Python Graph.set_pos方法代码示例

# 需要导入模块: from sage.graphs.graph import Graph [as 别名]
# 或者: from sage.graphs.graph.Graph import set_pos [as 别名]
def make_graph(M):
    """
    Code extracted from Sage's elliptic curve isogeny class (reshaped
    in the case maxdegree==12)
    """
    from sage.schemes.elliptic_curves.ell_curve_isogeny import fill_isogeny_matrix, unfill_isogeny_matrix
    from sage.graphs.graph import Graph
    n = M.nrows() # = M.ncols()
    G = Graph(unfill_isogeny_matrix(M), format='weighted_adjacency_matrix')
    MM = fill_isogeny_matrix(M)
    # The maximum degree classifies the shape of the isogeny
    # graph, though the number of vertices is often enough.
    # This only holds over Q, so this code will need to change
    # once other isogeny classes are implemented.
    if n == 1:
        # one vertex
        pass
    elif n == 2:
        # one edge, two vertices.  We align horizontally and put
        # the lower number on the left vertex.
        G.set_pos(pos={0:[-0.5,0],1:[0.5,0]})
    else:
        maxdegree = max(max(MM))
        if n == 3:
            # o--o--o
            centervert = [i for i in range(3) if max(MM.row(i)) < maxdegree][0]
            other = [i for i in range(3) if i != centervert]
            G.set_pos(pos={centervert:[0,0],other[0]:[-1,0],other[1]:[1,0]})
        elif maxdegree == 4:
            # o--o<8
            centervert = [i for i in range(4) if max(MM.row(i)) < maxdegree][0]
            other = [i for i in range(4) if i != centervert]
            G.set_pos(pos={centervert:[0,0],other[0]:[0,1],other[1]:[-0.8660254,-0.5],other[2]:[0.8660254,-0.5]})
        elif maxdegree == 27:
            # o--o--o--o
            centers = [i for i in range(4) if list(MM.row(i)).count(3) == 2]
            left = [j for j in range(4) if MM[centers[0],j] == 3 and j not in centers][0]
            right = [j for j in range(4) if MM[centers[1],j] == 3 and j not in centers][0]
            G.set_pos(pos={left:[-1.5,0],centers[0]:[-0.5,0],centers[1]:[0.5,0],right:[1.5,0]})
        elif n == 4:
            # square
            opp = [i for i in range(1,4) if not MM[0,i].is_prime()][0]
            other = [i for i in range(1,4) if i != opp]
            G.set_pos(pos={0:[1,1],other[0]:[-1,1],opp:[-1,-1],other[1]:[1,-1]})
        elif maxdegree == 8:
            # 8>o--o<8
            centers = [i for i in range(6) if list(MM.row(i)).count(2) == 3]
            left = [j for j in range(6) if MM[centers[0],j] == 2 and j not in centers]
            right = [j for j in range(6) if MM[centers[1],j] == 2 and j not in centers]
            G.set_pos(pos={centers[0]:[-0.5,0],left[0]:[-1,0.8660254],left[1]:[-1,-0.8660254],centers[1]:[0.5,0],right[0]:[1,0.8660254],right[1]:[1,-0.8660254]})
        elif maxdegree == 18:
            # two squares joined on an edge
            centers = [i for i in range(6) if list(MM.row(i)).count(3) == 2]
            top = [j for j in range(6) if MM[centers[0],j] == 3]
            bl = [j for j in range(6) if MM[top[0],j] == 2][0]
            br = [j for j in range(6) if MM[top[1],j] == 2][0]
            G.set_pos(pos={centers[0]:[0,0.5],centers[1]:[0,-0.5],top[0]:[-1,0.5],top[1]:[1,0.5],bl:[-1,-0.5],br:[1,-0.5]})
        elif maxdegree == 16:
            # tree from bottom, 3 regular except for the leaves.
            centers = [i for i in range(8) if list(MM.row(i)).count(2) == 3]
            center = [i for i in centers if len([j for j in centers if MM[i,j] == 2]) == 2][0]
            centers.remove(center)
            bottom = [j for j in range(8) if MM[center,j] == 2 and j not in centers][0]
            left = [j for j in range(8) if MM[centers[0],j] == 2 and j != center]
            right = [j for j in range(8) if MM[centers[1],j] == 2 and j != center]
            G.set_pos(pos={center:[0,0],bottom:[0,-1],centers[0]:[-0.8660254,0.5],centers[1]:[0.8660254,0.5],left[0]:[-0.8660254,1.5],right[0]:[0.8660254,1.5],left[1]:[-1.7320508,0],right[1]:[1.7320508,0]})
        elif maxdegree == 12:
            # tent
            centers = [i for i in range(8) if list(MM.row(i)).count(2) == 3]
            left = [j for j in range(8) if MM[centers[0],j] == 2]
            right = []
            for i in range(3):
                right.append([j for j in range(8) if MM[centers[1],j] == 2 and MM[left[i],j] == 3][0])
            G.set_pos(pos={centers[0]:[-0.3,0],centers[1]:[0.3,0],
                           left[0]:[-0.14,0.15], right[0]:[0.14,0.15],
                           left[1]:[-0.14,-0.15],right[1]:[0.14,-0.15],
                           left[2]:[-0.14,-0.3],right[2]:[0.14,-0.3]})

