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""" |
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Functions for hashing graphs to strings. |
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Isomorphic graphs should be assigned identical hashes. |
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For now, only Weisfeiler-Lehman hashing is implemented. |
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""" |
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from collections import Counter, defaultdict |
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from hashlib import blake2b |
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import networkx as nx |
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__all__ = ["weisfeiler_lehman_graph_hash", "weisfeiler_lehman_subgraph_hashes"] |
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def _hash_label(label, digest_size): |
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return blake2b(label.encode("ascii"), digest_size=digest_size).hexdigest() |
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def _init_node_labels(G, edge_attr, node_attr): |
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if node_attr: |
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return {u: str(dd[node_attr]) for u, dd in G.nodes(data=True)} |
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elif edge_attr: |
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return {u: "" for u in G} |
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else: |
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return {u: str(deg) for u, deg in G.degree()} |
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def _neighborhood_aggregate(G, node, node_labels, edge_attr=None): |
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""" |
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Compute new labels for given node by aggregating |
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the labels of each node's neighbors. |
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""" |
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label_list = [] |
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for nbr in G.neighbors(node): |
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prefix = "" if edge_attr is None else str(G[node][nbr][edge_attr]) |
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label_list.append(prefix + node_labels[nbr]) |
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return node_labels[node] + "".join(sorted(label_list)) |
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@nx._dispatchable(edge_attrs={"edge_attr": None}, node_attrs="node_attr") |
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def weisfeiler_lehman_graph_hash( |
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G, edge_attr=None, node_attr=None, iterations=3, digest_size=16 |
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): |
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"""Return Weisfeiler Lehman (WL) graph hash. |
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The function iteratively aggregates and hashes neighborhoods of each node. |
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After each node's neighbors are hashed to obtain updated node labels, |
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a hashed histogram of resulting labels is returned as the final hash. |
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Hashes are identical for isomorphic graphs and strong guarantees that |
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non-isomorphic graphs will get different hashes. See [1]_ for details. |
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If no node or edge attributes are provided, the degree of each node |
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is used as its initial label. |
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Otherwise, node and/or edge labels are used to compute the hash. |
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Parameters |
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---------- |
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G : graph |
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The graph to be hashed. |
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Can have node and/or edge attributes. Can also have no attributes. |
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edge_attr : string, optional (default=None) |
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The key in edge attribute dictionary to be used for hashing. |
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If None, edge labels are ignored. |
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node_attr: string, optional (default=None) |
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The key in node attribute dictionary to be used for hashing. |
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If None, and no edge_attr given, use the degrees of the nodes as labels. |
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iterations: int, optional (default=3) |
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Number of neighbor aggregations to perform. |
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Should be larger for larger graphs. |
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digest_size: int, optional (default=16) |
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Size (in bits) of blake2b hash digest to use for hashing node labels. |
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Returns |
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------- |
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h : string |
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Hexadecimal string corresponding to hash of the input graph. |
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Examples |
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-------- |
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Two graphs with edge attributes that are isomorphic, except for |
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differences in the edge labels. |
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>>> G1 = nx.Graph() |
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>>> G1.add_edges_from( |
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... [ |
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... (1, 2, {"label": "A"}), |
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... (2, 3, {"label": "A"}), |
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... (3, 1, {"label": "A"}), |
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... (1, 4, {"label": "B"}), |
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... ] |
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... ) |
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>>> G2 = nx.Graph() |
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>>> G2.add_edges_from( |
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... [ |
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... (5, 6, {"label": "B"}), |
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... (6, 7, {"label": "A"}), |
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... (7, 5, {"label": "A"}), |
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... (7, 8, {"label": "A"}), |
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... ] |
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... ) |
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Omitting the `edge_attr` option, results in identical hashes. |
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>>> nx.weisfeiler_lehman_graph_hash(G1) |
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'7bc4dde9a09d0b94c5097b219891d81a' |
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>>> nx.weisfeiler_lehman_graph_hash(G2) |
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'7bc4dde9a09d0b94c5097b219891d81a' |
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With edge labels, the graphs are no longer assigned |
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the same hash digest. |
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>>> nx.weisfeiler_lehman_graph_hash(G1, edge_attr="label") |
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'c653d85538bcf041d88c011f4f905f10' |
