# Imbalanced positive/negative edges - graph link prediction

I am using a graph autoencoder to perform link prediction on a graph. The issue is that the number of negative (absent) edges is about 100 times the number of positive (existing) edges. To deal with the imbalance of data, I use a positive weight of 100 in the computation of the BCE loss.
I get a very high AUC and AP (88% for both), but the balanced accuracy ((tp+tn)/2) is only 50% (as a random classifier).
Also, after converting the probability estimates to binary predictions, I compute the confusion matrix and get the following results:
TN = 0.0
FP = 1.0
FN = 0.0
TP = 1.0

The binary predictions show that the model always predicts 1 although both negative and positive edges are taken into account in the training.

Can anyone understand why the accuracy is 50%?

Your model is now overfitting the positive classes so you might need to reduce the `pos_weight` value.
How did the confusion matrix look before adding the weight?

I reduced the pos_weight to 1, the accuracy is still 50% and the confusion matrix is the same.

So it seems that even though you have a quite large class imbalance, your model still overfits to the minority class without any loss weighing?
This seems yo be quite an interesting use case, as you would usually see the opposite.
Could you try to decrease the `pos_weight` even further and see if the model starts to predict the majority class at one point?

If that doesn’t help, could you post the code you are using to get the predictions and calculate the confusion matrix, as you might have a bug there.

Yes, I also thought the imbalance would lead the model to predict more negative cases than positive ones.
In my case I use a graph variational auto-encoder (GVAE) to reconstruct an adjacency matrix (link prediction task). Therefore the loss function is the sum of 2 terms: the BCE loss (reconstruction term) and the KL divergence (regularisation term).
Here’s the implementation of the loss function:

``````def loss_function(preds, labels, mu, logvar, n_nodes, pos_weight):

BCE_loss = F.binary_cross_entropy_with_logits(preds, labels, pos_weight=pos_weight)
KLD_loss = -0.5 / n_nodes * torch.mean(torch.sum(1 + 2 * logvar - mu.pow(2) - logvar.exp().pow(2), 1))

return BCE_loss + KLD_loss
``````

I get a balanced accuracy=0.5, F1score=0.33 and the confusion matrix is tn=0, fp=1, fn=0, tp=1. The code to compute these metrics is:

``````p = np.array([1 if  p>0.5 else 0 for p in preds_all]).reshape((-1,1))
y = labels_all.reshape((-1,1))
c = confusion_matrix(y, p, normalize='true')
tn, fp, fn, tp = c.ravel()
balanced_acc = (tp+tn)/2
f1_sco = f1_score(y, p, average='weighted')
``````

I tried several values for pos_weight and still get the same results.

Also, the AUC and AP scores are very high in the first epochs:

Thanks again for your help.

I think there still might be a bug, as your model is apparently even with a `pos_weight<1` still predicting only the minority class.
I don’t know, how `preds_all` was calculated, but could you post the model definition, the training routine, as well as all necessary input shapes and other arguments to execute the script?

I use an implementation of a graph variational auto-encoder (GVAE) that is available at this link: “https://github.com/zfjsail/gae-pytorch/tree/master/gae”.

The adjacency matrix is derived from the graph data object. Since there are no node features, `data.x` is filled with ones (represents an identity matrix).

My code is:

# DATA PRE-PROCESSING:

``````import networkx as nx
import torch_geometric
from torch_geometric.datasets import Planetoid
from scipy.sparse import coo_matrix, hstack, vstack
!pip install tensorboardX
from tensorboardX import SummaryWriter

data = GI_data_list[1]
print("graph data: ", data)

HPA_graph = nx.Graph()
edgeinfo = data.edge_index
src = edgeinfo[0].cpu().numpy()
dst = edgeinfo[1].cpu().numpy()
edgelist = zip(src,dst)
for i,j in edgelist:

A = nx.adjacency_matrix(HPA_graph) # ADJACENCY MATRIX USED IN TRAINING

non_edges = list(nx.non_edges(HPA_graph))
e0 = [tup[0] for tup in non_edges]
e1 = [tup[1] for tup in non_edges]
neg_edge_list = np.vstack((e0, e1))
neg_edge_index = torch.as_tensor(neg_edge_list, dtype=torch.long)
``````

GRAPH DATA FORMAT:

The adjacency matrix derived from the edge_index (edge list) has the shape (3163, 3163).