    G.relabel(range(1,n+1))
    return G

# 需要导入模块: from sage.graphs.graph import Graph [as 别名]
# 或者: from sage.graphs.graph.Graph import set_pos [as 别名]
def RandomBipartite(n1,n2, p):
    r"""
    Returns a bipartite graph with `n1+n2` vertices
    such that any edge from `[n1]` to `[n2]` exists
    with probability `p`.

    INPUT:

        - ``n1,n2`` : Cardinalities of the two sets
        - ``p``   : Probability for an edge to exist


    EXAMPLE::

        sage: g=graphs.RandomBipartite(5,2,0.5)
        sage: g.vertices()
        [(0, 0), (0, 1), (0, 2), (0, 3), (0, 4), (1, 0), (1, 1)]

    TESTS::

        sage: g=graphs.RandomBipartite(5,-3,0.5)
        Traceback (most recent call last):
        ...
        ValueError: n1 and n2 should be integers strictly greater than 0
        sage: g=graphs.RandomBipartite(5,3,1.5)
        Traceback (most recent call last):
        ...
        ValueError: Parameter p is a probability, and so should be a real value between 0 and 1

    Trac ticket #12155::

        sage: graphs.RandomBipartite(5,6,.2).complement()
        complement(Random bipartite graph of size 5+6 with edge probability 0.200000000000000): Graph on 11 vertices
    """
    if not (p>=0 and p<=1):
        raise ValueError, "Parameter p is a probability, and so should be a real value between 0 and 1"
    if not (n1>0 and n2>0):
        raise ValueError, "n1 and n2 should be integers strictly greater than 0"

    from numpy.random import uniform

    g=Graph(name="Random bipartite graph of size "+str(n1) +"+"+str(n2)+" with edge probability "+str(p))

    S1=[(0,i) for i in range(n1)]
    S2=[(1,i) for i in range(n2)]
    g.add_vertices(S1)
    g.add_vertices(S2)

    for w in range(n2):
        for v in range(n1):
            if uniform()<=p :
                g.add_edge((0,v),(1,w))

    pos = {}
    for i in range(n1):
        pos[(0,i)] = (0, i/(n1-1.0))
    for i in range(n2):
        pos[(1,i)] = (1, i/(n2-1.0))

    g.set_pos(pos)

    return g

# 需要导入模块: from sage.graphs.graph import Graph [as 别名]
# 或者: from sage.graphs.graph.Graph import set_pos [as 别名]
    def graph(self):
        """
        Returns a graph whose vertices correspond to curves in this class, and whose edges correspond to prime degree isogenies.

        .. note:

            There are only finitely many possible isogeny graphs for
            curves over Q [M78].  This function tries to lay out the graph
            nicely by special casing each isogeny graph.

        .. note:

            The vertices are labeled 1 to n rather than 0 to n-1 to
            correspond to LMFDB and Cremona labels.

