"""
.. _model-line-graph:
Line Graph Neural Network
=========================
**Author**: `Qi Huang `_, Yu Gai,
`Minjie Wang `_, Zheng Zhang
"""
###########################################################################################
#
# In :doc:`GCN <1_gcn>` , we demonstrate how to classify nodes on an input
# graph in a semi-supervised setting, using graph convolutional neural network
# as embedding mechanism for graph features.
# In this tutorial, we shift our focus to community detection problem. The
# task of community detection, i.e. graph clustering, consists of partitioning
# the vertices in a graph into clusters in which nodes are more "similar" to
# one another.
#
# To generalize GNN to supervised community detection, Chen et al. introduced
# a line-graph based variation of graph neural network in
# `Supervised Community Detection with Line Graph Neural Networks `__.
# One of the highlight of their model is
# to augment the vanilla graph neural network(GNN) architecture to operate on
# the line graph of edge adjacencies, defined with non-backtracking operator.
#
# In addition to its high performance, LGNN offers an opportunity to
# illustrate how DGL can implement an advanced graph algorithm by flexibly
# mixing vanilla tensor operations, sparse-matrix multiplication and message-
# passing APIs.
#
# In the following sections, we will go through community detection, line
# graph, LGNN, and its implementation.
#
# Supervised Community Detection Task on CORA
# --------------------------------------------
# Community Detection
# ~~~~~~~~~~~~~~~~~~~~
# In community detection task, we cluster "similar" nodes instead of
# "labeling" them. The node similarity is typically described as higher inner
# density in each cluster.
#
# What's the difference between community detection and node classification？
# Comparing to node classification, community detection focuses on retrieving
# cluster information in the graph, rather than assigning a specific label to
# a node. For example, as long as a node is clustered with its community
# members, it doesn't matter whether the node is assigned as "community A",
# or "community B", while assigning all "great movies" to label "bad movies"
# will be a disaster in a movie network classification task.
#
# What's the difference then, between a community detection algorithm and
# other clustering algorithm such as k-means? Community detection algorithm operates on
# graph-structured data. Comparing to k-means, community detection leverages
# graph structure, instead of simply clustering nodes based on their
# features.
#
# CORA
# ~~~~~
# To be consistent with Graph Convolutional Network tutorial,
# we use `CORA `__
# to illustrate a simple community detection task. To refresh our memory,
# CORA is a scientific publication dataset, with 2708 papers belonging to 7
# different machine learning sub-fields. Here, we formulate CORA as a
# directed graph, with each node being a paper, and each edge being a
# citation link (A->B means A cites B). Here is a visualization of the whole
# CORA dataset.
#
# .. figure:: https://i.imgur.com/X404Byc.png
# :alt: cora
# :height: 400px
# :width: 500px
# :align: center
#
# CORA naturally contains 7 "classes", and statistics below show that each
# "class" does satisfy our assumption of community, i.e. nodes of same class
# class have higher connection probability among them than with nodes of different class.
# The following code snippet verifies that there are more intra-class edges
# than inter-class:
import torch
import torch as th
import torch.nn as nn
import torch.nn.functional as F
import dgl
from dgl.data import citation_graph as citegrh
data = citegrh.load_cora()
G = dgl.DGLGraph(data.graph)
labels = th.tensor(data.labels)
# find all the nodes labeled with class 0
label0_nodes = th.nonzero(labels == 0).squeeze()
# find all the edges pointing to class 0 nodes
src, _ = G.in_edges(label0_nodes)
src_labels = labels[src]
# find all the edges whose both endpoints are in class 0
intra_src = th.nonzero(src_labels == 0)
print('Intra-class edges percent: %.4f' % (len(intra_src) / len(src_labels)))
###########################################################################################
# Binary Community Subgraph from CORA -- a Toy Dataset
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Without loss of generality, in this tutorial we limit the scope of our
