Numpy 后端示例：发现子图

此示例展示了如何将较小的图匹配到较大图的子集。

# Author: Runzhong Wang <runzhong.wang@sjtu.edu.cn>
#         Qi Liu <purewhite@sjtu.edu.cn>
#
# License: Mulan PSL v2 License

注意

以下求解器包含在此示例中：

• rrwm() (经典求解器)

• ipfp() (经典求解器)

• sm() (经典求解器)

• ngm() (神经网络求解器)

import numpy as np # numpy backend
import pygmtools as pygm
import matplotlib.pyplot as plt # for plotting
from matplotlib.patches import ConnectionPatch # for plotting matching result
import networkx as nx # for plotting graphs
pygm.set_backend('numpy') # set default backend for pygmtools
np.random.seed(1) # fix random seed

生成更大的图

num_nodes2 = 10
A2 = np.random.rand(num_nodes2, num_nodes2)
A2 = (A2 + A2.T > 1.) * (A2 + A2.T) / 2
np.fill_diagonal(A2, 0)
n2 = np.array([num_nodes2])

生成较小的图

num_nodes1 = 5
G2 = nx.from_numpy_array(A2)
pos2 = nx.spring_layout(G2)
pos2_t = np.array([pos2[_] for _ in range(num_nodes2)])
selected = [0] # build G1 as a cluster in visualization
unselected = list(range(1, num_nodes2))
while len(selected) < num_nodes1:
    dist = np.sum(np.sum(np.abs(np.expand_dims(pos2_t[selected], 1) - np.expand_dims(pos2_t[unselected], 0)), axis=-1), axis=0)
    select_id = unselected[np.argmin(dist).item()] # find the closest node from unselected
    selected.append(select_id)
    unselected.remove(select_id)
selected.sort()
A1 = A2[selected, :][:, selected]
X_gt = np.eye(num_nodes2)[selected, :]
n1 = np.array([num_nodes1])

可视化图表

G1 = nx.from_numpy_array(A1)
pos1 = {_: pos2[selected[_]] for _ in range(num_nodes1)}
color1 = ['#FF5733' for _ in range(num_nodes1)]
color2 = ['#FF5733' if _ in selected else '#1f78b4' for _ in range(num_nodes2)]
plt.figure(figsize=(8, 4))
plt.subplot(1, 2, 1)
plt.title('Subgraph 1')
plt.gca().margins(0.4)
nx.draw_networkx(G1, pos=pos1, node_color=color1)
plt.subplot(1, 2, 2)
plt.title('Graph 2')
nx.draw_networkx(G2, pos=pos2, node_color=color2)

然后我们展示了如何通过图匹配自动发现匹配。

构建亲和矩阵

为了匹配较大的图和较小的图，我们遵循二次分配问题（QAP）的公式：

\[\begin{split}&\max_{\mathbf{X}} \ \texttt{vec}(\mathbf{X})^\top \mathbf{K} \texttt{vec}(\mathbf{X})\\ s.t. \quad &\mathbf{X} \in \{0, 1\}^{n_1\times n_2}, \ \mathbf{X}\mathbf{1} = \mathbf{1}, \ \mathbf{X}^\top\mathbf{1} \leq \mathbf{1}\end{split}\]

其中第一步是构建亲和矩阵（\(\mathbf{K}\)）

conn1, edge1 = pygm.utils.dense_to_sparse(A1)
conn2, edge2 = pygm.utils.dense_to_sparse(A2)
import functools
gaussian_aff = functools.partial(pygm.utils.gaussian_aff_fn, sigma=.001) # set affinity function
K = pygm.utils.build_aff_mat(None, edge1, conn1, None, edge2, conn2, n1, None, n2, None, edge_aff_fn=gaussian_aff)

亲和矩阵的可视化。对于具有\(N_1\)和\(N_2\)节点的图匹配问题，亲和矩阵有\(N_1N_2\times N_1N_2\)个元素，因为每个图中分别有\(N_1^2\)和\(N_2^2\)条边。

注意

在这个例子中，亲和矩阵的对角线元素为空，因为没有节点特征。

plt.figure(figsize=(4, 4))
plt.title(f'Affinity Matrix (size: {K.shape[0]}$\\times${K.shape[1]})')
plt.imshow(K, cmap='Blues')

<matplotlib.image.AxesImage object at 0x7f2254aa6a70>

使用RRWM求解器解决图匹配问题

请参阅rrwm()以获取API参考。

X = pygm.rrwm(K, n1, n2)

RRWM 的输出是一个软匹配矩阵。可视化：

plt.figure(figsize=(8, 4))
plt.subplot(1, 2, 1)
plt.title('RRWM Soft Matching Matrix')
plt.imshow(X, cmap='Blues')
plt.subplot(1, 2, 2)
plt.title('Ground Truth Matching Matrix')
plt.imshow(X_gt, cmap='Blues')

