Python networkx.chordal_graph_cliques() Examples

def get_clique_tree(nodes, edges):
    """Given a set of int nodes i and edges (i,j), returns an nx.Graph object G
    which is a clique tree, where:
        - G.node[i]['members'] contains the set of original nodes in the ith
            maximal clique
        - G[i][j]['members'] contains the set of original nodes in the seperator
            set between maximal cliques i and j

    Note: This method is currently only implemented for chordal graphs; TODO:
    add a step to triangulate non-chordal graphs.
    """
    # Form the original graph G1
    G1 = nx.Graph()
    G1.add_nodes_from(nodes)
    G1.add_edges_from(edges)

    # Check if graph is chordal
    # TODO: Add step to triangulate graph if not
    if not nx.is_chordal(G1):
        raise NotImplementedError("Graph triangulation not implemented.")

    # Create maximal clique graph G2
    # Each node is a maximal clique C_i
    # Let w = |C_i \cap C_j|; C_i, C_j have an edge with weight w if w > 0
    G2 = nx.Graph()
    for i, c in enumerate(nx.chordal_graph_cliques(G1)):
        G2.add_node(i, members=c)
    for i in G2.nodes:
        for j in G2.nodes:
            S = G2.node[i]["members"].intersection(G2.node[j]["members"])
            w = len(S)
            if w > 0:
                G2.add_edge(i, j, weight=w, members=S)

    # Return a minimum spanning tree of G2
    return nx.minimum_spanning_tree(G2) 

def chordal_graph_cliques(G):
    """Returns the set of maximal cliques of a chordal graph.
    
    The algorithm breaks the graph in connected components and performs a 
    maximum cardinality search in each component to get the cliques.
    
    Parameters
    ----------
    G : graph
      A NetworkX graph 
    
    Returns
    -------
    cliques : A set containing the maximal cliques in G.
    
    Raises
    ------
    NetworkXError
        The algorithm does not support DiGraph, MultiGraph and MultiDiGraph. 
        If the input graph is an instance of one of these classes, a
        NetworkXError is raised.
        The algorithm can only be applied to chordal graphs. If the
        input graph is found to be non-chordal, a NetworkXError is raised.
        
    Examples
    --------
    >>> import networkx as nx
    >>> e= [(1,2),(1,3),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6),(7,8)]
    >>> G = nx.Graph(e)
    >>> G.add_node(9)    
    >>> setlist = nx.chordal_graph_cliques(G)
    """
    if not is_chordal(G):
        raise nx.NetworkXError("Input graph is not chordal.")
    
    cliques = set()
    for C in nx.connected.connected_component_subgraphs(G):
        cliques |= _connected_chordal_graph_cliques(C)
        
    return cliques 

def chordal_graph_cliques(G):
    """Returns the set of maximal cliques of a chordal graph.

    The algorithm breaks the graph in connected components and performs a
    maximum cardinality search in each component to get the cliques.

    Parameters
    ----------
    G : graph
      A NetworkX graph

    Returns
    -------
    cliques : A set containing the maximal cliques in G.

    Raises
    ------
    NetworkXError
        The algorithm does not support DiGraph, MultiGraph and MultiDiGraph.
        If the input graph is an instance of one of these classes, a
        :exc:`NetworkXError` is raised.
        The algorithm can only be applied to chordal graphs. If the
        input graph is found to be non-chordal, a :exc:`NetworkXError` is raised.

    Examples
    --------
    >>> import networkx as nx
    >>> e= [(1,2),(1,3),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6),(7,8)]
    >>> G = nx.Graph(e)
    >>> G.add_node(9)
    >>> setlist = nx.chordal_graph_cliques(G)
    """
    if not is_chordal(G):
        raise nx.NetworkXError("Input graph is not chordal.")

    cliques = set()
    for C in nx.connected.connected_component_subgraphs(G):
        cliques |= _connected_chordal_graph_cliques(C)

    return cliques 

def test_chordal_find_cliques(self):
        cliques = set([frozenset([9]),frozenset([7,8]),frozenset([1,2,3]),
                       frozenset([2,3,4]),frozenset([3,4,5,6])])
        assert_equal(nx.chordal_graph_cliques(self.chordal_G),cliques) 

def test_chordal_find_cliques_path(self):
        G = nx.path_graph(10)
        cliqueset = nx.chordal_graph_cliques(G)
        for (u,v) in G.edges_iter():
            assert_true(frozenset([u,v]) in cliqueset 
                        or frozenset([v,u]) in cliqueset) 

def test_chordal_find_cliquesCC(self):
        cliques = set([frozenset([1,2,3]),frozenset([2,3,4]),
                       frozenset([3,4,5,6])])
        assert_equal(nx.chordal_graph_cliques(self.connected_chordal_G),cliques) 

def test_chordal_find_cliquesCC(self):
        cliques = set([frozenset([1, 2, 3]), frozenset([2, 3, 4]),
                       frozenset([3, 4, 5, 6])])
        assert_equal(nx.chordal_graph_cliques(self.connected_chordal_G), cliques) 

