"""
Functions for hashing graphs to strings.
Isomorphic graphs should be assigned identical hashes.
For now, only Weisfeiler-Lehman hashing is implemented.
"""
import warnings
from collections import Counter, defaultdict
from hashlib import blake2b
import networkx as nx
__all__ = ["weisfeiler_lehman_graph_hash", "weisfeiler_lehman_subgraph_hashes"]
def _hash_label(label, digest_size):
return blake2b(label.encode("ascii"), digest_size=digest_size).hexdigest()
def _init_node_labels(G, edge_attr, node_attr):
if node_attr:
return {u: str(dd[node_attr]) for u, dd in G.nodes(data=True)}
elif edge_attr:
return {u: "" for u in G}
else:
warnings.warn(
"The hashes produced for graphs without node or edge attributes"
"changed in v3.5 due to a bugfix (see documentation).",
UserWarning,
stacklevel=2,
)
if nx.is_directed(G):
return {u: str(G.in_degree(u)) + "_" + str(G.out_degree(u)) for u in G}
else:
return {u: str(deg) for u, deg in G.degree()}
def _neighborhood_aggregate_undirected(G, node, node_labels, edge_attr=None):
"""
Compute new labels for given node in an undirected graph by aggregating
the labels of each node's neighbors.
"""
label_list = []
for nbr in G.neighbors(node):
prefix = "" if edge_attr is None else str(G[node][nbr][edge_attr])
label_list.append(prefix + node_labels[nbr])
return node_labels[node] + "".join(sorted(label_list))
def _neighborhood_aggregate_directed(G, node, node_labels, edge_attr=None):
"""
Compute new labels for given node in a directed graph by aggregating
the labels of each node's neighbors.
"""
successor_labels = []
for nbr in G.successors(node):
prefix = "s_" + "" if edge_attr is None else str(G[node][nbr][edge_attr])
successor_labels.append(prefix + node_labels[nbr])
predecessor_labels = []
for nbr in G.predecessors(node):
prefix = "p_" + "" if edge_attr is None else str(G[nbr][node][edge_attr])
predecessor_labels.append(prefix + node_labels[nbr])
return (
node_labels[node]
+ "".join(sorted(successor_labels))
+ "".join(sorted(predecessor_labels))
)
[docs]
@nx.utils.not_implemented_for("multigraph")
@nx._dispatchable(edge_attrs={"edge_attr": None}, node_attrs="node_attr")
def weisfeiler_lehman_graph_hash(
G, edge_attr=None, node_attr=None, iterations=3, digest_size=16
):
"""Return Weisfeiler Lehman (WL) graph hash.
.. Warning:: Hash values for directed graphs and graphs without edge or
node attributes have changed in v3.5. In previous versions,
directed graphs did not distinguish in- and outgoing edges. Also,
graphs without attributes set initial states such that effectively
one extra iteration of WL occurred than indicated by `iterations`.
For undirected graphs without node or edge labels, the old
hashes can be obtained by increasing the iteration count by one.
For more details, see `issue #7806
<https://github.com/networkx/networkx/issues/7806>`_.
The function iteratively aggregates and hashes neighborhoods of each node.
After each node's neighbors are hashed to obtain updated node labels,
a hashed histogram of resulting labels is returned as the final hash.
Hashes are identical for isomorphic graphs and strong guarantees that
non-isomorphic graphs will get different hashes. See [1]_ for details.
If no node or edge attributes are provided, the degree of each node
is used as its initial label.
Otherwise, node and/or edge labels are used to compute the hash.
Parameters
----------
G : graph
The graph to be hashed.
Can have node and/or edge attributes. Can also have no attributes.
edge_attr : string, optional (default=None)
The key in edge attribute dictionary to be used for hashing.
If None, edge labels are ignored.
node_attr: string, optional (default=None)
The key in node attribute dictionary to be used for hashing.
If None, and no edge_attr given, use the degrees of the nodes as labels.
iterations: int, optional (default=3)
Number of neighbor aggregations to perform.