        EXAMPLES::

            sage: isocls = EllipticCurve('15a3').isogeny_class(use_tuple=False)
            sage: G = isocls.graph()
            sage: sorted(G._pos.items())
            [(1, [-0.8660254, 0.5]), (2, [-0.8660254, 1.5]), (3, [-1.7320508, 0]), (4, [0, 0]), (5, [0, -1]), (6, [0.8660254, 0.5]), (7, [0.8660254, 1.5]), (8, [1.7320508, 0])]

        REFERENCES:

        .. [M78] B. Mazur.  Rational Isogenies of Prime Degree.
          *Inventiones mathematicae* 44,129-162 (1978).
        """
        from sage.graphs.graph import Graph
        M = self.matrix(fill = False)
        n = M.nrows() # = M.ncols()
        G = Graph(M, format='weighted_adjacency_matrix')
        N = self.matrix(fill = True)
        D = dict([(v,self.curves[v]) for v in G.vertices()])
        # The maximum degree classifies the shape of the isogeny
        # graph, though the number of vertices is often enough.
        # This only holds over Q, so this code will need to change
        # once other isogeny classes are implemented.
        if n == 1:
            # one vertex
            pass
        elif n == 2:
            # one edge, two vertices.  We align horizontally and put
            # the lower number on the left vertex.
            G.set_pos(pos={0:[-0.5,0],1:[0.5,0]})
        else:
            maxdegree = max(max(N))
            if n == 3:
                # o--o--o
                centervert = [i for i in range(3) if max(N.row(i)) < maxdegree][0]
                other = [i for i in range(3) if i != centervert]
                G.set_pos(pos={centervert:[0,0],other[0]:[-1,0],other[1]:[1,0]})
            elif maxdegree == 4:
                # o--o<8
                centervert = [i for i in range(4) if max(N.row(i)) < maxdegree][0]
                other = [i for i in range(4) if i != centervert]
                G.set_pos(pos={centervert:[0,0],other[0]:[0,1],other[1]:[-0.8660254,-0.5],other[2]:[0.8660254,-0.5]})
            elif maxdegree == 27:
                # o--o--o--o
                centers = [i for i in range(4) if list(N.row(i)).count(3) == 2]
                left = [j for j in range(4) if N[centers[0],j] == 3 and j not in centers][0]
                right = [j for j in range(4) if N[centers[1],j] == 3 and j not in centers][0]
                G.set_pos(pos={left:[-1.5,0],centers[0]:[-0.5,0],centers[1]:[0.5,0],right:[1.5,0]})
            elif n == 4:
                # square
                opp = [i for i in range(1,4) if not N[0,i].is_prime()][0]
                other = [i for i in range(1,4) if i != opp]
                G.set_pos(pos={0:[1,1],other[0]:[-1,1],opp:[-1,-1],other[1]:[1,-1]})
            elif maxdegree == 8:
                # 8>o--o<8
                centers = [i for i in range(6) if list(N.row(i)).count(2) == 3]
                left = [j for j in range(6) if N[centers[0],j] == 2 and j not in centers]
                right = [j for j in range(6) if N[centers[1],j] == 2 and j not in centers]
                G.set_pos(pos={centers[0]:[-0.5,0],left[0]:[-1,0.8660254],left[1]:[-1,-0.8660254],centers[1]:[0.5,0],right[0]:[1,0.8660254],right[1]:[1,-0.8660254]})
            elif maxdegree == 18:
                # two squares joined on an edge
                centers = [i for i in range(6) if list(N.row(i)).count(3) == 2]
                top = [j for j in range(6) if N[centers[0],j] == 3]
                bl = [j for j in range(6) if N[top[0],j] == 2][0]
                br = [j for j in range(6) if N[top[1],j] == 2][0]
                G.set_pos(pos={centers[0]:[0,0.5],centers[1]:[0,-0.5],top[0]:[-1,0.5],top[1]:[1,0.5],bl:[-1,-0.5],br:[1,-0.5]})
            elif maxdegree == 16:
                # tree from bottom, 3 regular except for the leaves.
                centers = [i for i in range(8) if list(N.row(i)).count(2) == 3]
                center = [i for i in centers if len([j for j in centers if N[i,j] == 2]) == 2][0]
                centers.remove(center)
                bottom = [j for j in range(8) if N[center,j] == 2 and j not in centers][0]
                left = [j for j in range(8) if N[centers[0],j] == 2 and j != center]
                right = [j for j in range(8) if N[centers[1],j] == 2 and j != center]
                G.set_pos(pos={center:[0,0],bottom:[0,-1],centers[0]:[-0.8660254,0.5],centers[1]:[0.8660254,0.5],left[0]:[-0.8660254,1.5],right[0]:[0.8660254,1.5],left[1]:[-1.7320508,0],right[1]:[1.7320508,0]})
            elif maxdegree == 12:
                # tent
                centers = [i for i in range(8) if list(N.row(i)).count(2) == 3]
                left = [j for j in range(8) if N[centers[0],j] == 2]
                right = []
                for i in range(3):
                    right.append([j for j in range(8) if N[centers[1],j] == 2 and N[left[i],j] == 3][0])
                G.set_pos(pos={centers[0]:[-0.75,0],centers[1]:[0.75,0],left[0]:[-0.75,1],right[0]:[0.75,1],left[1]:[-1.25,-0.75],right[1]:[0.25,-0.75],left[2]:[-0.25,-0.25],right[2]:[1.25,-0.25]})
        G.set_vertices(D)
        G.relabel(range(1,n+1))
        return G

