"""
Functions for hashing graphs to strings.
Isomorphic graphs should be assigned identical hashes.
For now, only Weisfeiler-Lehman hashing is implemented.
"""
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:
return {u: str(deg) for u, deg in G.degree()}
def _neighborhood_aggregate(G, node, node_labels, edge_attr=None):
"""
Compute new labels for given node 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))
[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.
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 bits) of blake2b hash digest to use for hashing node labels.
Returns
-------
h : string
Hexadecimal string corresponding to hash of the input graph.
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)
'7bc4dde9a09d0b94c5097b219891d81a'
>>> nx.weisfeiler_lehman_graph_hash(G2)
'7bc4dde9a09d0b94c5097b219891d81a'
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
"""
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
# set initial node labels
node_labels = _init_node_labels(G, edge_attr, node_attr)
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.
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 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 bits) of blake2b hash digest to use for hashing node labels.
The default size is 16 bits.
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.
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=3, digest_size=8
... )
>>> g2_hashes = nx.weisfeiler_lehman_subgraph_hashes(
... G2, iterations=3, 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]
['a93b64973cfc8897', 'db1b43ae35a1878f', '57872a7d2059c1c0']
>>> g2_hashes[5]
['a93b64973cfc8897', 'db1b43ae35a1878f', '1716d2a4012fa4bc']
The first 2 WL subgraph hashes match. From this we can conclude that it's very
likely the neighborhood of 2 hops around these nodes are isomorphic.
However the 3-hop neighborhoods of ``G1`` and ``G2`` are not isomorphic since the
3rd 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
"""
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
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)
for _ in range(iterations):
node_labels = weisfeiler_lehman_step(
G, node_labels, node_subgraph_hashes, edge_attr
)
return dict(node_subgraph_hashes)
