注意
跳转到末尾 以下载完整示例代码。
融合Gromov-Wasserstein求解器的比较
该示例说明了使用4种不同求解器计算属性图的FGW，基于条件梯度[24]、Sinkhorn投影[12, 51]和交替Bregman投影[63, 64]来估计距离。
我们生成两个图，遵循随机块模型，并赋予节点特征，并计算它们的FGW匹配。
[12] Gabriel Peyré, Marco Cuturi 和 Justin Solomon (2016)， “Gromov-Wasserstein 平均核和距离矩阵”。 国际机器学习会议 (ICML)。
[24] Vayer Titouan, Chapel Laetitia, Flamary Rémi, Tavenard Romain 和 Courty Nicolas “适用于图的结构化数据的最优运输” 国际机器学习会议 (ICML). 2019.
[51] Xu, H., Luo, D., Zha, H., & Duke, L. C. (2019). “Gromov-wasserstein学习用于图匹配和节点嵌入”。 在国际机器学习大会（ICML），2019。
[63] Li, J., Tang, J., Kong, L., Liu, H., Li, J., So, A. M. C., & Blanchet, J. “一个用于图数据中Gromov-Wasserstein松弛的收敛单循环算法”。国际学习表征会议 (ICLR), 2023.
[64] 马, X., 朱, X., 王, Y., 林, Y., 赵, J., 马, L., & 朱, W. “融合的 Gromov-Wasserstein 图混合用于图级分类”。在第三十七届神经信息处理系统会议（NeurIPS），2023。
# Author: Cédric Vincent-Cuaz <cedvincentcuaz@gmail.com>
#
# License: MIT License
# sphinx_gallery_thumbnail_number = 1
import numpy as np
import matplotlib.pylab as pl
from ot.gromov import (
fused_gromov_wasserstein,
entropic_fused_gromov_wasserstein,
BAPG_fused_gromov_wasserstein,
)
import networkx
from networkx.generators.community import stochastic_block_model as sbm
from time import time
生成遵循2个和3个聚类的随机区块模型的两个图形。
np.random.seed(0)
N2 = 20 # 2 communities
N3 = 30 # 3 communities
p2 = [[1.0, 0.1], [0.1, 0.9]]
p3 = [[1.0, 0.1, 0.0], [0.1, 0.95, 0.1], [0.0, 0.1, 0.9]]
G2 = sbm(seed=0, sizes=[N2 // 2, N2 // 2], p=p2)
G3 = sbm(seed=0, sizes=[N3 // 3, N3 // 3, N3 // 3], p=p3)
part_G2 = [G2.nodes[i]["block"] for i in range(N2)]
part_G3 = [G3.nodes[i]["block"] for i in range(N3)]
C2 = networkx.to_numpy_array(G2)
C3 = networkx.to_numpy_array(G3)
# We add node features with given mean - by clusters
# and inversely proportional to clusters' intra-connectivity
F2 = np.zeros((N2, 1))
for i, c in enumerate(part_G2):
F2[i, 0] = np.random.normal(loc=c, scale=0.01)
F3 = np.zeros((N3, 1))
for i, c in enumerate(part_G3):
F3[i, 0] = np.random.normal(loc=2.0 - c, scale=0.01)
# Compute pairwise euclidean distance between node features
M = (F2**2).dot(np.ones((1, N3))) + np.ones((N2, 1)).dot((F3**2).T) - 2 * F2.dot(F3.T)
h2 = np.ones(C2.shape[0]) / C2.shape[0]
h3 = np.ones(C3.shape[0]) / C3.shape[0]
计算它们的融合 Gromov-Wasserstein 距离
alpha = 0.5
# Conditional Gradient algorithm
print("Conditional Gradient \n")
start_cg = time()
T_cg, log_cg = fused_gromov_wasserstein(
M, C2, C3, h2, h3, "square_loss", alpha=alpha, tol_rel=1e-9, verbose=True, log=True
)
end_cg = time()
time_cg = 1000 * (end_cg - start_cg)
# Proximal Point algorithm with Kullback-Leibler as proximal operator
print("Proximal Point Algorithm \n")
start_ppa = time()
T_ppa, log_ppa = entropic_fused_gromov_wasserstein(
M,
C2,
C3,
h2,
h3,
"square_loss",
alpha=alpha,
epsilon=1.0,
solver="PPA",
tol=1e-9,
log=True,
verbose=True,
warmstart=False,
numItermax=10,
)
end_ppa = time()
time_ppa = 1000 * (end_ppa - start_ppa)
