""" .. _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.