

Deepcopy – A “deepcopy” copies the graph structure as well as all data attributes and any objects they might contain. The entire graph object is new so that changes in the copy do not affect the original object. (see Python’s copy.deepcopy)

Data Reference (Shallow) – For a shallow copy the graph structure is copied but the edge, node and graph attribute dicts are references to those in the original graph. This saves time and memory but could cause confusion if you change an attribute in one graph and it changes the attribute in the other. NetworkX does not provide this level of shallow copy.

Independent Shallow – This copy creates new independent attribute dicts and then does a shallow copy of the attributes. That is, any attributes that are containers are shared between the new graph and the original. This is exactly what dict.copy() provides. You can obtain this style copy using:

>>> G = nx.path_graph(5)
>>> H = G.copy()
>>> H = G.copy(as_view=False)
>>> H = nx.Graph(G)
>>> H = G.__class__(G)

Fresh Data – For fresh data, the graph structure is copied while new empty data attribute dicts are created. The resulting graph is independent of the original and it has no edge, node or graph attributes. Fresh copies are not enabled. Instead use:

>>> H = G.__class__()
>>> H.add_nodes_from(G)
>>> H.add_edges_from(G.edges)

View – Inspired by dict-views, graph-views act like read-only versions of the original graph, providing a copy of the original structure without requiring any memory for copying the information.

See the Python copy module for more information on shallow and deep copies, https://docs.python.org/3/library/copy.html.

Parameters:

as_view (bool, optional (default=False)) – If True, the returned graph-view provides a read-only view of the original graph without actually copying any data.

Returns:

G – A copy of the graph.

Return type:

Graph

See also

to_directed

return a directed copy of the graph.

Examples

>>> G = nx.path_graph(4)  # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> H = G.copy()

edge_subgraph(edges)

Returns the subgraph induced by the specified edges.

The induced subgraph contains each edge in edges and each node incident to any one of those edges.

Parameters:

edges (iterable) – An iterable of edges in this graph.

Returns:

G – An edge-induced subgraph of this graph with the same edge attributes.

Return type:

Graph

Notes

The graph, edge, and node attributes in the returned subgraph view are references to the corresponding attributes in the original graph. The view is read-only.

To create a full graph version of the subgraph with its own copy of the edge or node attributes, use:

G.edge_subgraph(edges).copy()

Examples

>>> G = nx.path_graph(5)
>>> H = G.edge_subgraph([(0, 1), (3, 4)])
>>> list(H.nodes)
[0, 1, 3, 4]
>>> list(H.edges)
[(0, 1), (3, 4)]

get_edge_data(u, v, default=None)

Returns the attribute dictionary associated with edge (u, v).

This is identical to G[u][v] except the default is returned instead of an exception if the edge doesn’t exist.

Parameters:

• u (nodes)

• v (nodes)

• default (any Python object (default=None)) – Value to return if the edge (u, v) is not found.

Returns:

edge_dict – The edge attribute dictionary.

Return type:

dictionary

Examples

>>> G = nx.path_graph(4)  # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G[0][1]
{}

Warning: Assigning to G[u][v] is not permitted. But it is safe to assign attributes G[u][v][‘foo’]

>>> G[0][1]["weight"] = 7
>>> G[0][1]["weight"]
7
>>> G[1][0]["weight"]
7

>>> G = nx.path_graph(4)  # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.get_edge_data(0, 1)  # default edge data is {}
{}
>>> e = (0, 1)
>>> G.get_edge_data(*e)  # tuple form
{}
>>> G.get_edge_data("a", "b", default=0)  # edge not in graph, return 0
0

has_edge(u, v)

Returns True if the edge (u, v) is in the graph.

This is the same as v in G[u] without KeyError exceptions.

Parameters:

• u (nodes) – Nodes can be, for example, strings or numbers. Nodes must be hashable (and not None) Python objects.

• v (nodes) – Nodes can be, for example, strings or numbers. Nodes must be hashable (and not None) Python objects.

Returns:

edge_ind – True if edge is in the graph, False otherwise.

Return type:

bool

Examples

>>> G = nx.path_graph(4)  # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.has_edge(0, 1)  # using two nodes
True
>>> e = (0, 1)
>>> G.has_edge(*e)  #  e is a 2-tuple (u, v)
True
>>> e = (0, 1, {"weight": 7})
>>> G.has_edge(*e[:2])  # e is a 3-tuple (u, v, data_dictionary)
True

The following syntax are equivalent:

>>> G.has_edge(0, 1)
True
>>> 1 in G[0]  # though this gives KeyError if 0 not in G
True

has_node(n)

Returns True if the graph contains the node n.

