contracted_nodes#

contracted_nodes(G, u, v, self_loops=True, copy=True, *, store_contraction_as='contraction')[source]#

Returns the graph that results from contracting u and v.

Node contraction identifies the two nodes as a single node incident to any edge that was incident to the original two nodes. Information about the contracted nodes and any modified edges are stored on the output graph in a "contraction" attribute - see Examples for details.

Parameters:

GNetworkX graph: The graph whose nodes will be contracted.
u, vnodes: Must be nodes in G.
self_loopsBoolean: If this is True, any edges joining u and v in G become self-loops on the new node in the returned graph.
copyBoolean: If this is True (the default), make a copy of G and return that instead of directly changing G.
store_contraction_asstr or None, default=”contraction”: Name of the node/edge attribute where information about the contraction should be stored. By default information about the contracted node and any contracted edges is stored in a "contraction" attribute on the resulting node and edge. If None, information about the contracted nodes/edges and their data are not stored.

Returns:

Networkx graph: If copy is True, A new graph object of the same type as G (leaving G unmodified) with u and v identified in a single node. The right node v will be merged into the node u, so only u will appear in the returned graph. If copy is False, Modifies G with u and v identified in a single node. The right node v will be merged into the node u, so only u will appear in the returned graph.

See also

contracted_edge
quotient_graph

Notes

For multigraphs, the edge keys for the realigned edges may not be the same as the edge keys for the old edges. This is natural because edge keys are unique only within each pair of nodes.

This function is also available as identified_nodes.

Examples

Contracting two nonadjacent nodes of the cycle graph on four nodes C_4 yields the path graph (ignoring parallel edges):

>>> G = nx.cycle_graph(4)
>>> M = nx.contracted_nodes(G, 1, 3)
>>> P3 = nx.path_graph(3)
>>> nx.is_isomorphic(M, P3)
True

Information about the contracted nodes is stored on the resulting graph in a "contraction" attribute. For instance, the contracted node is stored as an attribute on u:

>>> H = nx.contracted_nodes(P3, 0, 2)
>>> H.nodes(data=True)
NodeDataView({0: {'contraction': {2: {}}}, 1: {}})

Any node attributes on the contracted node are also preserved:

>>> nx.set_node_attributes(P3, dict(enumerate("rgb")), name="color")
>>> P3.nodes(data=True)
NodeDataView({0: {'color': 'r'}, 1: {'color': 'g'}, 2: {'color': 'b'}})
>>> H = nx.contracted_nodes(P3, 0, 2)
>>> H.nodes[0]
{'color': 'r', 'contraction': {2: {'color': 'b'}}}

Edges are handled similarly: when u and v are adjacent to a third node w, the edge (v, w) will be contracted into the edge (u, w) with its attributes stored into a "contraction" attribute on edge (u, w):

>>> nx.set_edge_attributes(P3, {(0, 1): 10, (1, 2): 100}, name="weight")
>>> P3.edges(data=True)
EdgeDataView([(0, 1, {'weight': 10}), (1, 2, {'weight': 100})])
>>> H = nx.contracted_nodes(P3, 0, 2)
>>> H.edges(data=True)
EdgeDataView([(0, 1, {'weight': 10, 'contraction': {(2, 1): {'weight': 100}}})])

Attributes from contracted nodes/edges can be combined with those of the nodes/edges onto which they were contracted:

>>> # Concatenate colors of contracted nodes
>>> for u, cdict in H.nodes(data="contraction"):
...     if cdict is not None:
...         H.nodes[u]["color"] += "".join(n["color"] for n in cdict.values())
...         del H.nodes[u]["contraction"]  # Remove contraction attr (optional)
>>> H.nodes(data=True)
NodeDataView({0: {'color': 'rb'}, 1: {'color': 'g'}})
>>> # Sum contracted edge weights
>>> for u, v, cdict in H.edges(data="contraction"):
...     if cdict is not None:
...         H[u][v]["weight"] += sum(n["weight"] for n in cdict.values())
...         del H.edges[(u, v)]["contraction"]  # Remove contraction attr (optional)
>>> H.edges(data=True)
EdgeDataView([(0, 1, {'weight': 110})])

If G is a multigraph, then a new edge is added instead. Any edge attributes are still preserved:

>>> MG = nx.MultiGraph(P3)
>>> MH = nx.contracted_nodes(MG, 0, 2)
>>> MH.edges(keys=True, data=True)
MultiEdgeDataView([(0, 1, 0, {'weight': 10}), (0, 1, 1, {'weight': 100})])

If selfloops=True (the default), any edges adjoining u and v become self-loops on u in the resulting graph:

>>> G = nx.Graph([(1, 2)])
>>> H = nx.contracted_nodes(G, 1, 2)
>>> H.nodes, H.edges
(NodeView((1,)), EdgeView([(1, 1)]))
>>> H = nx.contracted_nodes(G, 1, 2, self_loops=False)
>>> H.nodes, H.edges
(NodeView((1,)), EdgeView([]))

Note however that any self loops in the original graph G are preserved:

>>> G = nx.Graph([(1, 2), (2, 2)])
>>> H = nx.contracted_nodes(G, 1, 2, self_loops=False)
>>> H.nodes, H.edges
(NodeView((1,)), EdgeView([(1, 1)]))

The same reasoning applies to MultiGraphs:

>>> MG = nx.MultiGraph([(1, 2), (2, 2)])
>>> # Edge (1, 1, 0) in MH corresponds to edge (2, 2) in MG
>>> MH = nx.contracted_nodes(MG, 1, 2, self_loops=False)
>>> MH.edges(keys=True)
MultiEdgeView([(1, 1, 0)])
>>> # MH has two (1, 1) edges - one from edge (2, 2) in MG, and one
>>> # resulting from the contraction of 2->1
>>> MH = nx.contracted_nodes(MG, 1, 2, self_loops=True)
>>> MH.edges(keys=True)
MultiEdgeView([(1, 1, 0), (1, 1, 1)])

In a MultiDiGraph with a self loop, the in and out edges will be treated separately as edges, so while contracting a node which has a self loop the contraction will add multiple edges:

>>> G = nx.MultiDiGraph([(1, 2), (2, 2)])
>>> H = nx.contracted_nodes(G, 1, 2)
>>> list(H.edges())  # edge 1->2, 2->2, 2<-2 from the original Graph G
[(1, 1), (1, 1), (1, 1)]
>>> H = nx.contracted_nodes(G, 1, 2, self_loops=False)
>>> list(H.edges())  # edge 2->2, 2<-2 from the original Graph G
[(1, 1), (1, 1)]