# task to binary community detection.
#
# .. note::
#
# To create a toy binary-community dataset from CORA, We first extract
# all two-class pairs from the original CORA 7 classes. For each pair, we
# treat each class as one community, and find the largest subgraph that
# at least contain one cross-community edge as the training example. As
# a result, there are a total of 21 training samples in this mini-dataset.
#
# Here we visualize one of the training samples and its community structure:
import networkx as nx
import matplotlib.pyplot as plt
train_set = dgl.data.CoraBinary()
G1, pmpd1, label1 = train_set[1]
nx_G1 = G1.to_networkx()
def visualize(labels, g):
pos = nx.spring_layout(g, seed=1)
plt.figure(figsize=(8, 8))
plt.axis('off')
nx.draw_networkx(g, pos=pos, node_size=50, cmap=plt.get_cmap('coolwarm'),
node_color=labels, edge_color='k',
arrows=False, width=0.5, style='dotted', with_labels=False)
visualize(label1, nx_G1)
###########################################################################################
# Interested readers can go to the original paper to see how to generalize
# to multi communities case.
#
# Community Detection in a Supervised Setting
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Community Detection problem could be tackled with both supervised and
# unsupervised approaches. Same as the original paper, we formulate
# Community Detection in a supervised setting as follows:
#
# - Each training example consists of :math:`(G, L)`, where :math:`G` is a
# directed graph :math:`(V, E)`. For each node :math:`v` in :math:`V`, we
# assign a ground truth community label :math:`z_v \in \{0,1\}`.
# - The parameterized model :math:`f(G, \theta)` predicts a label set
# :math:`\tilde{Z} = f(G)` for nodes :math:`V`.
# - For each example :math:`(G,L)`, the model learns to minimize a specially
# designed loss function (equivariant loss) :math:`L_{equivariant} =
# (\tilde{Z}，Z)`
#
# .. note::
#
# In this supervised setting, the model naturally predicts a "label" for
# each community. However, community assignment should be equivariant to
# label permutations. To achieve this, in each forward process, we take
# the minimum among losses calculated from all possible permutations of
# labels.
#
# Mathematically, this means
# :math:`L_{equivariant} = \underset{\pi \in S_c} {min}-\log(\hat{\pi}, \pi)`,
# where :math:`S_c` is the set of all permutations of labels, and
# :math:`\hat{\pi}` is the set of predicted labels,
# :math:`- \log(\hat{\pi},\pi)` denotes negative log likelihood.
#
# For instance, for a toy graph with node :math:`\{1,2,3,4\}` and
# community assignment :math:`\{A, A, A, B\}`, with each node's label
# :math:`l \in \{0,1\}`,The group of all possible permutations
# :math:`S_c = \{\{0,0,0,1\}, \{1,1,1,0\}\}`.
#
# Line Graph Neural network: key ideas
# ------------------------------------
# An key innovation in this paper is the use of line-graph.
# Unlike models in previous tutorials, message passing happens not only on the
# original graph, e.g. the binary community subgraph from CORA, but also on the
# line-graph associated with the original graph.
#
# What's a line-graph ?
# ~~~~~~~~~~~~~~~~~~~~~
# In graph theory, line graph is a graph representation that encodes the
# edge adjacency structure in the original graph.
#
# Specifically, a line-graph :math:`L(G)` turns an edge of the original graph `G`
# into a node. This is illustrated with the graph below (taken from the
# paper)
#
# .. figure:: https://i.imgur.com/4WO5jEm.png
# :alt: lg
# :align: center
#
# Here, :math:`e_{A}:= （i\rightarrow j）` and :math:`e_{B}:= (j\rightarrow k)`
# are two edges in the original graph :math:`G`. In line graph :math:`G_L`,
# they correspond to nodes :math:`v^{l}_{A}, v^{l}_{B}`.