Should be larger for larger graphs.
digest_size: int, optional (default=16)
Size (in bytes) of blake2b hash digest to use for hashing node labels.
Returns
-------
h : string
Hexadecimal string corresponding to hash of `G` (length ``2 * digest_size``).
Raises
------
ValueError
If `iterations` is not a positve number.
Examples
--------
Two graphs with edge attributes that are isomorphic, except for
differences in the edge labels.
>>> G1 = nx.Graph()
>>> G1.add_edges_from(
... [
... (1, 2, {"label": "A"}),
... (2, 3, {"label": "A"}),
... (3, 1, {"label": "A"}),
... (1, 4, {"label": "B"}),
... ]
... )
>>> G2 = nx.Graph()
>>> G2.add_edges_from(
... [
... (5, 6, {"label": "B"}),
... (6, 7, {"label": "A"}),
... (7, 5, {"label": "A"}),
... (7, 8, {"label": "A"}),
... ]
... )
Omitting the `edge_attr` option, results in identical hashes.
>>> nx.weisfeiler_lehman_graph_hash(G1)
'c045439172215f49e0bef8c3d26c6b61'
>>> nx.weisfeiler_lehman_graph_hash(G2)
'c045439172215f49e0bef8c3d26c6b61'
With edge labels, the graphs are no longer assigned
the same hash digest.
>>> nx.weisfeiler_lehman_graph_hash(G1, edge_attr="label")
'c653d85538bcf041d88c011f4f905f10'
>>> nx.weisfeiler_lehman_graph_hash(G2, edge_attr="label")
'3dcd84af1ca855d0eff3c978d88e7ec7'
Notes
-----
To return the WL hashes of each subgraph of a graph, use
`weisfeiler_lehman_subgraph_hashes`
Similarity between hashes does not imply similarity between graphs.
References
----------
.. [1] Shervashidze, Nino, Pascal Schweitzer, Erik Jan Van Leeuwen,
Kurt Mehlhorn, and Karsten M. Borgwardt. Weisfeiler Lehman
Graph Kernels. Journal of Machine Learning Research. 2011.
http://www.jmlr.org/papers/volume12/shervashidze11a/shervashidze11a.pdf
See also
--------
weisfeiler_lehman_subgraph_hashes
"""
if G.is_directed():
_neighborhood_aggregate = _neighborhood_aggregate_directed
warnings.warn(
"The hashes produced for directed graphs changed in version v3.5"
" due to a bugfix to track in and out edges separately (see documentation).",
UserWarning,
stacklevel=2,
)
else:
_neighborhood_aggregate = _neighborhood_aggregate_undirected
def weisfeiler_lehman_step(G, labels, edge_attr=None):
"""
Apply neighborhood aggregation to each node
in the graph.
Computes a dictionary with labels for each node.
"""
new_labels = {}
for node in G.nodes():
label = _neighborhood_aggregate(G, node, labels, edge_attr=edge_attr)
new_labels[node] = _hash_label(label, digest_size)
return new_labels
if iterations <= 0:
raise ValueError("The WL algorithm requires that `iterations` be positive")
# set initial node labels
node_labels = _init_node_labels(G, edge_attr, node_attr)
# If the graph has no attributes, initial labels are the nodes' degrees.
# This is equivalent to doing the first iterations of WL.
if not edge_attr and not node_attr:
iterations -= 1
subgraph_hash_counts = []
for _ in range(iterations):
node_labels = weisfeiler_lehman_step(G, node_labels, edge_attr=edge_attr)
counter = Counter(node_labels.values())
# sort the counter, extend total counts
subgraph_hash_counts.extend(sorted(counter.items(), key=lambda x: x[0]))
# hash the final counter
return _hash_label(str(tuple(subgraph_hash_counts)), digest_size)
[docs]
@nx.utils.not_implemented_for("multigraph")
@nx._dispatchable(edge_attrs={"edge_attr": None}, node_attrs="node_attr")
def weisfeiler_lehman_subgraph_hashes(
G,
edge_attr=None,
node_attr=None,
iterations=3,
digest_size=16,
include_initial_labels=False,
):
"""
Return a dictionary of subgraph hashes by node.
