Answering my own question here, as I hope it will be useful to some readers.
Scikit-learn is primarily designed to deal with vector structured data. Hence, if you want to perform label propagation/label spreading on graph-structured data, you're probably better off reimplementing the method yourself rather than using Scikit interface.
Here is an implementation of Label Propagation and Label Spreading in PyTorch.
The two methods overall follow the same algorithmic steps, with variations on how the adjacency matrix is normalized and how the labels are propagated at each step. Let's, therefore, create a base class for our two models.
from abc import abstractmethod
import torch
class BaseLabelPropagation:
"""Base class for label propagation models.
Parameters
----------
adj_matrix: torch.FloatTensor
Adjacency matrix of the graph.
"""
def __init__(self, adj_matrix):
self.norm_adj_matrix = self._normalize(adj_matrix)
self.n_nodes = adj_matrix.size(0)
self.one_hot_labels = None
self.n_classes = None
self.labeled_mask = None
self.predictions = None
@staticmethod
@abstractmethod
def _normalize(adj_matrix):
raise NotImplementedError("_normalize must be implemented")
@abstractmethod
def _propagate(self):
raise NotImplementedError("_propagate must be implemented")
def _one_hot_encode(self, labels):
# Get the number of classes
classes = torch.unique(labels)
classes = classes[classes != -1]
self.n_classes = classes.size(0)
# One-hot encode labeled data instances and zero rows corresponding to unlabeled instances
unlabeled_mask = (labels == -1)
labels = labels.clone() # defensive copying
labels[unlabeled_mask] = 0
self.one_hot_labels = torch.zeros((self.n_nodes, self.n_classes), dtype=torch.float)
self.one_hot_labels = self.one_hot_labels.scatter(1, labels.unsqueeze(1), 1)
self.one_hot_labels[unlabeled_mask, 0] = 0
self.labeled_mask = ~unlabeled_mask
def fit(self, labels, max_iter, tol):
"""Fits a semi-supervised learning label propagation model.
labels: torch.LongTensor
Tensor of size n_nodes indicating the class number of each node.
Unlabeled nodes are denoted with -1.
max_iter: int
Maximum number of iterations allowed.
tol: float
Convergence tolerance: threshold to consider the system at steady state.
"""
self._one_hot_encode(labels)
self.predictions = self.one_hot_labels.clone()
prev_predictions = torch.zeros((self.n_nodes, self.n_classes), dtype=torch.float)
for i in range(max_iter):
# Stop iterations if the system is considered at a steady state
variation = torch.abs(self.predictions - prev_predictions).sum().item()
if variation < tol:
print(f"The method stopped after {i} iterations, variation={variation:.4f}.")
break
prev_predictions = self.predictions
self._propagate()
def predict(self):
return self.predictions
def predict_classes(self):
return self.predictions.max(dim=1).indices
The model takes as input the adjacency matrix of the graph as well as the labels of the nodes. The labels are in the form of a vector of an integer indicating the class number of each node with a -1 at the position of unlabeled nodes.
Label Propagation algorithm is presented below.
$$
\begin{array}{l}{
\mathbf{W} \text {: adjacency matrix of the graph} \\
\text { Compute the diagonal degree matrix } \mathbf{D} \text { by } \mathbf{D}_{i i} \leftarrow \sum_{j} W_{i j}} \\ {\text { Initialize } \hat{Y}^{(0)} \leftarrow\left(y_{1}, \ldots, y_{l}, 0,0, \ldots, 0\right)} \\ {\text { Iterate }} \\ {\text { 1. } \hat{Y}^{(t+1)} \leftarrow \mathbf{D}^{-1} \mathbf{W} \hat{Y}^{(t)}} \\ {\text { 2. } \hat{Y}_{l}^{(t+1)} \leftarrow Y_{l}} \\ {\text { until convergence to } \hat{Y}^{(\infty)}} \\ {\text { Label point } x_{i} \text { by the sign of } \hat{y}_{i}^{(\infty)}}\end{array}
$$
From Xiaojin Zhu and Zoubin Ghahramani. Learning from labeled and unlabeled data with label propagation. Technical Report CMU-CALD-02-107, Carnegie Mellon University, 2002
We get the following implementation.