#
# The next natural question is, how to connect nodes in line-graph？ How to
# connect two "edges"? Here, we use the following connection rule:
#
# Two nodes :math:`v^{l}_{A}`, :math:`v^{l}_{B}` in `lg` are connected if
# the corresponding two edges :math:`e_{A}, e_{B}` in `g` share one and only
# one node:
# :math:`e_{A}`'s destination node is :math:`e_{B}`'s source node
# (:math:`j`).
#
# .. note::
#
# Mathematically, this definition corresponds to a notion called non-backtracking
# operator:
# :math:`B_{(i \rightarrow j), (\hat{i} \rightarrow \hat{j})}`
# :math:`= \begin{cases}
# 1 \text{ if } j = \hat{i}, \hat{j} \neq i\\
# 0 \text{ otherwise} \end{cases}`
# where an edge is formed if :math:`B_{node1, node2} = 1`.
#
#
# One layer in LGNN -- algorithm structure
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#
# LGNN chains up a series of line-graph neural network layers. The graph
# representation :math:`x` and its line-graph companion :math:`y` evolve with
# the dataflow as follows,
#
# .. figure:: https://i.imgur.com/bZGGIGp.png
# :alt: alg
# :align: center
#
# At the :math:`k`-th layer, the :math:`i`-th neuron of the :math:`l`-th
# channel updates its embedding :math:`x^{(k+1)}_{i,l}` with:
#
# .. math::
# \begin{split}
# x^{(k+1)}_{i,l} ={}&\rho[x^{(k)}_{i}\theta^{(k)}_{1,l}
# +(Dx^{(k)})_{i}\theta^{(k)}_{2,l} \\
# &+\sum^{J-1}_{j=0}(A^{2^{j}}x^{k})_{i}\theta^{(k)}_{3+j,l}\\
# &+[\{\text{Pm},\text{Pd}\}y^{(k)}]_{i}\theta^{(k)}_{3+J,l}] \\
# &+\text{skip-connection}
# \qquad i \in V, l = 1,2,3, ... b_{k+1}/2
# \end{split}
#
# Then, the line-graph representation :math:`y^{(k+1)}_{i,l}` with,
#
# .. math::
#
# \begin{split}
# y^{(k+1)}_{i',l^{'}} = {}&\rho[y^{(k)}_{i^{'}}\gamma^{(k)}_{1,l^{'}}+
# (D_{L(G)}y^{(k)})_{i^{'}}\gamma^{(k)}_{2,l^{'}}\\
# &+\sum^{J-1}_{j=0}(A_{L(G)}^{2^{j}}y^{k})_{i}\gamma^{(k)}_{3+j,l^{'}}\\
# &+[\{\text{Pm},\text{Pd}\}^{T}x^{(k+1)}]_{i^{'}}\gamma^{(k)}_{3+J,l^{'}}]\\
# &+\text{skip-connection}
# \qquad i^{'} \in V_{l}, l^{'} = 1,2,3, ... b^{'}_{k+1}/2
# \end{split}
#
# Where :math:`\text{skip-connection}` refers to performing the same operation without the non-linearity
# :math:`\rho`, and with linear projection :math:`\theta_\{\frac{b_{k+1}}{2} + 1, ..., b_{k+1}-1, b_{k+1}\}`
# and :math:`\gamma_\{\frac{b_{k+1}}{2} + 1, ..., b_{k+1}-1, b_{k+1}\}`.
#
# Implement LGNN in DGL
# ---------------------
# General idea
# ~~~~~~~~~~~~
# The above equations look intimidating. However, we observe the following:
#
# - The two equations are symmetric and can be implemented as two instances
# of the same class with different parameters.
# Mainly, the first equation operates on graph representation :math:`x`,
# whereas the second operates on line-graph
# representation :math:`y`. Let us denote this abstraction as :math:`f`. Then
# the first is :math:`f(x,y; \theta_x)`, and the second
# is :math:`f(y,x, \theta_y)`. That is, they are parameterized to compute
# representations of the original graph and its
# companion line graph, respectively.
#
# - Each equation consists of 4 terms (take the first one as an example):
#
# - :math:`x^{(k)}\theta^{(k)}_{1,l}`, a linear projection of previous
# layer's output :math:`x^{(k)}`, denote as :math:`\text{prev}(x)`.
# - :math:`(Dx^{(k)})\theta^{(k)}_{2,l}`, a linear projection of degree
# operator on :math:`x^{(k)}`, denote as :math:`\text{deg}(x)`.