# 需要导入模块: from sage.graphs.graph import Graph [as 别名]
# 或者: from sage.graphs.graph.Graph import set_pos [as 别名]
def CompleteMultipartiteGraph(l):
    r"""
    Returns a complete multipartite graph.

    INPUT:

    - ``l`` -- a list of integers : the respective sizes
      of the components.

    EXAMPLE:

    A complete tripartite graph with sets of sizes
    `5, 6, 8`::

        sage: g = graphs.CompleteMultipartiteGraph([5, 6, 8]); g
        Multipartite Graph with set sizes [5, 6, 8]: Graph on 19 vertices

    It clearly has a chromatic number of 3::

        sage: g.chromatic_number()
        3
    """
    
    n = sum(l) #getting the number of vertices
    r = len(l) #getting the number of partitions
    positions = {}

    if r > 2: #position code gives bad results on bipartite or isolated graphs

        '''
        Produce a layout of the vertices so that vertices in the same
        vertex set are adjecent and clearly separated from vertices in other
        vertex sets.

        This is done by calculating the vertices of an r-gon then
        calculating the slope between adjacent vertices. We then 'walk'
        around the r-gon placing graph vertices in regular intervals between 
        adjacent vertices of the r-gon.

        Makes a nicely organized graph like in this picture: 
        https://commons.wikimedia.org/wiki/File:Turan_13-4.svg
        '''

        points = [[cos(2*pi*i/r),sin(2*pi*i/r)] for i in range(r)]
        slopes = [[(points[(i+1)%r][0]-points[i%r][0]),
                   (points[(i+1)%r][1]-points[i%r][1])] for i in range(r)]

        counter = 0

        for i in range(len(l)):
            vertex_set_size = l[i]+1
            for j in range(1,vertex_set_size):
                x = points[i][0]+slopes[i][0]*j/(vertex_set_size)
                y = points[i][1]+slopes[i][1]*j/(vertex_set_size)
                positions[counter] = (x,y)
                counter += 1

    g = Graph()
    for i in l:
        g = g + CompleteGraph(i)

    g = g.complement()
    g.set_pos(positions)
    g.name("Multipartite Graph with set sizes "+str(l))



    return g

# 需要导入模块: from sage.graphs.graph import Graph [as 别名]
# 或者: from sage.graphs.graph.Graph import set_pos [as 别名]
def CycleGraph(n):
    r"""
    Return a cycle graph with n nodes.

    A cycle graph is a basic structure which is also typically called an
    `n`-gon.

    PLOTTING: Upon construction, the position dictionary is filled to override
    the spring-layout algorithm. By convention, each cycle graph will be
    displayed with the first (0) node at the top, with the rest following in a
    counterclockwise manner.

    The cycle graph is a good opportunity to compare efficiency of filling a
    position dictionary vs. using the spring-layout algorithm for
    plotting. Because the cycle graph is very symmetric, the resulting plots
    should be similar (in cases of small `n`).

    Filling the position dictionary in advance adds `O(n)` to the constructor.