class LabelPropagation(BaseLabelPropagation):
def __init__(self, adj_matrix):
super().__init__(adj_matrix)
@staticmethod
def _normalize(adj_matrix):
"""Computes D^-1 * W"""
degs = adj_matrix.sum(dim=1)
degs[degs == 0] = 1 # avoid division by 0 error
return adj_matrix / degs[:, None]
def _propagate(self):
self.predictions = torch.matmul(self.norm_adj_matrix, self.predictions)
# Put back already known labels
self.predictions[self.labeled_mask] = self.one_hot_labels[self.labeled_mask]
def fit(self, labels, max_iter=1000, tol=1e-3):
super().fit(labels, max_iter, tol)
Label Spreading algorithm is:
$$
\begin{array}{l}{
\mathbf{W} \text {: adjacency matrix of the graph} \\ \text { Compute the diagonal degree matrix } \mathbf{D} \text { by } \mathbf{D}_{i i} \leftarrow \sum_{j} W_{i j}} \\ {\text { Compute the normalized graph Laplacian } \\ \mathcal{L} \leftarrow \mathbf{D}^{-1 / 2} \mathbf{W} \mathbf{D}^{-1 / 2}} \\ {\text { Initialize } \hat{Y}^{(0)} \leftarrow\left(y_{1}, \ldots, y_{l}, 0,0, \ldots, 0\right)} \\ {\text { Choose a parameter } \alpha \in[0,1)} \\ {\text { Iterate } \hat{Y}(t+1) \leftarrow \alpha \mathcal{L} \hat{Y}^{(t)}+(1-\alpha) \hat{Y}^{(0)} \text { until convergence to } \hat{Y}^{(\infty)}} \\ {\text { Label point } x_{i} \text { by the sign of } \hat{y}_{i}^{(\infty)}}
\end{array}
$$
From Dengyong Zhou, Olivier Bousquet, Thomas Navin Lal, Jason Weston, Bernhard Schoelkopf. Learning with local and global consistency (2004)
The implementation is, therefore, the following.
class LabelSpreading(BaseLabelPropagation):
def __init__(self, adj_matrix):
super().__init__(adj_matrix)
self.alpha = None
@staticmethod
def _normalize(adj_matrix):
"""Computes D^-1/2 * W * D^-1/2"""
degs = adj_matrix.sum(dim=1)
norm = torch.pow(degs, -0.5)
norm[torch.isinf(norm)] = 1
return adj_matrix * norm[:, None] * norm[None, :]
def _propagate(self):
self.predictions = (
self.alpha * torch.matmul(self.norm_adj_matrix, self.predictions)
+ (1 - self.alpha) * self.one_hot_labels
)
def fit(self, labels, max_iter=1000, tol=1e-3, alpha=0.5):
"""
Parameters
----------
alpha: float
Clamping factor.
"""
self.alpha = alpha
super().fit(labels, max_iter, tol)
Let's now test our propagation models on synthetic data. To do so, we choose to use a caveman graph.
import pandas as pd
import numpy as np
import networkx as nx
import matplotlib.pyplot as plt
# Create caveman graph
n_cliques = 4
size_cliques = 10
caveman_graph = nx.connected_caveman_graph(n_cliques, size_cliques)
adj_matrix = nx.adjacency_matrix(caveman_graph).toarray()
# Create labels
labels = np.full(n_cliques * size_cliques, -1.)
# Only one node per clique is labeled. Each clique belongs to a different class.
labels[0] = 0
labels[size_cliques] = 1
labels[size_cliques * 2] = 2
labels[size_cliques * 3] = 3
# Create input tensors
adj_matrix_t = torch.FloatTensor(adj_matrix)
labels_t = torch.LongTensor(labels)
# Learn with Label Propagation
label_propagation = LabelPropagation(adj_matrix_t)
label_propagation.fit(labels_t)
label_propagation_output_labels = label_propagation.predict_classes()
# Learn with Label Spreading
label_spreading = LabelSpreading(adj_matrix_t)
label_spreading.fit(labels_t, alpha=0.8)
label_spreading_output_labels = label_spreading.predict_classes()
# Plot graphs
color_map = {-1: "orange", 0: "blue", 1: "green", 2: "red", 3: "cyan"}
input_labels_colors = [color_map[l] for l in labels]
lprop_labels_colors = [color_map[l] for l in label_propagation_output_labels.numpy()]
lspread_labels_colors = [color_map[l] for l in label_spreading_output_labels.numpy()]
plt.figure(figsize=(14, 6))
ax1 = plt.subplot(1, 4, 1)
ax2 = plt.subplot(1, 4, 2)
ax3 = plt.subplot(1, 4, 3)
ax1.title.set_text("Raw data (4 classes)")
ax2.title.set_text("Label Propagation")
ax3.title.set_text("Label Spreading")
pos = nx.spring_layout(caveman_graph)
nx.draw(caveman_graph, ax=ax1, pos=pos, node_color=input_labels_colors, node_size=50)
nx.draw(caveman_graph, ax=ax2, pos=pos, node_color=lprop_labels_colors, node_size=50)
nx.draw(caveman_graph, ax=ax3, pos=pos, node_color=lspread_labels_colors, node_size=50)
# Legend
ax4 = plt.subplot(1, 4, 4)
ax4.axis("off")
legend_colors = ["orange", "blue", "green", "red", "cyan"]
legend_labels = ["unlabeled", "class 0", "class 1", "class 2", "class 3"]
dummy_legend = [ax4.plot([], [], ls='-', c=c)[0] for c in legend_colors]
plt.legend(dummy_legend, legend_labels)
plt.show()
The implemented models work correctly and allow to detect the communities in the graph.
Note: The propagation methods presented are meant to be used on undirected graphs.
The code is available as an interactive Jupyter notebook here.