# - :math:`\sum^{J-1}_{j=0}(A^{2^{j}}x^{(k)})\theta^{(k)}_{3+j,l}`,
# a summation of :math:`2^{j}` adjacency operator on :math:`x^{(k)}`,
# denote as :math:`\text{radius}(x)`
# - :math:`[\{Pm,Pd\}y^{(k)}]\theta^{(k)}_{3+J,l}`, fusing another
# graph's embedding information using incidence matrix
# :math:`\{Pm, Pd\}`, followed with a linear projection,
# denote as :math:`\text{fuse}(y)`.
#
# - In addition, each of the terms are performed again with different
# parameters, and without the nonlinearity after the sum.
# Therefore, :math:`f` could be written as:
#
# .. math::
# \begin{split}
# f(x^{(k)},y^{(k)}) = {}\rho[&\text{prev}(x^{(k-1)}) + \text{deg}(x^{(k-1)}) +\text{radius}(x^{k-1})
# +\text{fuse}(y^{(k)})]\\
# +&\text{prev}(x^{(k-1)}) + \text{deg}(x^{(k-1)}) +\text{radius}(x^{k-1}) +\text{fuse}(y^{(k)})
# \end{split}
#
# - Two equations are chained up in the following order :
#
# .. math::
# \begin{split}
# x^{(k+1)} = {}& f(x^{(k)}, y^{(k)})\\
# y^{(k+1)} = {}& f(y^{(k)}, x^{(k+1)})
# \end{split}
#
# With these observations, we proceed to implementation.
# The important point is we are to use different strategies for these terms.
#
# .. note::
# For a detailed explanation of :math:`\{Pm, Pd\}`, please go to `Advanced Topic`_.
#
# Implementing :math:`\text{prev}` and :math:`\text{deg}` as tensor operation
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Since linear projection and degree operation are both simply matrix
# multiplication, we can write them as PyTorch tensor operation.
#
# In ``__init__``, we define the projection variables:
#
# ::
#
# self.linear_prev = nn.Linear(in_feats, out_feats)
# self.linear_deg = nn.Linear(in_feats, out_feats)
#
#
# In ``forward()``, :math:`\text{prev}` and :math:`\text{deg}` are the same
# as any other PyTorch tensor operations.
#
# ::
#
# prev_proj = self.linear_prev(feat_a)
# deg_proj = self.linear_deg(deg * feat_a)
#
# Implementing :math:`\text{radius}` as message passing in DGL
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# As discussed in GCN tutorial, we can formulate one adjacency operator as
# doing one step message passing. As a generalization, :math:`2^j` adjacency
# operations can be formulated as performing :math:`2^j` step of message
# passing. Therefore, the summation is equivalent to summing nodes'
# representation of :math:`2^j, j=0, 1, 2..` step message passing, i.e.
# gathering information in :math:`2^{j}` neighbourhood of each node.
#
# In ``__init__``, we define the projection variables used in each
# :math:`2^j` steps of message passing:
#
# ::
#
# self.linear_radius = nn.ModuleList(
# [nn.Linear(in_feats, out_feats) for i in range(radius)])
#
# In ``__forward__``, we use following function ``aggregate_radius()`` to
# gather data from multiple hop. Note that the ``update_all`` is called
# multiple times.
# Return a list containing features gathered from multiple radius.
import dgl.function as fn
def aggregate_radius(radius, g, z):
# initializing list to collect message passing result
z_list = []
g.ndata['z'] = z
# pulling message from 1-hop neighbourhood
g.update_all(fn.copy_src(src='z', out='m'), fn.sum(msg='m', out='z'))
z_list.append(g.ndata['z'])
for i in range(radius - 1):
for j in range(2 ** i):
#pulling message from 2^j neighborhood
g.update_all(fn.copy_src(src='z', out='m'), fn.sum(msg='m', out='z'))
z_list.append(g.ndata['z'])
return z_list
#########################################################################
# Implementing :math:`\text{fuse}` as sparse matrix multiplication
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# :math:`\{Pm, Pd\}` is a sparse matrix with only two non-zero entries on