Identical to n in G

Parameters:

n (node)

Examples

>>> G = nx.path_graph(3)  # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.has_node(0)
True

It is more readable and simpler to use

>>> 0 in G
True

is_directed()

Returns True if graph is directed, False otherwise.

is_multigraph()

Returns True if graph is a multigraph, False otherwise.

nbunch_iter(nbunch=None)

Returns an iterator over nodes contained in nbunch that are also in the graph.

The nodes in nbunch are checked for membership in the graph and if not are silently ignored.

Parameters:

nbunch (single node, container, or all nodes (default= all nodes)) – The view will only report edges incident to these nodes.

Returns:

niter – An iterator over nodes in nbunch that are also in the graph. If nbunch is None, iterate over all nodes in the graph.

Return type:

iterator

Raises:

NetworkXError – If nbunch is not a node or sequence of nodes. If a node in nbunch is not hashable.

See also

Graph.__iter__

Notes

When nbunch is an iterator, the returned iterator yields values directly from nbunch, becoming exhausted when nbunch is exhausted.

To test whether nbunch is a single node, one can use “if nbunch in self:”, even after processing with this routine.

If nbunch is not a node or a (possibly empty) sequence/iterator or None, a NetworkXError is raised. Also, if any object in nbunch is not hashable, a NetworkXError is raised.

neighbors(n)

Returns an iterator over all neighbors of node n.

This is identical to iter(G[n])

Parameters:

n (node) – A node in the graph

Returns:

neighbors – An iterator over all neighbors of node n

Return type:

iterator

Raises:

NetworkXError – If the node n is not in the graph.

Examples

>>> G = nx.path_graph(4)  # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> [n for n in G.neighbors(0)]
[1]

Notes

Alternate ways to access the neighbors are G.adj[n] or G[n]:

>>> G = nx.Graph()  # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_edge("a", "b", weight=7)
>>> G["a"]
AtlasView({'b': {'weight': 7}})
>>> G = nx.path_graph(4)
>>> [n for n in G[0]]
[1]

number_of_edges(u=None, v=None)

Returns the number of edges between two nodes.

Parameters:

• u (nodes, optional (default=all edges)) – If u and v are specified, return the number of edges between u and v. Otherwise return the total number of all edges.

• v (nodes, optional (default=all edges)) – If u and v are specified, return the number of edges between u and v. Otherwise return the total number of all edges.

Returns:

nedges – The number of edges in the graph. If nodes u and v are specified return the number of edges between those nodes. If the graph is directed, this only returns the number of edges from u to v.

Return type:

int

See also

size

Examples

For undirected graphs, this method counts the total number of edges in the graph:

>>> G = nx.path_graph(4)
>>> G.number_of_edges()
3

If you specify two nodes, this counts the total number of edges joining the two nodes:

>>> G.number_of_edges(0, 1)
1

For directed graphs, this method can count the total number of directed edges from u to v:

>>> G = nx.DiGraph()
>>> G.add_edge(0, 1)
>>> G.add_edge(1, 0)
>>> G.number_of_edges(0, 1)
1

number_of_nodes()

Returns the number of nodes in the graph.

Returns:

nnodes – The number of nodes in the graph.

Return type:

int

See also

order

identical method

__len__

identical method

Examples

>>> G = nx.path_graph(3)  # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.number_of_nodes()
3

order()

Returns the number of nodes in the graph.

Returns:

nnodes – The number of nodes in the graph.

Return type:

int

See also

number_of_nodes

identical method

__len__

identical method

Examples

>>> G = nx.path_graph(3)  # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.order()
3

remove_edge(u, v)

Remove the edge between u and v.

Parameters:

• u (nodes) – Remove the edge between nodes u and v.

• v (nodes) – Remove the edge between nodes u and v.

Raises:

NetworkXError – If there is not an edge between u and v.

See also

remove_edges_from

remove a collection of edges

Examples

>>> G = nx.path_graph(4)  # or DiGraph, etc
>>> G.remove_edge(0, 1)
>>> e = (1, 2)
>>> G.remove_edge(*e)  # unpacks e from an edge tuple
>>> e = (2, 3, {"weight": 7})  # an edge with attribute data
>>> G.remove_edge(*e[:2])  # select first part of edge tuple

remove_edges_from(ebunch)

Remove all edges specified in ebunch.