.. Warning:: Hash values for directed graphs have changed in version
v3.5. In previous versions, directed graphs did not distinguish in-
and outgoing edges.
Graphs without attributes previously performed an extra iteration of
WL at initialisation, which was not visible in the output of this
function. This hash value is now included in the returned dictionary,
shifting the other calculated hashes one position to the right. To
obtain the same last subgraph hash, increase the number of iterations
by one.
For more details, see `issue #7806
<https://github.com/networkx/networkx/issues/7806>`_.
Dictionary keys are nodes in `G`, and values are a list of hashes.
Each hash corresponds to a subgraph rooted at a given node u in `G`.
Lists of subgraph hashes are sorted in increasing order of depth from
their root node, with the hash at index i corresponding to a subgraph
of nodes at most i-hops (i edges) distance from u. Thus, each list will contain
`iterations` elements - a hash for a subgraph at each depth. If
`include_initial_labels` is set to `True`, each list will additionally
have contain a hash of the initial node label (or equivalently a
subgraph of depth 0) prepended, totalling ``iterations + 1`` elements.
The function iteratively aggregates and hashes neighborhoods of each node.
This is achieved for each step by replacing for each node its label from
the previous iteration with its hashed 1-hop neighborhood aggregate.
The new node label is then appended to a list of node labels for each
node.
To aggregate neighborhoods for a node $u$ at each step, all labels of
nodes adjacent to $u$ are concatenated. If the `edge_attr` parameter is set,
labels for each neighboring node are prefixed with the value of this attribute
along the connecting edge from this neighbor to node $u$. The resulting string
is then hashed to compress this information into a fixed digest size.
Thus, at the i-th iteration, nodes within i hops influence any given
hashed node label. We can therefore say that at depth $i$ for node $u$
we have a hash for a subgraph induced by the i-hop neighborhood of $u$.
The output can be used to create general Weisfeiler-Lehman graph kernels,
or generate features for graphs or nodes - for example to generate 'words' in
a graph as seen in the 'graph2vec' algorithm.
See [1]_ & [2]_ respectively for details.
Hashes are identical for isomorphic subgraphs and there exist strong
guarantees that non-isomorphic graphs will get different hashes.
See [1]_ for details.
If no node or edge attributes are provided, the degree of each node
is used as its initial label.
Otherwise, node and/or edge labels are used to compute the hash.
Parameters
----------
G : graph
The graph to be hashed.
Can have node and/or edge attributes. Can also have no attributes.
edge_attr : string, optional (default=None)
The key in edge attribute dictionary to be used for hashing.
If None, edge labels are ignored.
node_attr : string, optional (default=None)
The key in node attribute dictionary to be used for hashing.
If None, and no edge_attr given, use the degrees of the nodes as labels.
If None, and edge_attr is given, each node starts with an identical label.
iterations : int, optional (default=3)
Number of neighbor aggregations to perform.
Should be larger for larger graphs.
digest_size : int, optional (default=16)
Size (in bytes) of blake2b hash digest to use for hashing node labels.
The default size is 16 bytes.
include_initial_labels : bool, optional (default=False)
If True, include the hashed initial node label as the first subgraph
hash for each node.
Returns
-------
node_subgraph_hashes : dict
A dictionary with each key given by a node in G, and each value given
by the subgraph hashes in order of depth from the key node.
Hashes are hexadecimal strings (hence ``2 * digest_size`` long).
Raises
------
ValueError
If `iterations` is not a positve number.
