88 lines
3.0 KiB
Python
88 lines
3.0 KiB
Python
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r"""Function for computing a junction tree of a graph."""
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from itertools import combinations
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import networkx as nx
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from networkx.algorithms import chordal_graph_cliques, complete_to_chordal_graph, moral
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from networkx.utils import not_implemented_for
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__all__ = ["junction_tree"]
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@not_implemented_for("multigraph")
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def junction_tree(G):
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r"""Returns a junction tree of a given graph.
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A junction tree (or clique tree) is constructed from a (un)directed graph G.
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The tree is constructed based on a moralized and triangulated version of G.
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The tree's nodes consist of maximal cliques and sepsets of the revised graph.
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The sepset of two cliques is the intersection of the nodes of these cliques,
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e.g. the sepset of (A,B,C) and (A,C,E,F) is (A,C). These nodes are often called
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"variables" in this literature. The tree is bipartitie with each sepset
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connected to its two cliques.
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Junction Trees are not unique as the order of clique consideration determines
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which sepsets are included.
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The junction tree algorithm consists of five steps [1]_:
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1. Moralize the graph
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2. Triangulate the graph
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3. Find maximal cliques
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4. Build the tree from cliques, connecting cliques with shared
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nodes, set edge-weight to number of shared variables
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5. Find maximum spanning tree
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Parameters
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----------
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G : networkx.Graph
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Directed or undirected graph.
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Returns
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-------
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junction_tree : networkx.Graph
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The corresponding junction tree of `G`.
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Raises
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------
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NetworkXNotImplemented
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Raised if `G` is an instance of `MultiGraph` or `MultiDiGraph`.
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References
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----------
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.. [1] Junction tree algorithm:
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https://en.wikipedia.org/wiki/Junction_tree_algorithm
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.. [2] Finn V. Jensen and Frank Jensen. 1994. Optimal
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junction trees. In Proceedings of the Tenth international
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conference on Uncertainty in artificial intelligence (UAI’94).
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Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 360–366.
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"""
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clique_graph = nx.Graph()
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if G.is_directed():
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G = moral.moral_graph(G)
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chordal_graph, _ = complete_to_chordal_graph(G)
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cliques = [tuple(sorted(i)) for i in chordal_graph_cliques(chordal_graph)]
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clique_graph.add_nodes_from(cliques, type="clique")
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for edge in combinations(cliques, 2):
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set_edge_0 = set(edge[0])
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set_edge_1 = set(edge[1])
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if not set_edge_0.isdisjoint(set_edge_1):
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sepset = tuple(sorted(set_edge_0.intersection(set_edge_1)))
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clique_graph.add_edge(edge[0], edge[1], weight=len(sepset), sepset=sepset)
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junction_tree = nx.maximum_spanning_tree(clique_graph)
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for edge in list(junction_tree.edges(data=True)):
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junction_tree.add_node(edge[2]["sepset"], type="sepset")
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junction_tree.add_edge(edge[0], edge[2]["sepset"])
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junction_tree.add_edge(edge[1], edge[2]["sepset"])
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junction_tree.remove_edge(edge[0], edge[1])
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return junction_tree
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