Parameters:

ebunch (list or container of edge tuples) –

Each edge given in the list or container will be removed from the graph. The edges can be:

• 2-tuples (u, v) edge between u and v.

• 3-tuples (u, v, k) where k is ignored.

See also

remove_edge

remove a single edge

Notes

Will fail silently if an edge in ebunch is not in the graph.

Examples

>>> G = nx.path_graph(4)  # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> ebunch = [(1, 2), (2, 3)]
>>> G.remove_edges_from(ebunch)

remove_node(n)

Remove node n.

Removes the node n and all adjacent edges. Attempting to remove a nonexistent node will raise an exception.

Parameters:

n (node) – A node in the graph

Raises:

NetworkXError – If n is not in the graph.

See also

remove_nodes_from

Examples

>>> G = nx.path_graph(3)  # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> list(G.edges)
[(0, 1), (1, 2)]
>>> G.remove_node(1)
>>> list(G.edges)
[]

remove_nodes_from(nodes)

Remove multiple nodes.

Parameters:

nodes (iterable container) – A container of nodes (list, dict, set, etc.). If a node in the container is not in the graph it is silently ignored.

See also

remove_node

Notes

When removing nodes from an iterator over the graph you are changing, a RuntimeError will be raised with message: RuntimeError: dictionary changed size during iteration. This happens when the graph’s underlying dictionary is modified during iteration. To avoid this error, evaluate the iterator into a separate object, e.g. by using list(iterator_of_nodes), and pass this object to G.remove_nodes_from.

Examples

>>> G = nx.path_graph(3)  # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> e = list(G.nodes)
>>> e
[0, 1, 2]
>>> G.remove_nodes_from(e)
>>> list(G.nodes)
[]

Evaluate an iterator over a graph if using it to modify the same graph

>>> G = nx.Graph([(0, 1), (1, 2), (3, 4)])
>>> # this command will fail, as the graph's dict is modified during iteration
>>> # G.remove_nodes_from(n for n in G.nodes if n < 2)
>>> # this command will work, since the dictionary underlying graph is not modified
>>> G.remove_nodes_from(list(n for n in G.nodes if n < 2))

size(weight=None)

Returns the number of edges or total of all edge weights.

Parameters:

weight (string or None, optional (default=None)) – The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1.

Returns:

size – The number of edges or (if weight keyword is provided) the total weight sum.

If weight is None, returns an int. Otherwise a float (or more general numeric if the weights are more general).

Return type:

numeric

See also

number_of_edges

Examples

>>> G = nx.path_graph(4)  # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.size()
3

>>> G = nx.Graph()  # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_edge("a", "b", weight=2)
>>> G.add_edge("b", "c", weight=4)
>>> G.size()
2
>>> G.size(weight="weight")
6.0

subgraph(nodes)

Returns a SubGraph view of the subgraph induced on nodes.

The induced subgraph of the graph contains the nodes in nodes and the edges between those nodes.

Parameters:

nodes (list, iterable) – A container of nodes which will be iterated through once.

Returns:

G – A subgraph view of the graph. The graph structure cannot be changed but node/edge attributes can and are shared with the original graph.

Return type:

SubGraph View

Notes

The graph, edge and node attributes are shared with the original graph. Changes to the graph structure is ruled out by the view, but changes to attributes are reflected in the original graph.

To create a subgraph with its own copy of the edge/node attributes use: G.subgraph(nodes).copy()

For an inplace reduction of a graph to a subgraph you can remove nodes: G.remove_nodes_from([n for n in G if n not in set(nodes)])

Subgraph views are sometimes NOT what you want. In most cases where you want to do more than simply look at the induced edges, it makes more sense to just create the subgraph as its own graph with code like:

# Create a subgraph SG based on a (possibly multigraph) G
SG = G.__class__()
SG.add_nodes_from((n, G.nodes[n]) for n in largest_wcc)
if SG.is_multigraph():
    SG.add_edges_from(
        (n, nbr, key, d)
        for n, nbrs in G.adj.items()
        if n in largest_wcc
        for nbr, keydict in nbrs.items()
        if nbr in largest_wcc
        for key, d in keydict.items()
    )
else:
    SG.add_edges_from(
        (n, nbr, d)
        for n, nbrs in G.adj.items()
        if n in largest_wcc
        for nbr, d in nbrs.items()
        if nbr in largest_wcc
    )
SG.graph.update(G.graph)

Examples

>>> G = nx.path_graph(4)  # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> H = G.subgraph([0, 1, 2])
>>> list(H.edges)
[(0, 1), (1, 2)]

to_directed(as_view=False)

Returns a directed representation of the graph.