# Projected Gradient algorithm with entropic regularization
print("Projected Gradient Descent \n")
start_pgd = time()
T_pgd, log_pgd = entropic_fused_gromov_wasserstein(
M,
C2,
C3,
h2,
h3,
"square_loss",
alpha=alpha,
epsilon=0.01,
solver="PGD",
tol=1e-9,
log=True,
verbose=True,
warmstart=False,
numItermax=10,
)
end_pgd = time()
time_pgd = 1000 * (end_pgd - start_pgd)
# Alternated Bregman Projected Gradient algorithm with Kullback-Leibler as proximal operator
print("Bregman Alternated Projected Gradient \n")
start_bapg = time()
T_bapg, log_bapg = BAPG_fused_gromov_wasserstein(
M,
C2,
C3,
h2,
h3,
"square_loss",
alpha=alpha,
epsilon=1.0,
tol=1e-9,
marginal_loss=True,
verbose=True,
log=True,
)
end_bapg = time()
time_bapg = 1000 * (end_bapg - start_bapg)
print(
"Fused Gromov-Wasserstein distance estimated with Conditional Gradient solver: "
+ str(log_cg["fgw_dist"])
)
print(
"Fused Gromov-Wasserstein distance estimated with Proximal Point solver: "
+ str(log_ppa["fgw_dist"])
)
print(
"Entropic Fused Gromov-Wasserstein distance estimated with Projected Gradient solver: "
+ str(log_pgd["fgw_dist"])
)
print(
"Fused Gromov-Wasserstein distance estimated with Projected Gradient solver: "
+ str(log_bapg["fgw_dist"])
)
# compute OT sparsity level
T_cg_sparsity = 100 * (T_cg == 0.0).astype(np.float64).sum() / (N2 * N3)
T_ppa_sparsity = 100 * (T_ppa == 0.0).astype(np.float64).sum() / (N2 * N3)
T_pgd_sparsity = 100 * (T_pgd == 0.0).astype(np.float64).sum() / (N2 * N3)
T_bapg_sparsity = 100 * (T_bapg == 0.0).astype(np.float64).sum() / (N2 * N3)
# Methods using Sinkhorn/Bregman projections tend to produce feasibility errors on the
# marginal constraints
err_cg = np.linalg.norm(T_cg.sum(1) - h2) + np.linalg.norm(T_cg.sum(0) - h3)
err_ppa = np.linalg.norm(T_ppa.sum(1) - h2) + np.linalg.norm(T_ppa.sum(0) - h3)
err_pgd = np.linalg.norm(T_pgd.sum(1) - h2) + np.linalg.norm(T_pgd.sum(0) - h3)
err_bapg = np.linalg.norm(T_bapg.sum(1) - h2) + np.linalg.norm(T_bapg.sum(0) - h3)
Conditional Gradient
It. |Loss |Relative loss|Absolute loss
------------------------------------------------
0|8.271184e-01|0.000000e+00|0.000000e+00
1|4.211305e-01|9.640431e-01|4.059879e-01
2|4.024660e-01|4.637523e-02|1.866445e-02
3|3.936346e-01|2.243555e-02|8.831410e-03
4|3.891614e-01|1.149450e-02|4.473216e-03
5|3.854134e-01|9.724554e-03|3.747973e-03
6|3.850574e-01|9.244899e-04|3.559817e-04
7|3.841819e-01|2.279017e-03|8.755571e-04
8|3.819396e-01|5.870728e-03|2.242264e-03
9|3.784264e-01|9.283767e-03|3.513222e-03
10|3.772225e-01|3.191339e-03|1.203845e-03
11|3.764565e-01|2.034974e-03|7.660790e-04
12|3.761179e-01|9.000610e-04|3.385291e-04
13|3.761179e-01|0.000000e+00|0.000000e+00
Proximal Point Algorithm
/home/circleci/project/ot/bregman/_sinkhorn.py:667: UserWarning: Sinkhorn did not converge. You might want to increase the number of iterations `numItermax` or the regularization parameter `reg`.