def test_chordal_find_cliques(self):
        cliques = set([frozenset([9]), frozenset([7, 8]), frozenset([1, 2, 3]),
                       frozenset([2, 3, 4]), frozenset([3, 4, 5, 6])])
        assert_equal(nx.chordal_graph_cliques(self.chordal_G), cliques) 

def test_chordal_find_cliques_path(self):
        G = nx.path_graph(10)
        cliqueset = nx.chordal_graph_cliques(G)
        for (u, v) in G.edges():
            assert_true(frozenset([u, v]) in cliqueset
                        or frozenset([v, u]) in cliqueset) 

def test_chordal_find_cliquesCC(self):
        cliques = set([frozenset([1, 2, 3]), frozenset([2, 3, 4]),
                       frozenset([3, 4, 5, 6])])
        assert_equal(nx.chordal_graph_cliques(self.connected_chordal_G), cliques) 

def chordal_graph_cliques(G):
    """Returns the set of maximal cliques of a chordal graph.

    The algorithm breaks the graph in connected components and performs a
    maximum cardinality search in each component to get the cliques.

    Parameters
    ----------
    G : graph
      A NetworkX graph

    Returns
    -------
    cliques : A set containing the maximal cliques in G.

    Raises
    ------
    NetworkXError
        The algorithm does not support DiGraph, MultiGraph and MultiDiGraph.
        If the input graph is an instance of one of these classes, a
        :exc:`NetworkXError` is raised.
        The algorithm can only be applied to chordal graphs. If the
        input graph is found to be non-chordal, a :exc:`NetworkXError` is raised.

    Examples
    --------
    >>> import networkx as nx
    >>> e= [(1,2),(1,3),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6),(7,8)]
    >>> G = nx.Graph(e)
    >>> G.add_node(9)
    >>> setlist = nx.chordal_graph_cliques(G)
    """
    if not is_chordal(G):
        raise nx.NetworkXError("Input graph is not chordal.")

    cliques = set()
    for C in nx.connected.connected_component_subgraphs(G):
        cliques |= _connected_chordal_graph_cliques(C)

    return cliques 

def test_chordal_find_cliques_path(self):
        G = nx.path_graph(10)
        cliqueset = nx.chordal_graph_cliques(G)
        for (u, v) in G.edges():
            assert_true(frozenset([u, v]) in cliqueset
                        or frozenset([v, u]) in cliqueset) 

def test_chordal_find_cliques(self):
        cliques = set([frozenset([9]), frozenset([7, 8]), frozenset([1, 2, 3]),
                       frozenset([2, 3, 4]), frozenset([3, 4, 5, 6])])
        assert_equal(nx.chordal_graph_cliques(self.chordal_G), cliques) 

def chordal_graph_treewidth(G):
    """Returns the treewidth of the chordal graph G.

    Parameters
    ----------
    G : graph
      A NetworkX graph

    Returns
    -------
    treewidth : int
        The size of the largest clique in the graph minus one.

    Raises
    ------
    NetworkXError
        The algorithm does not support DiGraph, MultiGraph and MultiDiGraph.
        If the input graph is an instance of one of these classes, a
        :exc:`NetworkXError` is raised.
        The algorithm can only be applied to chordal graphs. If
        the input graph is found to be non-chordal, a :exc:`NetworkXError` is raised.

    Examples
    --------
    >>> import networkx as nx
    >>> e = [(1,2),(1,3),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6),(7,8)]
    >>> G = nx.Graph(e)
    >>> G.add_node(9)
    >>> nx.chordal_graph_treewidth(G)
    3

    References
    ----------
    .. [1] https://en.wikipedia.org/wiki/Tree_decomposition#Treewidth
    """
    if not is_chordal(G):
        raise nx.NetworkXError("Input graph is not chordal.")

    max_clique = -1
    for clique in nx.chordal_graph_cliques(G):
        max_clique = max(max_clique, len(clique))
    return max_clique - 1 

def chordal_graph_treewidth(G):
    """Returns the treewidth of the chordal graph G.

    Parameters
    ----------
    G : graph
      A NetworkX graph

    Returns
    -------
    treewidth : int
        The size of the largest clique in the graph minus one.

    Raises
    ------
    NetworkXError
        The algorithm does not support DiGraph, MultiGraph and MultiDiGraph.
        If the input graph is an instance of one of these classes, a
        :exc:`NetworkXError` is raised.
        The algorithm can only be applied to chordal graphs. If
        the input graph is found to be non-chordal, a :exc:`NetworkXError` is raised.