Returns:

G – A directed graph with the same name, same nodes, and with each edge (u, v, data) replaced by two directed edges (u, v, data) and (v, u, data).

Return type:

DiGraph

Notes

This returns a “deepcopy” of the edge, node, and graph attributes which attempts to completely copy all of the data and references.

This is in contrast to the similar D=DiGraph(G) which returns a shallow copy of the data.

See the Python copy module for more information on shallow and deep copies, https://docs.python.org/3/library/copy.html.

Warning: If you have subclassed Graph to use dict-like objects in the data structure, those changes do not transfer to the DiGraph created by this method.

Examples

>>> G = nx.Graph()  # or MultiGraph, etc
>>> G.add_edge(0, 1)
>>> H = G.to_directed()
>>> list(H.edges)
[(0, 1), (1, 0)]

If already directed, return a (deep) copy

>>> G = nx.DiGraph()  # or MultiDiGraph, etc
>>> G.add_edge(0, 1)
>>> H = G.to_directed()
>>> list(H.edges)
[(0, 1)]

to_directed_class()

Returns the class to use for empty directed copies.

If you subclass the base classes, use this to designate what directed class to use for to_directed() copies.

to_undirected(as_view=False)

Returns an undirected copy of the graph.

Parameters:

as_view (bool (optional, default=False)) – If True return a view of the original undirected graph.

Returns:

G – A deepcopy of the graph.

Return type:

Graph/MultiGraph

See also

Graph, copy, add_edge, add_edges_from

Notes

This returns a “deepcopy” of the edge, node, and graph attributes which attempts to completely copy all of the data and references.

This is in contrast to the similar G = nx.DiGraph(D) which returns a shallow copy of the data.

See the Python copy module for more information on shallow and deep copies, https://docs.python.org/3/library/copy.html.

Warning: If you have subclassed DiGraph to use dict-like objects in the data structure, those changes do not transfer to the Graph created by this method.

Examples

>>> G = nx.path_graph(2)  # or MultiGraph, etc
>>> H = G.to_directed()
>>> list(H.edges)
[(0, 1), (1, 0)]
>>> G2 = H.to_undirected()
>>> list(G2.edges)
[(0, 1)]

to_undirected_class()

Returns the class to use for empty undirected copies.

If you subclass the base classes, use this to designate what directed class to use for to_directed() copies.

update(edges=None, nodes=None)

Update the graph using nodes/edges/graphs as input.

Like dict.update, this method takes a graph as input, adding the graph’s nodes and edges to this graph. It can also take two inputs: edges and nodes. Finally it can take either edges or nodes. To specify only nodes the keyword nodes must be used.

The collections of edges and nodes are treated similarly to the add_edges_from/add_nodes_from methods. When iterated, they should yield 2-tuples (u, v) or 3-tuples (u, v, datadict).

Parameters:

• edges (Graph object, collection of edges, or None) – The first parameter can be a graph or some edges. If it has attributes nodes and edges, then it is taken to be a Graph-like object and those attributes are used as collections of nodes and edges to be added to the graph. If the first parameter does not have those attributes, it is treated as a collection of edges and added to the graph. If the first argument is None, no edges are added.

• nodes (collection of nodes, or None) – The second parameter is treated as a collection of nodes to be added to the graph unless it is None. If edges is None and nodes is None an exception is raised. If the first parameter is a Graph, then nodes is ignored.