warnings.warn(
It. |Err
-------------------
0|1.536990e-02|
10|7.472502e-04|
20|6.129779e-04|
30|5.936118e-04|
40|6.334121e-04|
50|6.852583e-04|
60|7.134797e-04|
70|7.177453e-04|
80|7.236970e-04|
90|7.465712e-04|
100|7.858590e-04|
110|8.312306e-04|
120|8.789776e-04|
130|9.493878e-04|
140|1.029873e-03|
150|1.095678e-03|
160|1.177744e-03|
170|1.257392e-03|
180|1.242762e-03|
190|1.125684e-03|
It. |Err
-------------------
200|9.770321e-04|
210|8.368455e-04|
220|7.094853e-04|
230|6.159374e-04|
240|5.564338e-04|
250|4.974174e-04|
260|4.297098e-04|
270|3.639787e-04|
280|3.106686e-04|
290|2.716049e-04|
300|2.433353e-04|
310|2.227167e-04|
320|2.080617e-04|
330|1.980812e-04|
340|1.912468e-04|
350|1.858901e-04|
360|1.805235e-04|
370|1.740559e-04|
380|1.658655e-04|
390|1.557834e-04|
It. |Err
-------------------
400|1.440242e-04|
410|1.310793e-04|
420|1.175881e-04|
/home/circleci/project/ot/backend.py:1176: RuntimeWarning: divide by zero encountered in log
return np.log(a)
430|1.042086e-04|
440|9.151311e-05|
450|7.992882e-05|
460|6.972309e-05|
470|6.101506e-05|
480|5.379844e-05|
490|4.797194e-05|
500|4.337741e-05|
510|3.984020e-05|
520|3.720212e-05|
530|3.534028e-05|
540|3.417109e-05|
550|3.364311e-05|
560|3.372351e-05|
570|3.438312e-05|
580|3.558368e-05|
590|3.726958e-05|
It. |Err
-------------------
600|3.936418e-05|
610|4.176966e-05|
620|4.436877e-05|
630|4.702796e-05|
640|4.960175e-05|
650|5.193888e-05|
660|5.389046e-05|
670|5.531979e-05|
680|5.611295e-05|
690|5.618871e-05|
700|5.550627e-05|
710|5.406954e-05|
720|5.192695e-05|
730|4.916691e-05|
740|4.590932e-05|
750|4.229430e-05|
760|3.846976e-05|
770|3.457933e-05|
780|3.075230e-05|
790|2.709643e-05|
It. |Err
-------------------
800|2.369414e-05|
810|2.060184e-05|
820|1.785186e-05|
830|1.545590e-05|
840|1.340916e-05|
850|1.169442e-05|
860|1.028566e-05|
870|9.150926e-06|
880|8.254840e-06|
890|7.560795e-06|
900|7.033105e-06|
910|6.638947e-06|
920|6.349865e-06|
930|6.142583e-06|
940|5.999120e-06|
950|5.906371e-06|
960|5.855388e-06|
970|5.840590e-06|
980|5.859006e-06|
990|5.909627e-06|
Projected Gradient Descent
It. |Err
-------------------
0|4.981055e-02|
10|1.110756e-01|
20|1.139916e-01|
30|1.158952e-01|
40|1.159648e-01|
50|1.159715e-01|
60|1.159728e-01|
70|1.159732e-01|
80|1.159732e-01|
90|1.159733e-01|
100|1.159733e-01|
110|1.159733e-01|
120|1.159733e-01|
130|1.159733e-01|
140|1.159733e-01|
150|1.159733e-01|
160|1.159733e-01|
170|1.159733e-01|
180|1.159733e-01|
190|1.159733e-01|
It. |Err
-------------------
200|1.159733e-01|
210|1.159733e-01|
220|1.159733e-01|
230|1.159733e-01|
240|1.159733e-01|
250|1.159733e-01|
260|1.159733e-01|
270|1.159733e-01|
280|1.159733e-01|
290|1.159733e-01|
300|1.159733e-01|
310|1.159733e-01|
320|1.159733e-01|
330|1.159733e-01|
340|1.159733e-01|
350|1.159733e-01|
360|1.159733e-01|
370|1.159733e-01|
380|1.159733e-01|
390|1.159733e-01|
It. |Err
-------------------
400|1.159733e-01|
410|1.159733e-01|
420|1.159733e-01|
430|1.159733e-01|