POSITIVE WEIGHT FACTOR:

``````w_factor = (neg_edge_index.shape[1]/edgeinfo.shape[1])
``````

w_factor = 79.3

# GRAPH CONVOLUTION CLASS:

``````import torch
import torch.nn.functional as F
from torch.nn.modules.module import Module
from torch.nn.parameter import Parameter

class GraphConvolution(Module):

def __init__(self, in_features, out_features, dropout):
super(GraphConvolution, self).__init__()
self.in_features = in_features
self.out_features = out_features
self.dropout = dropout
self.weight = Parameter(torch.DoubleTensor(in_features, out_features))
self.reset_parameters()

def reset_parameters(self):
torch.nn.init.xavier_uniform_(self.weight)

def forward(self, input, adj):
input = F.dropout(input, self.dropout, self.training)
support = torch.mm(input, self.weight)
output = torch.spmm(adj, support)
return output

def __repr__(self):
return self.__class__.__name__ + ' (' \
+ str(self.in_features) + ' -> ' \
+ str(self.out_features) + ')'
``````

# MODEL:

``````import torch
import torch.nn as nn
import torch.nn.functional as F

class GCNModelVAE(nn.Module):

def __init__(self, input_feat_dim, hidden_dim1, hidden_dim2, dropout):
super(GCNModelVAE, self).__init__()
self.gc1 = GraphConvolution(input_feat_dim, hidden_dim1, dropout)
self.gc2 = GraphConvolution(hidden_dim1, hidden_dim2, dropout)
self.gc3 = GraphConvolution(hidden_dim1, hidden_dim2, dropout)
self.dc = InnerProductDecoder(dropout)

def encode(self, x, adj):
hidden1 = F.relu(self.gc1(x, adj))

def reparameterize(self, mu, logvar):
if self.training:
std = torch.exp(logvar)
eps = torch.randn_like(std)
else:
return mu

def forward(self, x, adj):
mu, logvar = self.encode(x, adj)
z = self.reparameterize(mu, logvar)
return self.dc(z), mu, logvar

class InnerProductDecoder(nn.Module):
"""Decoder for using inner product for prediction."""

def __init__(self, dropout):
super(InnerProductDecoder, self).__init__()
self.dropout = dropout
self.act = torch.sigmoid

def forward(self, z):
z = F.dropout(z, self.dropout, training=self.training)
adj = self.act(torch.mm(z, z.t()))
``````

# LOSS FUNCTION:

``````import torch
import torch.nn.modules.loss
import torch.nn.functional as F

def loss_function(preds, labels, mu, logvar, n_nodes, norm, weight):

cost = norm * F.binary_cross_entropy_with_logits(preds, labels, pos_weight=weight)

KLD = -0.5 / n_nodes * torch.mean(torch.sum(
1 + 2 * logvar - mu.pow(2) - logvar.exp().pow(2), 1))

return cost + KLD
``````

# UTILS:

``````import pickle as pkl
import networkx as nx
import numpy as np
import scipy.sparse as sp
import torch
from sklearn.metrics import roc_auc_score, average_precision_score
from sklearn.metrics import confusion_matrix
from sklearn.metrics import f1_score

def sparse_to_tuple(sparse_mx):

if not sp.isspmatrix_coo(sparse_mx):
sparse_mx = sparse_mx.tocoo()
coords = np.vstack((sparse_mx.row, sparse_mx.col)).transpose()
values = sparse_mx.data
shape = sparse_mx.shape
return coords, values, shape

# Function to build test set with 10% positive links

# Remove diagonal elements
# Check that diag is zero:
# assert np.diag(adj.todense()).sum() == 0

num_test = int(np.floor(edges.shape[0] / 10.))
num_val = int(np.floor(edges.shape[0] / 20.))

all_edge_idx = list(range(edges.shape[0]))
np.random.shuffle(all_edge_idx)
val_edge_idx = all_edge_idx[:num_val]
test_edge_idx = all_edge_idx[num_val:(num_val + num_test)]
test_edges = edges[test_edge_idx]
val_edges = edges[val_edge_idx]
train_edges = np.delete(edges, np.hstack([test_edge_idx, val_edge_idx]), axis=0)

def ismember(a, b, tol=5):
rows_close = np.all(np.round(a - b[:, None], tol) == 0, axis=-1)
return np.any(rows_close)

test_edges_false = []
while len(test_edges_false) < len(test_edges):
idx_i = np.random.randint(0, adj.shape[0])
idx_j = np.random.randint(0, adj.shape[0])
if idx_i == idx_j:
continue
if ismember([idx_i, idx_j], edges_all):
continue
if test_edges_false:
if ismember([idx_j, idx_i], np.array(test_edges_false)):
continue
if ismember([idx_i, idx_j], np.array(test_edges_false)):
continue
test_edges_false.append([idx_i, idx_j])

val_edges_false = []
while len(val_edges_false) < len(val_edges):
idx_i = np.random.randint(0, adj.shape[0])
idx_j = np.random.randint(0, adj.shape[0])
if idx_i == idx_j:
continue
if ismember([idx_i, idx_j], train_edges):
continue
if ismember([idx_j, idx_i], train_edges):
continue
if ismember([idx_i, idx_j], val_edges):
continue
if ismember([idx_j, idx_i], val_edges):
continue
if val_edges_false:
if ismember([idx_j, idx_i], np.array(val_edges_false)):
continue
if ismember([idx_i, idx_j], np.array(val_edges_false)):
continue
val_edges_false.append([idx_i, idx_j])

data = np.ones(train_edges.shape[0])

# Re-build adj matrix
adj_train = sp.csr_matrix((data, (train_edges[:, 0], train_edges[:, 1])),     shape=adj.shape)

# NOTE: these edge lists only contain single direction of edge!
return adj_train, train_edges, val_edges, val_edges_false, test_edges,     test_edges_false

degree_mat_inv_sqrt = sp.diags(np.power(rowsum, -0.5).flatten())

def sparse_mx_to_torch_sparse_tensor(sparse_mx):

"""Convert a scipy sparse matrix to a torch sparse tensor."""
sparse_mx = sparse_mx.tocoo().astype(np.float64)
indices = torch.from_numpy(
np.vstack((sparse_mx.row, sparse_mx.col)).astype(np.int64))
values = torch.from_numpy(sparse_mx.data)
shape = torch.Size(sparse_mx.shape)

def get_roc_score(emb, adj_orig, edges_pos, edges_neg):

def sigmoid(x):
x = np.float128(x)
return 1 / (1 + np.exp(-x))

# Predict on test set of edges
adj_rec = np.dot(emb, emb.T)
preds = []
pos = []
for e in edges_pos:

preds_neg = []
neg = []
for e in edges_neg:

preds_all = np.hstack([preds, preds_neg])
labels_all = np.hstack([np.ones(len(preds)), np.zeros(len(preds))])

p = np.array([1 if  p>0.5 else 0 for p in preds_all]).reshape((-1,1))
y = labels_all.reshape((-1,1))

roc_score = roc_auc_score(labels_all, preds_all)
ap_score = average_precision_score(labels_all, preds_all)

c = confusion_matrix(y, p, normalize='true')
# print("confusion matrix: ", c)
tn, fp, fn, tp = c.ravel()
# print("tn: ", tn, "fp: ", fp, "fn: ", fn, "tp: ", tp)
balanced_acc = (tp+tn)/2
f1_sco = f1_score(y, p, average='weighted')

return roc_score, ap_score, balanced_acc, f1_sco
``````