    Examples
    --------
    >>> import networkx as nx
    >>> e = [(1,2),(1,3),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6),(7,8)]
    >>> G = nx.Graph(e)
    >>> G.add_node(9)
    >>> nx.chordal_graph_treewidth(G)
    3

    References
    ----------
    .. [1] https://en.wikipedia.org/wiki/Tree_decomposition#Treewidth
    """
    if not is_chordal(G):
        raise nx.NetworkXError("Input graph is not chordal.")

    max_clique = -1
    for clique in nx.chordal_graph_cliques(G):
        max_clique = max(max_clique, len(clique))
    return max_clique - 1 

def initialize_tree(self):
        """
        Initialize the structure of a clique tree, using
        the following steps:
            - Moralize graph (i.e. marry parents)
            - Triangulate graph (i.e. make graph chordal)
            - Get max cliques (i.e. community/clique detection)
            - Max spanning tree over sepset cardinality (i.e. create tree)
        
        """
        ### MORALIZE GRAPH & MAKE IT CHORDAL ###
        chordal_G = make_chordal(self.bn) # must return a networkx object
        V = chordal_G.nodes()

        ### GET MAX CLIQUES FROM CHORDAL GRAPH ###
        C = {} # key = vertex, value = clique object
        max_cliques = reversed(list(nx.chordal_graph_cliques(chordal_G)))
        for v_idx,clique in enumerate(max_cliques):
            C[v_idx] = Clique(set(clique))

        ### MAXIMUM SPANNING TREE OVER COMPLETE GRAPH TO MAKE A TREE ###
        weighted_edge_dict = dict([(c_idx,{}) for c_idx in range(len(C))])
        for i in range(len(C)):
            for j in range(len(C)):
                if i!=j:
                    intersect_cardinality = len(C[i].sepset(C[j]))
                    weighted_edge_dict[i][j] = -1*intersect_cardinality
        mst_G = mst(weighted_edge_dict)
        ### SET V,E,C ###
        self.E = mst_G # dictionary
        self.V = mst_G.keys() # list
        self.C = C

        
        ### ASSIGN EACH FACTOR TO ONE CLIQUE ONLY ###
        v_a = dict([(rv, False) for rv in self.bn.nodes()])
        for clique in self.C.values():
            temp_scope = []
            for var in v_a:
                if v_a[var] == False and set(self.bn.scope(var)).issubset(clique.scope):
                    temp_scope.append(var)
                    v_a[var] = True
            clique._F = Factorization(self.bn, temp_scope)

        ### COMPUTE INITIAL POTENTIAL FOR EACH FACTOR ###
        # - i.e. multiply all of its assigned factors together
        for i, clique in self.C.items():
            if len(self.parents(i)) == 0:
                clique.is_ready = True
            clique.initialize_psi() 

def chordal_graph_treewidth(G):
    """Returns the treewidth of the chordal graph G.
 
    Parameters
    ----------
    G : graph
      A NetworkX graph 
    
    Returns
    -------
    treewidth : int
	The size of the largest clique in the graph minus one.
    
    Raises
    ------
    NetworkXError
        The algorithm does not support DiGraph, MultiGraph and MultiDiGraph. 
        If the input graph is an instance of one of these classes, a
        NetworkXError is raised.
        The algorithm can only be applied to chordal graphs. If
        the input graph is found to be non-chordal, a NetworkXError is raised.
        
    Examples
    --------
    >>> import networkx as nx
    >>> e = [(1,2),(1,3),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6),(7,8)]
    >>> G = nx.Graph(e)
    >>> G.add_node(9)    
    >>> nx.chordal_graph_treewidth(G)
    3

    References
    ----------
    .. [1] http://en.wikipedia.org/wiki/Tree_decomposition#Treewidth
    """
    if not is_chordal(G):
        raise nx.NetworkXError("Input graph is not chordal.")
    
    max_clique = -1
    for clique in nx.chordal_graph_cliques(G):
        max_clique = max(max_clique,len(clique))
    return max_clique - 1 

def get_clique_tree(nodes: Iterable[int], edges: List[Tuple[int, int]]) -> nx.Graph:
    """
    Given a set of int nodes i and edges (i,j), returns a clique tree.

    Clique tree is an object G for which:
    - G.node[i]['members'] contains the set of original nodes in the ith
        maximal clique
    - G[i][j]['members'] contains the set of original nodes in the seperator
        set between maximal cliques i and j

    Note: This method is currently only implemented for chordal graphs; TODO:
    add a step to triangulate non-chordal graphs.

    Parameters
    ----------
    nodes
        A list of nodes indices
    edges
        A list of tuples, where each tuple has indices for connected nodes

    Returns
    -------
    networkx.Graph
        An object G representing clique tree
    """
    # Form the original graph G1
    G1 = nx.Graph()
    G1.add_nodes_from(nodes)
    G1.add_edges_from(edges)

    # Check if graph is chordal
    # TODO: Add step to triangulate graph if not
    if not nx.is_chordal(G1):
        raise NotImplementedError("Graph triangulation not implemented.")

    # Create maximal clique graph G2
    # Each node is a maximal clique C_i
    # Let w = |C_i \cap C_j|; C_i, C_j have an edge with weight w if w > 0
    G2 = nx.Graph()
    for i, c in enumerate(nx.chordal_graph_cliques(G1)):
        G2.add_node(i, members=c)
    for i in G2.nodes:
        for j in G2.nodes:
            S = G2.node[i]["members"].intersection(G2.node[j]["members"])
            w = len(S)
            if w > 0:
                G2.add_edge(i, j, weight=w, members=S)

    # Return a minimum spanning tree of G2
    return nx.minimum_spanning_tree(G2)