440|1.159733e-01|
450|1.159733e-01|
460|1.159733e-01|
470|1.159733e-01|
480|1.159733e-01|
490|1.159733e-01|
500|1.159733e-01|
510|1.159733e-01|
520|1.159733e-01|
530|1.159733e-01|
540|1.159733e-01|
550|1.159733e-01|
560|1.159733e-01|
570|1.159733e-01|
580|1.159733e-01|
590|1.159733e-01|
It. |Err
-------------------
600|1.159733e-01|
610|1.159733e-01|
620|1.159733e-01|
630|1.159733e-01|
640|1.159733e-01|
650|1.159733e-01|
660|1.159733e-01|
670|1.159733e-01|
680|1.159733e-01|
690|1.159733e-01|
700|1.159733e-01|
710|1.159733e-01|
720|1.159733e-01|
730|1.159733e-01|
740|1.159733e-01|
750|1.159733e-01|
760|1.159733e-01|
770|1.159733e-01|
780|1.159733e-01|
790|1.159733e-01|
It. |Err
-------------------
800|1.159733e-01|
810|1.159733e-01|
820|1.159733e-01|
830|1.159733e-01|
840|1.159733e-01|
850|1.159733e-01|
860|1.159733e-01|
870|1.159733e-01|
880|1.159733e-01|
890|1.159733e-01|
900|1.159733e-01|
910|1.159733e-01|
920|1.159733e-01|
930|1.159733e-01|
940|1.159733e-01|
950|1.159733e-01|
960|1.159733e-01|
970|1.159733e-01|
980|1.159733e-01|
990|1.159733e-01|
Bregman Alternated Projected Gradient
It. |Err
-------------------
0|2.710197e-02|
10|7.722091e-04|
20|1.141907e-03|
30|1.705361e-03|
40|2.217013e-03|
50|2.780133e-03|
60|2.754405e-03|
70|2.618865e-03|
80|2.155365e-03|
90|1.467885e-03|
100|1.234502e-03|
110|1.427166e-03|
120|1.580081e-03|
130|1.480216e-03|
140|1.016301e-03|
150|7.489908e-04|
160|5.527976e-04|
170|3.724086e-04|
180|2.505381e-04|
190|1.890667e-04|
It. |Err
-------------------
200|1.660689e-04|
210|1.596126e-04|
220|1.580415e-04|
230|1.573214e-04|
240|1.554877e-04|
250|1.506710e-04|
260|1.413884e-04|
270|1.273670e-04|
280|1.099326e-04|
290|9.144168e-05|
300|7.415348e-05|
310|5.942604e-05|
320|4.761838e-05|
330|3.844073e-05|
340|3.135983e-05|
350|2.585715e-05|
360|2.152819e-05|
370|1.808727e-05|
380|1.533873e-05|
390|1.314610e-05|
It. |Err
-------------------
400|1.140922e-05|
410|1.004962e-05|
420|9.001810e-06|
430|8.208638e-06|
440|7.619328e-06|
450|7.189212e-06|
460|6.880145e-06|
470|6.660812e-06|
480|6.506527e-06|
490|6.398538e-06|
500|6.323025e-06|
510|6.270041e-06|
520|6.232558e-06|
530|6.205670e-06|
540|6.185991e-06|
550|6.171195e-06|
560|6.159695e-06|
570|6.150412e-06|
580|6.142611e-06|
590|6.135798e-06|
It. |Err
-------------------
600|6.129637e-06|
610|6.123902e-06|
620|6.118440e-06|
630|6.113149e-06|
640|6.107960e-06|
650|6.102825e-06|
660|6.097715e-06|
670|6.092609e-06|
680|6.087492e-06|
690|6.082355e-06|
700|6.077194e-06|
710|6.072003e-06|
720|6.066779e-06|
730|6.061523e-06|
740|6.056231e-06|
750|6.050905e-06|
760|6.045543e-06|
770|6.040145e-06|
780|6.034711e-06|
790|6.029242e-06|
It. |Err
-------------------
800|6.023737e-06|
810|6.018197e-06|
820|6.012621e-06|
830|6.007010e-06|
840|6.001365e-06|
850|5.995684e-06|
860|5.989968e-06|
870|5.984218e-06|
880|5.978434e-06|
890|5.972615e-06|
900|5.966762e-06|
910|5.960875e-06|
920|5.954955e-06|
930|5.949001e-06|
940|5.943013e-06|
950|5.936992e-06|
