weisfeiler_lehman_subgraph_hashes

weisfeiler_lehman_subgraph_hashes(G, edge_attr=None, node_attr=None, iterations=3, digest_size=16)

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 + 1 elements - a hash for a subgraph at each depth, and additionally a hash of the initial node label (or equivalently a subgraph of depth 0)

The function iteratively aggregates and hashes neighbourhoods 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 at each step for a node \(n\), all labels of nodes adjacent to \(n\) 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 \(n\). 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 \(n\) we have a hash for a subgraph induced by the \(2i\)-hop neighborhood of \(n\).

The output can be used to 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, default=None

The key in edge attribute dictionary to be used for hashing. If None, edge labels are ignored.

node_attr: string, 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, default=3

Number of neighbor aggregations to perform. Should be larger for larger graphs.

digest_size: int, default=16

Size (in bits) of blake2b hash digest to use for hashing node labels. The default size is 16 bits

Returns:

node_subgraph_hashesdict

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.

See also

weisfeiler_lehman_graph_hash

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] (1,2)

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

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 4 hops around these nodes are isomorphic: each iteration aggregates 1-hop neighbourhoods meaning hashes at depth \(n\) are influenced by every node within \(2n\) hops.

However the neighborhood of 6 hops is no longer isomorphic since their 3rd hash does not match.

These nodes may be candidates to be classified together since their local topology is similar.