# Paddle Backend Example: Discovering Subgraphs

This example shows how to match a smaller graph to a subset of a larger graph.

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


Note

The following solvers are included in this example:

import paddle # paddle 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
import warnings
warnings.filterwarnings("ignore")

pygm.set_backend('paddle') # set default backend for pygmtools

_ = paddle.seed(1) # fix random seed


## Generate the larger graph

num_nodes2 = 10
A2 = (A2 + A2.t() > 1.) / 2 * (A2 + A2.t())


## Generate the smaller graph

num_nodes1 = 5
G2 = nx.from_numpy_array(A2.numpy())
pos2 = nx.spring_layout(G2)
pos2_t = paddle.to_tensor([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:
select_id = unselected[paddle.argmin(dist).item()] # find the closest node from unselected
selected.append(select_id)
unselected.remove(select_id)
selected.sort()
A1 = A2[selected].T[selected].T # A1 = A2[selected, :][:, selected]


## Visualize the graphs

G1 = nx.from_numpy_array(A1.numpy())
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)


We then show how to automatically discover the matching by graph matching.

## Build affinity matrix

To match the larger graph and the smaller graph, we follow the formulation of Quadratic Assignment Problem (QAP):

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

where the first step is to build the affinity matrix ($$\mathbf{K}$$)

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)


Visualization of the affinity matrix. For graph matching problem with $$N_1$$ and $$N_2$$ nodes, the affinity matrix has $$N_1N_2\times N_1N_2$$ elements because there are $$N_1^2$$ and $$N_2^2$$ edges in each graph, respectively.

Note

The diagonal elements of the affinity matrix is empty because there is no node features in this example.

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

<matplotlib.image.AxesImage object at 0x7f0ea85f5210>


## Solve graph matching problem by RRWM solver

See rrwm() for the API reference.

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


The output of RRWM is a soft matching matrix. Visualization:

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

<matplotlib.image.AxesImage object at 0x7f0ea90a3e50>


## Get the discrete matching matrix

Hungarian algorithm is then adopted to reach a discrete matching matrix

X = pygm.hungarian(X)


Visualization of the discrete matching matrix:

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

<matplotlib.image.AxesImage object at 0x7f0ee1b5eaa0>


## Match the subgraph

Draw the matching:

plt.figure(figsize=(8, 4))
plt.suptitle(f'RRWM Matching Result (acc={((X * X_gt).sum()/ X_gt.sum()).item():.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):
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")


## Other solvers are also available

### Classic IPFP solver

See ipfp() for the API reference.

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


Visualization of IPFP matching result:

plt.figure(figsize=(8, 4))
plt.suptitle(f'IPFP Matching Result (acc={((X * X_gt).sum()/ X_gt.sum()).item():.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):
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")


### Classic SM solver

See sm() for the API reference.

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


Visualization of SM matching result:

plt.figure(figsize=(8, 4))
plt.suptitle(f'SM Matching Result (acc={((X * X_gt).sum()/ X_gt.sum()).item():.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):
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")


### NGM neural network solver

See ngm() for the API reference.

Note

The NGM solvers are pretrained on a different problem setting, so their performance may seem inferior. To improve their performance, you may change the way of building affinity matrices, or try finetuning NGM on the new problem.

with paddle.set_grad_enabled(False):
X = pygm.ngm(K, n1, n2, pretrain='voc')
X = pygm.hungarian(X)


Visualization of NGM matching result:

plt.figure(figsize=(8, 4))
plt.suptitle(f'NGM Matching Result (acc={((X * X_gt).sum()/ X_gt.sum()).item():.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):