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