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>>> nx.weisfeiler_lehman_graph_hash(G2, edge_attr="label") |
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'3dcd84af1ca855d0eff3c978d88e7ec7' |
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Notes |
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----- |
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To return the WL hashes of each subgraph of a graph, use |
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`weisfeiler_lehman_subgraph_hashes` |
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Similarity between hashes does not imply similarity between graphs. |
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References |
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---------- |
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.. [1] Shervashidze, Nino, Pascal Schweitzer, Erik Jan Van Leeuwen, |
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Kurt Mehlhorn, and Karsten M. Borgwardt. Weisfeiler Lehman |
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Graph Kernels. Journal of Machine Learning Research. 2011. |
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http://www.jmlr.org/papers/volume12/shervashidze11a/shervashidze11a.pdf |
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See also |
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-------- |
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weisfeiler_lehman_subgraph_hashes |
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""" |
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def weisfeiler_lehman_step(G, labels, edge_attr=None): |
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""" |
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Apply neighborhood aggregation to each node |
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in the graph. |
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Computes a dictionary with labels for each node. |
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""" |
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new_labels = {} |
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for node in G.nodes(): |
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label = _neighborhood_aggregate(G, node, labels, edge_attr=edge_attr) |
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new_labels[node] = _hash_label(label, digest_size) |
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return new_labels |
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node_labels = _init_node_labels(G, edge_attr, node_attr) |
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subgraph_hash_counts = [] |
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for _ in range(iterations): |
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node_labels = weisfeiler_lehman_step(G, node_labels, edge_attr=edge_attr) |
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counter = Counter(node_labels.values()) |
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subgraph_hash_counts.extend(sorted(counter.items(), key=lambda x: x[0])) |
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return _hash_label(str(tuple(subgraph_hash_counts)), digest_size) |
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@nx._dispatchable(edge_attrs={"edge_attr": None}, node_attrs="node_attr") |
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def weisfeiler_lehman_subgraph_hashes( |
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G, |
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edge_attr=None, |
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node_attr=None, |
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iterations=3, |
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digest_size=16, |
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include_initial_labels=False, |
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): |
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""" |
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Return a dictionary of subgraph hashes by node. |
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Dictionary keys are nodes in `G`, and values are a list of hashes. |
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Each hash corresponds to a subgraph rooted at a given node u in `G`. |
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Lists of subgraph hashes are sorted in increasing order of depth from |
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their root node, with the hash at index i corresponding to a subgraph |
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of nodes at most i edges distance from u. Thus, each list will contain |
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`iterations` elements - a hash for a subgraph at each depth. If |
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`include_initial_labels` is set to `True`, each list will additionally |
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have contain a hash of the initial node label (or equivalently a |
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subgraph of depth 0) prepended, totalling ``iterations + 1`` elements. |
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The function iteratively aggregates and hashes neighborhoods of each node. |
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This is achieved for each step by replacing for each node its label from |
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the previous iteration with its hashed 1-hop neighborhood aggregate. |
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The new node label is then appended to a list of node labels for each |
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node. |
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To aggregate neighborhoods for a node $u$ at each step, all labels of |
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nodes adjacent to $u$ are concatenated. If the `edge_attr` parameter is set, |
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labels for each neighboring node are prefixed with the value of this attribute |
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along the connecting edge from this neighbor to node $u$. The resulting string |
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is then hashed to compress this information into a fixed digest size. |
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Thus, at the $i$-th iteration, nodes within $i$ hops influence any given |
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hashed node label. We can therefore say that at depth $i$ for node $u$ |
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we have a hash for a subgraph induced by the $i$-hop neighborhood of $u$. |
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The output can be used to to create general Weisfeiler-Lehman graph kernels, |
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or generate features for graphs or nodes - for example to generate 'words' in |
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a graph as seen in the 'graph2vec' algorithm. |
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See [1]_ & [2]_ respectively for details. |
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Hashes are identical for isomorphic subgraphs and there exist strong |
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guarantees that non-isomorphic graphs will get different hashes. |
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See [1]_ for details. |
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If no node or edge attributes are provided, the degree of each node |
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is used as its initial label. |
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Otherwise, node and/or edge labels are used to compute the hash. |
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Parameters |
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---------- |
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|
G : graph |
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|
The graph to be hashed. |
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|
Can have node and/or edge attributes. Can also have no attributes. |
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|
edge_attr : string, optional (default=None) |
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The key in edge attribute dictionary to be used for hashing. |