960|5.930938e-06|
970|5.924851e-06|
980|5.918731e-06|
990|5.912579e-06|
Fused Gromov-Wasserstein distance estimated with Conditional Gradient solver: 0.3761179313933098
Fused Gromov-Wasserstein distance estimated with Proximal Point solver: 0.3671471715862438
Entropic Fused Gromov-Wasserstein distance estimated with Projected Gradient solver: 0.21736592892258025
Fused Gromov-Wasserstein distance estimated with Projected Gradient solver: 0.2576635020911173
融合的Gromov-Wasserstein匹配的可视化
我们给右侧的图中的节点上色 - 然后根据FGW匹配的最佳运输计划投影其节点颜色。我们根据发送的质量调整跨域链接的强度，如果发送的质量不为零，则添加最小强度0.1。对于每个匹配，所有节点的大小与其从OT计划的边际计算得到的质量成比例，以说明潜在的可行性错误。注意：颜色指的是簇 - 而不是节点特征
# Add weights on the edges for visualization later on
weight_intra_G2 = 5
weight_inter_G2 = 0.5
weight_intra_G3 = 1.0
weight_inter_G3 = 1.5
weightedG2 = networkx.Graph()
part_G2 = [G2.nodes[i]["block"] for i in range(N2)]
for node in G2.nodes():
weightedG2.add_node(node)
for i, j in G2.edges():
if part_G2[i] == part_G2[j]:
weightedG2.add_edge(i, j, weight=weight_intra_G2)
else:
weightedG2.add_edge(i, j, weight=weight_inter_G2)
weightedG3 = networkx.Graph()
part_G3 = [G3.nodes[i]["block"] for i in range(N3)]
for node in G3.nodes():
weightedG3.add_node(node)
for i, j in G3.edges():
if part_G3[i] == part_G3[j]:
weightedG3.add_edge(i, j, weight=weight_intra_G3)
else:
weightedG3.add_edge(i, j, weight=weight_inter_G3)
def draw_graph(
G,
C,
nodes_color_part,
Gweights=None,
pos=None,
edge_color="black",
node_size=None,
shiftx=0,
seed=0,
):
if pos is None:
pos = networkx.spring_layout(G, scale=1.0, seed=seed)
if shiftx != 0:
for k, v in pos.items():
v[0] = v[0] + shiftx
alpha_edge = 0.7
width_edge = 1.8
if Gweights is None:
networkx.draw_networkx_edges(
G, pos, width=width_edge, alpha=alpha_edge, edge_color=edge_color
)
else:
# We make more visible connections between activated nodes
n = len(Gweights)
edgelist_activated = []
edgelist_deactivated = []
for i in range(n):
for j in range(n):
if Gweights[i] * Gweights[j] * C[i, j] > 0:
edgelist_activated.append((i, j))
elif C[i, j] > 0:
edgelist_deactivated.append((i, j))
networkx.draw_networkx_edges(
G,
pos,
edgelist=edgelist_activated,
width=width_edge,
alpha=alpha_edge,
edge_color=edge_color,
)
networkx.draw_networkx_edges(
G,
pos,
edgelist=edgelist_deactivated,
width=width_edge,
alpha=0.1,
edge_color=edge_color,
)
if Gweights is None:
for node, node_color in enumerate(nodes_color_part):
networkx.draw_networkx_nodes(
G,
pos,
nodelist=[node],
node_size=node_size,
alpha=1,
node_color=node_color,
)
else:
scaled_Gweights = Gweights / (0.5 * Gweights.max())
nodes_size = node_size * scaled_Gweights
for node, node_color in enumerate(nodes_color_part):
networkx.draw_networkx_nodes(
G,
pos,
nodelist=[node],
node_size=nodes_size[node],
alpha=1,
node_color=node_color,
)
return pos
def draw_transp_colored_GW(
G1,
C1,
G2,
C2,
part_G1,
p1,
p2,
T,
pos1=None,
pos2=None,
shiftx=4,
switchx=False,
node_size=70,
seed_G1=0,
seed_G2=0,
):
starting_color = 0