# each column. Therefore, we construct it as a sparse matrix in the dataset,
# and implement :math:`\text{fuse}` as a sparse matrix multiplication.
#
# in ``__forward__``:
#
# ::
#
# fuse = self.linear_fuse(th.mm(pm_pd, feat_b))
#
# Completing :math:`f(x, y)`
# ~~~~~~~~~~~~~~~~~~~~~~~~~~
# Finally, we sum up all the terms together, pass it to skip connection and
# batch-norm.
#
# ::
#
# result = prev_proj + deg_proj + radius_proj + fuse
#
# Then pass result to skip connection:
#
# ::
#
# result = th.cat([result[:, :n], F.relu(result[:, n:])], 1)
#
# Then batch norm
#
# ::
#
# result = self.bn(result) #Batch Normalization.
#
#
# Below is the complete code for one LGNN layer's abstraction :math:`f(x,y)`
class LGNNCore(nn.Module):
def __init__(self, in_feats, out_feats, radius):
super(LGNNCore, self).__init__()
self.out_feats = out_feats
self.radius = radius
self.linear_prev = nn.Linear(in_feats, out_feats)
self.linear_deg = nn.Linear(in_feats, out_feats)
self.linear_radius = nn.ModuleList(
[nn.Linear(in_feats, out_feats) for i in range(radius)])
self.linear_fuse = nn.Linear(in_feats, out_feats)
self.bn = nn.BatchNorm1d(out_feats)
def forward(self, g, feat_a, feat_b, deg, pm_pd):
# term "prev"
prev_proj = self.linear_prev(feat_a)
# term "deg"
deg_proj = self.linear_deg(deg * feat_a)
# term "radius"
# aggregate 2^j-hop features
hop2j_list = aggregate_radius(self.radius, g, feat_a)
# apply linear transformation
hop2j_list = [linear(x) for linear, x in zip(self.linear_radius, hop2j_list)]
radius_proj = sum(hop2j_list)
# term "fuse"
fuse = self.linear_fuse(th.mm(pm_pd, feat_b))
# sum them together
result = prev_proj + deg_proj + radius_proj + fuse
# skip connection and batch norm
n = self.out_feats // 2
result = th.cat([result[:, :n], F.relu(result[:, n:])], 1)
result = self.bn(result)
return result
##############################################################################################################
# Chain up LGNN abstractions as a LGNN layer
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# To implement:
#
# .. math::
# \begin{split}
# x^{(k+1)} = {}& f(x^{(k)}, y^{(k)})\\
# y^{(k+1)} = {}& f(y^{(k)}, x^{(k+1)})
# \end{split}
#
# We chain up two ``LGNNCore`` instances with different parameter in the forward pass.
class LGNNLayer(nn.Module):
def __init__(self, in_feats, out_feats, radius):
super(LGNNLayer, self).__init__()
self.g_layer = LGNNCore(in_feats, out_feats, radius)
self.lg_layer = LGNNCore(in_feats, out_feats, radius)
def forward(self, g, lg, x, lg_x, deg_g, deg_lg, pm_pd):
next_x = self.g_layer(g, x, lg_x, deg_g, pm_pd)
pm_pd_y = th.transpose(pm_pd, 0, 1)
next_lg_x = self.lg_layer(lg, lg_x, x, deg_lg, pm_pd_y)
return next_x, next_lg_x
########################################################################################
# Chain up LGNN layers
# ~~~~~~~~~~~~~~~~~~~~
# We then define an LGNN with three hidden layers.
class LGNN(nn.Module):
def __init__(self, radius):
super(LGNN, self).__init__()
self.layer1 = LGNNLayer(1, 16, radius) # input is scalar feature
self.layer2 = LGNNLayer(16, 16, radius) # hidden size is 16
self.layer3 = LGNNLayer(16, 16, radius)
self.linear = nn.Linear(16, 2) # predice two classes
def forward(self, g, lg, pm_pd):
# compute the degrees
deg_g = g.in_degrees().float().unsqueeze(1)
deg_lg = lg.in_degrees().float().unsqueeze(1)
# use degree as the input feature
x, lg_x = deg_g, deg_lg
x, lg_x = self.layer1(g, lg, x, lg_x, deg_g, deg_lg, pm_pd)
x, lg_x = self.layer2(g, lg, x, lg_x, deg_g, deg_lg, pm_pd)
x, lg_x = self.layer3(g, lg, x, lg_x, deg_g, deg_lg, pm_pd)
return self.linear(x)
#########################################################################################
# Training and Inference
# -----------------------
# We first load the data
from torch.utils.data import DataLoader
training_loader = DataLoader(train_set,
batch_size=1,
collate_fn=train_set.collate_fn,
drop_last=True)
#######################################################################################