Examples

>>> G = nx.path_graph(5)
>>> G.update(nx.complete_graph(range(4, 10)))
>>> from itertools import combinations
>>> edges = (
...     (u, v, {"power": u * v})
...     for u, v in combinations(range(10, 20), 2)
...     if u * v < 225
... )
>>> nodes = [1000]  # for singleton, use a container
>>> G.update(edges, nodes)

Notes

It you want to update the graph using an adjacency structure it is straightforward to obtain the edges/nodes from adjacency. The following examples provide common cases, your adjacency may be slightly different and require tweaks of these examples:

>>> # dict-of-set/list/tuple
>>> adj = {1: {2, 3}, 2: {1, 3}, 3: {1, 2}}
>>> e = [(u, v) for u, nbrs in adj.items() for v in nbrs]
>>> G.update(edges=e, nodes=adj)

>>> DG = nx.DiGraph()
>>> # dict-of-dict-of-attribute
>>> adj = {1: {2: 1.3, 3: 0.7}, 2: {1: 1.4}, 3: {1: 0.7}}
>>> e = [
...     (u, v, {"weight": d})
...     for u, nbrs in adj.items()
...     for v, d in nbrs.items()
... ]
>>> DG.update(edges=e, nodes=adj)

>>> # dict-of-dict-of-dict
>>> adj = {1: {2: {"weight": 1.3}, 3: {"color": 0.7, "weight": 1.2}}}
>>> e = [
...     (u, v, {"weight": d})
...     for u, nbrs in adj.items()
...     for v, d in nbrs.items()
... ]
>>> DG.update(edges=e, nodes=adj)

>>> # predecessor adjacency (dict-of-set)
>>> pred = {1: {2, 3}, 2: {3}, 3: {3}}
>>> e = [(v, u) for u, nbrs in pred.items() for v in nbrs]

>>> # MultiGraph dict-of-dict-of-dict-of-attribute
>>> MDG = nx.MultiDiGraph()
>>> adj = {
...     1: {2: {0: {"weight": 1.3}, 1: {"weight": 1.2}}},
...     3: {2: {0: {"weight": 0.7}}},
... }
>>> e = [
...     (u, v, ekey, d)
...     for u, nbrs in adj.items()
...     for v, keydict in nbrs.items()
...     for ekey, d in keydict.items()
... ]
>>> MDG.update(edges=e)

See also

add_edges_from

add multiple edges to a graph

add_nodes_from

add multiple nodes to a graph

__annotations__ = {}

__networkx_backend__ = 'networkx'

property adj

Graph adjacency object holding the neighbors of each node.

This object is a read-only dict-like structure with node keys and neighbor-dict values. The neighbor-dict is keyed by neighbor to the edge-data-dict. So G.adj[3][2][‘color’] = ‘blue’ sets the color of the edge (3, 2) to “blue”.

Iterating over G.adj behaves like a dict. Useful idioms include for nbr, datadict in G.adj[n].items():.

The neighbor information is also provided by subscripting the graph. So for nbr, foovalue in G[node].data(‘foo’, default=1): works.

For directed graphs, G.adj holds outgoing (successor) info.

property degree

A DegreeView for the Graph as G.degree or G.degree().

The node degree is the number of edges adjacent to the node. The weighted node degree is the sum of the edge weights for edges incident to that node.

This object provides an iterator for (node, degree) as well as lookup for the degree for a single node.

Parameters:

• nbunch (single node, container, or all nodes (default= all nodes)) – The view will only report edges incident to these nodes.

• weight (string or None, optional (default=None)) – The name of an edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1. The degree is the sum of the edge weights adjacent to the node.

Returns:

If multiple nodes are requested (the default), returns a DegreeView mapping nodes to their degree. If a single node is requested, returns the degree of the node as an integer.

Return type:

DegreeView or int

Examples

>>> G = nx.path_graph(4)  # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.degree[0]  # node 0 has degree 1
1
>>> list(G.degree([0, 1, 2]))
[(0, 1), (1, 2), (2, 2)]

property edges

An EdgeView of the Graph as G.edges or G.edges().

edges(self, nbunch=None, data=False, default=None)

The EdgeView provides set-like operations on the edge-tuples as well as edge attribute lookup. When called, it also provides an EdgeDataView object which allows control of access to edge attributes (but does not provide set-like operations). Hence, G.edges[u, v][‘color’] provides the value of the color attribute for edge (u, v) while for (u, v, c) in G.edges.data(‘color’, default=’red’): iterates through all the edges yielding the color attribute with default ‘red’ if no color attribute exists.

Parameters:

• nbunch (single node, container, or all nodes (default= all nodes)) – The view will only report edges from these nodes.

• data (string or bool, optional (default=False)) – The edge attribute returned in 3-tuple (u, v, ddict[data]). If True, return edge attribute dict in 3-tuple (u, v, ddict). If False, return 2-tuple (u, v).

• default (value, optional (default=None)) – Value used for edges that don’t have the requested attribute. Only relevant if data is not True or False.