<matplotlib.image.AxesImage object at 0x7f21fd48d1e0>

获取离散匹配矩阵

然后采用匈牙利算法来达到一个离散的匹配矩阵

X = pygm.hungarian(X)

离散匹配矩阵的可视化：

plt.figure(figsize=(8, 4))
plt.subplot(1, 2, 1)
plt.title(f'RRWM Matching Matrix (acc={(X * X_gt).sum()/ X_gt.sum():.2f})')
plt.imshow(X, cmap='Blues')
plt.subplot(1, 2, 2)
plt.title('Ground Truth Matching Matrix')
plt.imshow(X_gt, cmap='Blues')

<matplotlib.image.AxesImage object at 0x7f22330d9240>

匹配子图

绘制匹配：

plt.figure(figsize=(8, 4))
plt.suptitle(f'RRWM Matching Result (acc={(X * X_gt).sum()/ X_gt.sum():.2f})')
ax1 = plt.subplot(1, 2, 1)
plt.title('Subgraph 1')
plt.gca().margins(0.4)
nx.draw_networkx(G1, pos=pos1, node_color=color1)
ax2 = plt.subplot(1, 2, 2)
plt.title('Graph 2')
nx.draw_networkx(G2, pos=pos2, node_color=color2)
for i in range(num_nodes1):
    j = np.argmax(X[i]).item()
    con = ConnectionPatch(xyA=pos1[i], xyB=pos2[j], coordsA="data", coordsB="data",
                          axesA=ax1, axesB=ax2, color="green" if X_gt[i,j] == 1 else "red")
    plt.gca().add_artist(con)

其他求解器也可用

经典IPFP求解器

请参阅ipfp()以获取API参考。

X = pygm.ipfp(K, n1, n2)

/home/wzever/pygmtools/pygmtools/numpy_backend.py:304: RuntimeWarning: invalid value encountered in divide
  t0 = alpha / beta

IPFP匹配结果的可视化：

plt.figure(figsize=(8, 4))
plt.suptitle(f'IPFP Matching Result (acc={(X * X_gt).sum()/ X_gt.sum():.2f})')
ax1 = plt.subplot(1, 2, 1)
plt.title('Subgraph 1')
plt.gca().margins(0.4)
nx.draw_networkx(G1, pos=pos1, node_color=color1)
ax2 = plt.subplot(1, 2, 2)
plt.title('Graph 2')
nx.draw_networkx(G2, pos=pos2, node_color=color2)
for i in range(num_nodes1):
    j = np.argmax(X[i]).item()
    con = ConnectionPatch(xyA=pos1[i], xyB=pos2[j], coordsA="data", coordsB="data",
                          axesA=ax1, axesB=ax2, color="green" if X_gt[i,j] == 1 else "red")
    plt.gca().add_artist(con)

经典SM求解器

请参阅sm()以获取API参考。

X = pygm.sm(K, n1, n2)
X = pygm.hungarian(X)

SM匹配结果的可视化：

plt.figure(figsize=(8, 4))
plt.suptitle(f'SM Matching Result (acc={(X * X_gt).sum()/ X_gt.sum():.2f})')
ax1 = plt.subplot(1, 2, 1)
plt.title('Subgraph 1')
plt.gca().margins(0.4)
nx.draw_networkx(G1, pos=pos1, node_color=color1)
ax2 = plt.subplot(1, 2, 2)
plt.title('Graph 2')
nx.draw_networkx(G2, pos=pos2, node_color=color2)
for i in range(num_nodes1):
    j = np.argmax(X[i]).item()
    con = ConnectionPatch(xyA=pos1[i], xyB=pos2[j], coordsA="data", coordsB="data",
                          axesA=ax1, axesB=ax2, color="green" if X_gt[i,j] == 1 else "red")
    plt.gca().add_artist(con)

NGM神经网络求解器

请参阅ngm()的API参考。

注意

NGM求解器是在不同的问题设置上预训练的，因此它们的性能可能看起来较差。 为了提高它们的性能，您可以改变构建亲和矩阵的方式，或者尝试在新的问题上微调NGM。

X = pygm.ngm(K, n1, n2, pretrain='voc')
X = pygm.hungarian(X)

NGM匹配结果的可视化：

plt.figure(figsize=(8, 4))
plt.suptitle(f'NGM Matching Result (acc={(X * X_gt).sum()/ X_gt.sum():.2f})')
ax1 = plt.subplot(1, 2, 1)
plt.title('Subgraph 1')
plt.gca().margins(0.4)
nx.draw_networkx(G1, pos=pos1, node_color=color1)
ax2 = plt.subplot(1, 2, 2)
plt.title('Graph 2')
nx.draw_networkx(G2, pos=pos2, node_color=color2)
for i in range(num_nodes1):
    j = np.argmax(X[i]).item()
    con = ConnectionPatch(xyA=pos1[i], xyB=pos2[j], coordsA="data", coordsB="data",
                          axesA=ax1, axesB=ax2, color="green" if X_gt[i,j] == 1 else "red")
    plt.gca().add_artist(con)