# We then define the main training loop. Note that each training sample contains
# three objects: a :class:`~dgl.DGLGraph`, a scipy sparse matrix ``pmpd`` and label
# array in ``numpy.ndarray``. We first generate the line graph using:
#
# ::
#
# lg = g.line_graph(backtracking=False)
#
# Note that ``backtracking=False`` is required to correctly simulate non-backtracking
# operation. We also define a utility function to convert the scipy sparse matrix to
# torch sparse tensor.
# create the model
model = LGNN(radius=3)
# define the optimizer
optimizer = th.optim.Adam(model.parameters(), lr=1e-2)
# a util function to convert a scipy.coo_matrix to torch.SparseFloat
def sparse2th(mat):
value = mat.data
indices = th.LongTensor([mat.row, mat.col])
tensor = th.sparse.FloatTensor(indices, th.from_numpy(value).float(), mat.shape)
return tensor
# train for 20 epochs
for i in range(20):
all_loss = []
all_acc = []
for [g, pmpd, label] in training_loader:
# Generate the line graph.
lg = g.line_graph(backtracking=False)
# Create torch tensors
pmpd = sparse2th(pmpd)
label = th.from_numpy(label)
# Forward
z = model(g, lg, pmpd)
# Calculate loss:
# Since there are only two communities, there are only two permutations
# of the community labels.
loss_perm1 = F.cross_entropy(z, label)
loss_perm2 = F.cross_entropy(z, 1 - label)
loss = th.min(loss_perm1, loss_perm2)
# Calculate accuracy:
_, pred = th.max(z, 1)
acc_perm1 = (pred == label).float().mean()
acc_perm2 = (pred == 1 - label).float().mean()
acc = th.max(acc_perm1, acc_perm2)
all_loss.append(loss.item())
all_acc.append(acc.item())
optimizer.zero_grad()
loss.backward()
optimizer.step()
niters = len(all_loss)
print("Epoch %d | loss %.4f | accuracy %.4f" % (i,
sum(all_loss) / niters, sum(all_acc) / niters))
#######################################################################################
# Visualize training progress
# -----------------------------
# We visualize the network's community prediction on one training example,
# together with the ground truth.
pmpd1 = sparse2th(pmpd1)
LG1 = G1.line_graph(backtracking=False)
z = model(G1, LG1, pmpd1)
_, pred = th.max(z, 1)
visualize(pred, nx_G1)
#######################################################################################
# Compared with the ground truth. Note that the color might be reversed for the
# two community as the model is to correctly predict the "partitioning".
visualize(label1, nx_G1)
#########################################
# Here is an animation to better understand the process. (40 epochs)
#
# .. figure:: https://i.imgur.com/KDUyE1S.gif
# :alt: lgnn-anim
#
# Advanced topic
# --------------
#
# Batching
# ~~~~~~~~
# LGNN takes a collection of different graphs.
# Thus, it's natural we use batching to explore parallelism.
# Why is it not done?
#
# As it turned out, we moved batching into the dataloader itself.
# In the ``collate_fn`` for PyTorch Dataloader, we batch graphs using DGL's
# batched_graph API. To refresh our memory, DGL batches graphs by merging them
# into a large graph, with each smaller graph's adjacency matrix being a block
# along the diagonal of the large graph's adjacency matrix. We concatenate
# :math`\{Pm,Pd\}` as block diagonal matrix in correspondence to DGL batched
# graph API.
def collate_fn(batch):
graphs, pmpds, labels = zip(*batch)
batched_graphs = dgl.batch(graphs)
batched_pmpds = sp.block_diag(pmpds)
batched_labels = np.concatenate(labels, axis=0)
return batched_graphs, batched_pmpds, batched_labels
######################################################################################
# You can check out the complete code
# `here `_.
#
# What's the business with :math:`\{Pm, Pd\}`?
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Roughly speaking, there is a relationship between how :math:`g` and
# :math:`lg` (the line graph) working together with loopy brief propagation.
# Here, we implement :math:`\{Pm, Pd\}` as scipy coo sparse matrix in the dataset,
# and stack them as tensors when batching. Another batching solution is to
# treat :math:`\{Pm, Pd\}` as the adjacency matrix of a bipartite graph, which maps
# line graph's feature to graph's, and vice versa.