draw_group.py

import networkx as nx
import pandas as pd
import numpy as np
import itertools
import src.cayley_dickson as KD

import graph_tool as gt
import graph_tool.draw as gtd

import argparse


def KD_table(order):
    reals = [1]
    X = pd.DataFrame(1, index=reals, columns=reals)
    for n in range(order):
        X = KD.KD_construction(X.index)
    return X


def construct_group(order):
    C = KD_table(order)
    members = list(C) + [-x for x in C]
    N = members[0].terms * 2
    G = np.zeros((N, N), dtype=int)
    for a, b in itertools.product(members, repeat=2):
        G[a.group_index(), b.group_index()] = (a * b).group_index()
    return G, C, members


def draw_cayley_graph(order):
    """From the Cayley multiplication table, determine the
    a group by right multiplying generators starting at index 1,
    e.g. the first non-real term."""

    G, KD_system, members = construct_group(order)
    N = len(G)

    def edge_matrix(idx):
        ex = np.zeros(G.shape, dtype=int)
        for k in range(len(G)):
            ex[k][G[k][idx]] = 1
        return ex

    C = []
    for k in range(1, N):
        C.append(edge_matrix(k))
        g = nx.from_numpy_matrix(sum(C))
        if nx.is_connected(g):
            break

    # Determine the loop structure
    g_loop = nx.from_numpy_matrix(C[0])
    loops = sorted(list(nx.connected_components(g_loop)))
    loops = [np.roll(x, -k) for k, x in enumerate(loops)]

    # loops always come in groups of four
    square = np.array([[-1, -1.0], [-1, 1], [1, 1], [1, -1]]) * (1 / np.sqrt(2))

    g = gt.Graph(directed=True)
    g.add_vertex(N)

    pos = g.new_vertex_property("vector<double>")
    label = g.new_vertex_property("string")
    for k, loop in enumerate(loops):
        for idx, r in zip(loop, square):
            v = g.vertex(idx)
            name = KD.cayley_index_name(members[idx])
            if name is None:
                name = "e{}".format(idx)

            pos[v] = r * (k + 1)
            label[v] = name

    edge_color = g.new_edge_property("vector<double>")
    print("{} generators found for the group".format(len(C)))

    for k, c in enumerate(C):
        edges = zip(*np.where(c))
        for e1, e2 in edges:
            ex = g.add_edge(e1, e2)
            edge_color[ex] = edge_color_set[k]

    f_output = (
        (args.f_output).format(order=args.order) + "." + args.save_format_extension
    )

    g_args = {
        "pos": pos,
        "vertex_text": label,
        "vertex_font_size": args.fontsize,
        "edge_color": edge_color,
        "vertex_size": 40,
        "output_size": (int(args.figsize),) * 2,
    }

    if not args.dont_save:
        print("Saving to {}, size {}".format(f_output, args.figsize))
        gtd.graph_draw(g, output=f_output, fmt=args.save_format_extension, **g_args)

    if not args.dont_show:
        gtd.graph_draw(g, **g_args)


brewer_set1_colors = [
    (0.89411765336990356, 0.10196078568696976, 0.10980392247438431, 0.9),
    (0.21602460800432691, 0.49487120380588606, 0.71987698697576341, 0.9),
    (0.30426760128900115, 0.68329106055054012, 0.29293349969620797, 0.9),
    (0.60083047361934883, 0.30814303335021526, 0.63169552298153153, 0.9),
    (1.0, 0.50591311045721465, 0.0031372549487095253, 0.9),
    (0.99315647868549117, 0.9870049982678657, 0.19915417450315812, 0.9),
    (0.65845446095747107, 0.34122261685483596, 0.1707958535236471, 0.9),
    (0.95850826852461868, 0.50846600392285513, 0.74492888871361229, 0.9),
] * 10
edge_color_set = brewer_set1_colors

if __name__ == "__main__":

    parser = argparse.ArgumentParser()
    parser.add_argument("--f_output", type=str, default="g{order}")
    parser.add_argument("-n", "--order", type=int, default=2)
    parser.add_argument("--dont_save", action="store_true")
    parser.add_argument("--dont_show", action="store_true")
    parser.add_argument("-e", "--save_format_extension", default="svg")
    parser.add_argument("-s", "--figsize", type=float, default=4.0)
    parser.add_argument("--fontsize", type=float, default=18)
    args = parser.parse_args()

    args.figsize *= 100

    draw_cayley_graph(args.order)