Returns:

edges – A view of edge attributes, usually it iterates over (u, v) or (u, v, d) tuples of edges, but can also be used for attribute lookup as edges[u, v][‘foo’].

Return type:

EdgeView

Notes

Nodes in nbunch that are not in the graph will be (quietly) ignored. For directed graphs this returns the out-edges.

Examples

>>> G = nx.path_graph(3)  # or MultiGraph, etc
>>> G.add_edge(2, 3, weight=5)
>>> [e for e in G.edges]
[(0, 1), (1, 2), (2, 3)]
>>> G.edges.data()  # default data is {} (empty dict)
EdgeDataView([(0, 1, {}), (1, 2, {}), (2, 3, {'weight': 5})])
>>> G.edges.data("weight", default=1)
EdgeDataView([(0, 1, 1), (1, 2, 1), (2, 3, 5)])
>>> G.edges([0, 3])  # only edges from these nodes
EdgeDataView([(0, 1), (3, 2)])
>>> G.edges(0)  # only edges from node 0
EdgeDataView([(0, 1)])

property name

String identifier of the graph.

This graph attribute appears in the attribute dict G.graph keyed by the string “name”. as well as an attribute (technically a property) G.name. This is entirely user controlled.

property nodes

A NodeView of the Graph as G.nodes or G.nodes().

Can be used as G.nodes for data lookup and for set-like operations. Can also be used as G.nodes(data=’color’, default=None) to return a NodeDataView which reports specific node data but no set operations. It presents a dict-like interface as well with G.nodes.items() iterating over (node, nodedata) 2-tuples and G.nodes[3][‘foo’] providing the value of the foo attribute for node 3. In addition, a view G.nodes.data(‘foo’) provides a dict-like interface to the foo attribute of each node. G.nodes.data(‘foo’, default=1) provides a default for nodes that do not have attribute foo.

Parameters:

• data (string or bool, optional (default=False)) – The node attribute returned in 2-tuple (n, ddict[data]). If True, return entire node attribute dict as (n, ddict). If False, return just the nodes n.

• default (value, optional (default=None)) – Value used for nodes that don’t have the requested attribute. Only relevant if data is not True or False.

Returns:

Allows set-like operations over the nodes as well as node attribute dict lookup and calling to get a NodeDataView. A NodeDataView iterates over (n, data) and has no set operations. A NodeView iterates over n and includes set operations.

When called, if data is False, an iterator over nodes. Otherwise an iterator of 2-tuples (node, attribute value) where the attribute is specified in data. If data is True then the attribute becomes the entire data dictionary.

Return type:

NodeView

Notes

If your node data is not needed, it is simpler and equivalent to use the expression for n in G, or list(G).

Examples

There are two simple ways of getting a list of all nodes in the graph:

>>> G = nx.path_graph(3)
>>> list(G.nodes)
[0, 1, 2]
>>> list(G)
[0, 1, 2]

To get the node data along with the nodes:

>>> G.add_node(1, time="5pm")
>>> G.nodes[0]["foo"] = "bar"
>>> list(G.nodes(data=True))
[(0, {'foo': 'bar'}), (1, {'time': '5pm'}), (2, {})]
>>> list(G.nodes.data())
[(0, {'foo': 'bar'}), (1, {'time': '5pm'}), (2, {})]

>>> list(G.nodes(data="foo"))
[(0, 'bar'), (1, None), (2, None)]
>>> list(G.nodes.data("foo"))
[(0, 'bar'), (1, None), (2, None)]

>>> list(G.nodes(data="time"))
[(0, None), (1, '5pm'), (2, None)]
>>> list(G.nodes.data("time"))
[(0, None), (1, '5pm'), (2, None)]

>>> list(G.nodes(data="time", default="Not Available"))
[(0, 'Not Available'), (1, '5pm'), (2, 'Not Available')]
>>> list(G.nodes.data("time", default="Not Available"))
[(0, 'Not Available'), (1, '5pm'), (2, 'Not Available')]

If some of your nodes have an attribute and the rest are assumed to have a default attribute value you can create a dictionary from node/attribute pairs using the default keyword argument to guarantee the value is never None:

>>> G = nx.Graph()
>>> G.add_node(0)
>>> G.add_node(1, weight=2)
>>> G.add_node(2, weight=3)
>>> dict(G.nodes(data="weight", default=1))
{0: 1, 1: 2, 2: 3}