# get graphs partition and their coloring
part1 = part_G1.copy()
unique_colors = ["C%s" % (starting_color + i) for i in np.unique(part1)]
nodes_color_part1 = []
for cluster in part1:
nodes_color_part1.append(unique_colors[cluster])
nodes_color_part2 = []
# T: getting colors assignment from argmin of columns
for i in range(len(G2.nodes())):
j = np.argmax(T[:, i])
nodes_color_part2.append(nodes_color_part1[j])
pos1 = draw_graph(
G1,
C1,
nodes_color_part1,
Gweights=p1,
pos=pos1,
node_size=node_size,
shiftx=0,
seed=seed_G1,
)
pos2 = draw_graph(
G2,
C2,
nodes_color_part2,
Gweights=p2,
pos=pos2,
node_size=node_size,
shiftx=shiftx,
seed=seed_G2,
)
for k1, v1 in pos1.items():
max_Tk1 = np.max(T[k1, :])
for k2, v2 in pos2.items():
if T[k1, k2] > 0:
pl.plot(
[pos1[k1][0], pos2[k2][0]],
[pos1[k1][1], pos2[k2][1]],
"-",
lw=0.7,
alpha=min(T[k1, k2] / max_Tk1 + 0.1, 1.0),
color=nodes_color_part1[k1],
)
return pos1, pos2
node_size = 40
fontsize = 13
seed_G2 = 0
seed_G3 = 4
pl.figure(2, figsize=(15, 3.5))
pl.clf()
pl.subplot(141)
pl.axis("off")
pl.title(
"(CG) FGW=%s\n \n OT sparsity = %s \n marg. error = %s \n runtime = %s"
% (
np.round(log_cg["fgw_dist"], 3),
str(np.round(T_cg_sparsity, 2)) + " %",
np.round(err_cg, 4),
str(np.round(time_cg, 2)) + " ms",
),
fontsize=fontsize,
)
pos1, pos2 = draw_transp_colored_GW(
weightedG2,
C2,
weightedG3,
C3,
part_G2,
p1=T_cg.sum(1),
p2=T_cg.sum(0),
T=T_cg,
shiftx=1.5,
node_size=node_size,
seed_G1=seed_G2,
seed_G2=seed_G3,
)
pl.subplot(142)
pl.axis("off")
pl.title(
"(PPA) FGW=%s\n \n OT sparsity = %s \n marg. error = %s \n runtime = %s"
% (
np.round(log_ppa["fgw_dist"], 3),
str(np.round(T_ppa_sparsity, 2)) + " %",
np.round(err_ppa, 4),
str(np.round(time_ppa, 2)) + " ms",
),
fontsize=fontsize,
)
pos1, pos2 = draw_transp_colored_GW(
weightedG2,
C2,
weightedG3,
C3,
part_G2,
p1=T_ppa.sum(1),
p2=T_ppa.sum(0),
T=T_ppa,
pos1=pos1,
pos2=pos2,
shiftx=0.0,
node_size=node_size,
seed_G1=0,
seed_G2=0,
)
pl.subplot(143)
pl.axis("off")
pl.title(
"(PGD) Entropic FGW=%s\n \n OT sparsity = %s \n marg. error = %s \n runtime = %s"
% (
np.round(log_pgd["fgw_dist"], 3),
str(np.round(T_pgd_sparsity, 2)) + " %",
np.round(err_pgd, 4),
str(np.round(time_pgd, 2)) + " ms",
),
fontsize=fontsize,
)
pos1, pos2 = draw_transp_colored_GW(
weightedG2,
C2,
weightedG3,
C3,
part_G2,
p1=T_pgd.sum(1),
p2=T_pgd.sum(0),
T=T_pgd,
pos1=pos1,
pos2=pos2,
shiftx=0.0,
node_size=node_size,
seed_G1=0,
seed_G2=0,
)
pl.subplot(144)
pl.axis("off")
pl.title(
"(BAPG) FGW=%s\n \n OT sparsity = %s \n marg. error = %s \n runtime = %s"
% (
np.round(log_bapg["fgw_dist"], 3),
str(np.round(T_bapg_sparsity, 2)) + " %",
np.round(err_bapg, 4),
str(np.round(time_bapg, 2)) + " ms",
),
fontsize=fontsize,
)
pos1, pos2 = draw_transp_colored_GW(
weightedG2,
C2,
weightedG3,
C3,
part_G2,
p1=T_bapg.sum(1),
p2=T_bapg.sum(0),
T=T_bapg,
pos1=pos1,
pos2=pos2,
shiftx=0.0,
node_size=node_size,
seed_G1=0,
seed_G2=0,
)
pl.tight_layout()
pl.show()
脚本的总运行时间： (0分钟 3.765秒)