    EXAMPLES: Compare plotting using the predefined layout and networkx::

        sage: import networkx
        sage: n = networkx.cycle_graph(23)
        sage: spring23 = Graph(n)
        sage: posdict23 = graphs.CycleGraph(23)
        sage: spring23.show() # long time
        sage: posdict23.show() # long time

    We next view many cycle graphs as a Sage graphics array. First we use the
    ``CycleGraph`` constructor, which fills in the position dictionary::

        sage: g = []
        sage: j = []
        sage: for i in range(9):
        ....:     k = graphs.CycleGraph(i+3)
        ....:     g.append(k)
        sage: for i in range(3):
        ....:     n = []
        ....:     for m in range(3):
        ....:         n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
        ....:     j.append(n)
        sage: G = sage.plot.graphics.GraphicsArray(j)
        sage: G.show() # long time

    Compare to plotting with the spring-layout algorithm::

        sage: g = []
        sage: j = []
        sage: for i in range(9):
        ....:     spr = networkx.cycle_graph(i+3)
        ....:     k = Graph(spr)
        ....:     g.append(k)
        sage: for i in range(3):
        ....:     n = []
        ....:     for m in range(3):
        ....:         n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
        ....:     j.append(n)
        sage: G = sage.plot.graphics.GraphicsArray(j)
        sage: G.show() # long time

    TESTS:

    The input parameter must be a positive integer::

        sage: G = graphs.CycleGraph(-1)
        Traceback (most recent call last):
        ...
        ValueError: parameter n must be a positive integer
    """
    if n < 0:
        raise ValueError("parameter n must be a positive integer")

    G = Graph(n, name="Cycle graph")
    if n == 1:
        G.set_pos({0:(0, 0)})
    else:
        G._circle_embedding(list(range(n)), angle=pi/2)
        G.add_cycle(list(range(n)))
    return G

# 需要导入模块: from sage.graphs.graph import Graph [as 别名]
# 或者: from sage.graphs.graph.Graph import set_pos [as 别名]
def StarGraph(n):
    r"""
    Return a star graph with `n+1` nodes.

    A Star graph is a basic structure where one node is connected to all other
    nodes.

    PLOTTING: Upon construction, the position dictionary is filled to override
    the spring-layout algorithm. By convention, each star graph will be
    displayed with the first (0) node in the center, the second node (1) at the
    top, with the rest following in a counterclockwise manner. (0) is the node
    connected to all other nodes.

    The star graph is a good opportunity to compare efficiency of filling a
    position dictionary vs. using the spring-layout algorithm for plotting. As
    far as display, the spring-layout should push all other nodes away from the
    (0) node, and thus look very similar to this constructor's positioning.

    EXAMPLES::

        sage: import networkx

    Compare the plots::

        sage: n = networkx.star_graph(23)
        sage: spring23 = Graph(n)
        sage: posdict23 = graphs.StarGraph(23)
        sage: spring23.show() # long time
        sage: posdict23.show() # long time

    View many star graphs as a Sage Graphics Array

    With this constructor (i.e., the position dictionary filled)

    ::

        sage: g = []
        sage: j = []
        sage: for i in range(9):
        ....:     k = graphs.StarGraph(i+3)
        ....:     g.append(k)
        sage: for i in range(3):
        ....:     n = []
        ....:     for m in range(3):
        ....:         n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
        ....:     j.append(n)
        sage: G = sage.plot.graphics.GraphicsArray(j)
        sage: G.show() # long time

    Compared to plotting with the spring-layout algorithm

    ::

        sage: g = []
        sage: j = []
        sage: for i in range(9):
        ....:     spr = networkx.star_graph(i+3)
        ....:     k = Graph(spr)
        ....:     g.append(k)
        sage: for i in range(3):
        ....:     n = []
        ....:     for m in range(3):
        ....:         n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
        ....:     j.append(n)
        sage: G = sage.plot.graphics.GraphicsArray(j)
        sage: G.show() # long time
    """
    G = Graph({0: list(range(1, n + 1))}, name="Star graph")
    G.set_pos({0:(0, 0)})
    G._circle_embedding(list(range(1, n + 1)), angle=pi/2)
    return G