# TRAINING:

``````from __future__ import division
from __future__ import print_function
import argparse
import numpy as np
import scipy.sparse as sp
import torch
from torch import optim

seed = int(42)
hidden1 = int(256)
hidden2 = int(128)
lr = 0.01
dropout = np.float64(0)

features = torch.as_tensor(data.x , dtype=torch.float64)
n_nodes, feat_dim = features.shape

#Store original adjacency matrix (without diagonal entries) for later

#Some preprocessing

``````
``````w = np.ones(adj_label.shape)
w = w*w_factor
pos_weight = torch.as_tensor(w, dtype=torch.float64)
``````
``````hidden_emb = None
epochs = int(900)
model = GCNModelVAE(feat_dim, hidden1, hidden2, dropout)
optimizer = optim.Adam(model.parameters(), lr=lr)

for epoch in range(epochs):

t = time.time()
model.train()
recovered, mu, logvar = model(features, adj_norm)
loss = loss_function(preds=recovered, labels=adj_label,
mu=mu, logvar=logvar, n_nodes=n_nodes,
norm=norm, weight=pos_weight)
loss.backward()
cur_loss = loss.item()
optimizer.step()

hidden_emb = mu.data.numpy()
roc_curr, ap_curr, balanced_acc, f1_sco = get_roc_score(hidden_emb, adj_orig, val_edges, val_edges_false)

print("Epoch:", '%04d' % (epoch + 1), "train_loss=", "{:.5f}".format(cur_loss),
"val_auc=", "{:.5f}".format(roc_curr),
"val_ap=", "{:.5f}".format(ap_curr),
"val_balanced_acc =", "{:.5f}".format(balanced_acc),
"val_F1_score =", "{:.5f}".format(f1_sco),
"time=", "{:.5f}".format(time.time() - t)
)

print("Optimization Finished!")

roc_score, ap_score, balanced_acc, f1_sco = get_roc_score(hidden_emb, adj_orig, test_edges, test_edges_false)
print('Test ROC AUC score: ' + str(roc_score))
print('Test AP score: ' + str(ap_score))
print('Test Bal_acc score: ' + str(balanced_acc))
print('Test F1 score: ' + str(f1_sco))``````

Hi @ptrblck,

So I managed to get a balanced accuracy of 80% and F1 score of 79% by adding actual features (biological features) to the network. I thought that the GNN model could learn the connective properties of the network without node features (i.e identity matrix), but it seems that they are essential in graph representation learning.
Now I encounter another issue with the loss. After training the model for a large number of iterations (400-500), the KLD loss suddenly increases and decreases again. This causes the accuracy to drop from 80% to 71%, before increasing again. I am thinking this could be due to a floating point arithmetic issue.

Here’s an illustration:

If you have any advice, please let me know.
Thanks again.

Why do you think so?

Unfortunately, I cannot execute the previous code snippet, as it’s using undefined data.

Have you managed to run it?