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If None, edge labels are ignored. |
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node_attr : string, optional (default=None) |
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The key in node attribute dictionary to be used for hashing. |
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If None, and no edge_attr given, use the degrees of the nodes as labels. |
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If None, and edge_attr is given, each node starts with an identical label. |
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iterations : int, optional (default=3) |
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Number of neighbor aggregations to perform. |
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Should be larger for larger graphs. |
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digest_size : int, optional (default=16) |
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Size (in bits) of blake2b hash digest to use for hashing node labels. |
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The default size is 16 bits. |
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include_initial_labels : bool, optional (default=False) |
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If True, include the hashed initial node label as the first subgraph |
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hash for each node. |
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Returns |
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------- |
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node_subgraph_hashes : dict |
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A dictionary with each key given by a node in G, and each value given |
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by the subgraph hashes in order of depth from the key node. |
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Examples |
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-------- |
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Finding similar nodes in different graphs: |
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>>> G1 = nx.Graph() |
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>>> G1.add_edges_from([(1, 2), (2, 3), (2, 4), (3, 5), (4, 6), (5, 7), (6, 7)]) |
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>>> G2 = nx.Graph() |
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>>> G2.add_edges_from([(1, 3), (2, 3), (1, 6), (1, 5), (4, 6)]) |
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>>> g1_hashes = nx.weisfeiler_lehman_subgraph_hashes(G1, iterations=3, digest_size=8) |
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>>> g2_hashes = nx.weisfeiler_lehman_subgraph_hashes(G2, iterations=3, digest_size=8) |
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Even though G1 and G2 are not isomorphic (they have different numbers of edges), |
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the hash sequence of depth 3 for node 1 in G1 and node 5 in G2 are similar: |
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>>> g1_hashes[1] |
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['a93b64973cfc8897', 'db1b43ae35a1878f', '57872a7d2059c1c0'] |
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>>> g2_hashes[5] |
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['a93b64973cfc8897', 'db1b43ae35a1878f', '1716d2a4012fa4bc'] |
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The first 2 WL subgraph hashes match. From this we can conclude that it's very |
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likely the neighborhood of 2 hops around these nodes are isomorphic. |
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However the 3-hop neighborhoods of ``G1`` and ``G2`` are not isomorphic since the |
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3rd hashes in the lists above are not equal. |
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These nodes may be candidates to be classified together since their local topology |
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is similar. |
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Notes |
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----- |
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To hash the full graph when subgraph hashes are not needed, use |
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`weisfeiler_lehman_graph_hash` for efficiency. |
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Similarity between hashes does not imply similarity between graphs. |
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References |
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---------- |
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.. [1] Shervashidze, Nino, Pascal Schweitzer, Erik Jan Van Leeuwen, |
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|
Kurt Mehlhorn, and Karsten M. Borgwardt. Weisfeiler Lehman |
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|
Graph Kernels. Journal of Machine Learning Research. 2011. |
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http://www.jmlr.org/papers/volume12/shervashidze11a/shervashidze11a.pdf |
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.. [2] Annamalai Narayanan, Mahinthan Chandramohan, Rajasekar Venkatesan, |
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Lihui Chen, Yang Liu and Shantanu Jaiswa. graph2vec: Learning |
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Distributed Representations of Graphs. arXiv. 2017 |
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https://arxiv.org/pdf/1707.05005.pdf |
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See also |
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-------- |
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weisfeiler_lehman_graph_hash |
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""" |
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def weisfeiler_lehman_step(G, labels, node_subgraph_hashes, edge_attr=None): |
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""" |
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Apply neighborhood aggregation to each node |
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in the graph. |
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Computes a dictionary with labels for each node. |
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Appends the new hashed label to the dictionary of subgraph hashes |
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originating from and indexed by each node in G |
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""" |
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new_labels = {} |
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for node in G.nodes(): |
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label = _neighborhood_aggregate(G, node, labels, edge_attr=edge_attr) |
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hashed_label = _hash_label(label, digest_size) |
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new_labels[node] = hashed_label |
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node_subgraph_hashes[node].append(hashed_label) |
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return new_labels |
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node_labels = _init_node_labels(G, edge_attr, node_attr) |
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if include_initial_labels: |
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node_subgraph_hashes = { |
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k: [_hash_label(v, digest_size)] for k, v in node_labels.items() |
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} |
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else: |
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node_subgraph_hashes = defaultdict(list) |
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for _ in range(iterations): |
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node_labels = weisfeiler_lehman_step( |
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G, node_labels, node_subgraph_hashes, edge_attr |
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) |
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return dict(node_subgraph_hashes) |
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|