Examples
--------
Finding similar nodes in different graphs:
>>> G1 = nx.Graph()
>>> G1.add_edges_from([(1, 2), (2, 3), (2, 4), (3, 5), (4, 6), (5, 7), (6, 7)])
>>> G2 = nx.Graph()
>>> G2.add_edges_from([(1, 3), (2, 3), (1, 6), (1, 5), (4, 6)])
>>> g1_hashes = nx.weisfeiler_lehman_subgraph_hashes(
... G1, iterations=4, digest_size=8
... )
>>> g2_hashes = nx.weisfeiler_lehman_subgraph_hashes(
... G2, iterations=4, digest_size=8
... )
Even though G1 and G2 are not isomorphic (they have different numbers of edges),
the hash sequence of depth 3 for node 1 in G1 and node 5 in G2 are similar:
>>> g1_hashes[1]
['f6fc42039fba3776', 'a93b64973cfc8897', 'db1b43ae35a1878f', '57872a7d2059c1c0']
>>> g2_hashes[5]
['f6fc42039fba3776', 'a93b64973cfc8897', 'db1b43ae35a1878f', '1716d2a4012fa4bc']
The first 3 WL subgraph hashes match. From this we can conclude that it's very
likely the neighborhood of 3 hops around these nodes are isomorphic.
However the 4-hop neighborhoods of ``G1`` and ``G2`` are not isomorphic since the
4th hashes in the lists above are not equal.
These nodes may be candidates to be classified together since their local topology
is similar.
Notes
-----
To hash the full graph when subgraph hashes are not needed, use
`weisfeiler_lehman_graph_hash` for efficiency.
Similarity between hashes does not imply similarity between graphs.
References
----------
.. [1] Shervashidze, Nino, Pascal Schweitzer, Erik Jan Van Leeuwen,
Kurt Mehlhorn, and Karsten M. Borgwardt. Weisfeiler Lehman
Graph Kernels. Journal of Machine Learning Research. 2011.
http://www.jmlr.org/papers/volume12/shervashidze11a/shervashidze11a.pdf
.. [2] Annamalai Narayanan, Mahinthan Chandramohan, Rajasekar Venkatesan,
Lihui Chen, Yang Liu and Shantanu Jaiswa. graph2vec: Learning
Distributed Representations of Graphs. arXiv. 2017
https://arxiv.org/pdf/1707.05005.pdf
See also
--------
weisfeiler_lehman_graph_hash
"""
if G.is_directed():
_neighborhood_aggregate = _neighborhood_aggregate_directed
warnings.warn(
"The hashes produced for directed graphs changed in v3.5"
" due to a bugfix (see documentation).",
UserWarning,
stacklevel=2,
)
else:
_neighborhood_aggregate = _neighborhood_aggregate_undirected
def weisfeiler_lehman_step(G, labels, node_subgraph_hashes, edge_attr=None):
"""
Apply neighborhood aggregation to each node
in the graph.
Computes a dictionary with labels for each node.
Appends the new hashed label to the dictionary of subgraph hashes
originating from and indexed by each node in G
"""
new_labels = {}
for node in G.nodes():
label = _neighborhood_aggregate(G, node, labels, edge_attr=edge_attr)
hashed_label = _hash_label(label, digest_size)
new_labels[node] = hashed_label
node_subgraph_hashes[node].append(hashed_label)
return new_labels
if iterations <= 0:
raise ValueError("The WL algorithm requires that `iterations` be positive")
node_labels = _init_node_labels(G, edge_attr, node_attr)
if include_initial_labels:
node_subgraph_hashes = {
k: [_hash_label(v, digest_size)] for k, v in node_labels.items()
}
else:
node_subgraph_hashes = defaultdict(list)
# If the graph has no attributes, initial labels are the nodes' degrees.
# This is equivalent to doing the first iterations of WL.
if not edge_attr and not node_attr:
iterations -= 1
for node in G.nodes():
hashed_label = _hash_label(node_labels[node], digest_size)
node_subgraph_hashes[node].append(hashed_label)
for _ in range(iterations):
node_labels = weisfeiler_lehman_step(
G, node_labels, node_subgraph_hashes, edge_attr
)
return dict(node_subgraph_hashes)
