import tensorlayerx as tlx
import numpy as np
from gammagl.mpops import unsorted_segment_mean
from gammagl.utils import degree
[docs]
def homophily(edge_index, y, batch=None, method: str = 'edge'):
r"""The homophily of a graph characterizes how likely nodes with the same
label are near each other in a graph.
There are many measures of homophily that fits this definition.
In particular:
- In the `"Beyond Homophily in Graph Neural Networks: Current Limitations
and Effective Designs" <https://arxiv.org/abs/2006.11468>`_ paper, the
homophily is the fraction of edges in a graph which connects nodes
that have the same class label:
.. math::
\frac{| \{ (v,w) : (v,w) \in \mathcal{E} \wedge y_v = y_w \} | }
{| \mathcal{E}|}
That measure is called the *edge homophily ratio*.
- In the `"Geom-GCN: Geometric Graph Convolutional Networks"
<https://arxiv.org/abs/2002.05287>`_ paper, edge homophily is normalized
across neighborhoods:
.. math::
\frac{1}{| \mathcal{V}|} \sum_{v \in \mathcal{V}} \frac{ | \{ (w,v) : w
\in \mathcal{N}(v) \wedge y_v = y_w \} | } { | \mathcal{N}(v)| }
That measure is called the *node homophily ratio*.
- In the `"Large-Scale Learning on Non-Homophilous Graphs: New Benchmarks
and Strong Simple Methods" <https://arxiv.org/abs/2110.14446>`_ paper,
edge homophily is modified to be insensitive to the number of classes
and size of each class:
.. math::
\frac{1}{C-1} \sum_{k=1}^{C} \max \left(0, h_k - \frac{| \mathcal{C}_k|}
{| \mathcal{V}|} \right)
where :math:`C` denotes the number of classes, :math:`| \mathcal{C}_k|`
denotes the number of nodes of class :math:`k`, and :math:`h_k` denotes
the edge homophily ratio of nodes of class :math:`k`.
Thus, that measure is called the *class insensitive edge homophily
ratio*.
Parameters
----------
edge_index: tensor
The graph connectivity.
y: tensor
The labels.
batch: tensor, optional
Batch vector\
:math:`\mathbf{b} \in {\{ 0, \ldots,B-1\}}^N`, which assigns
each node to a specific example. (default: :obj:`None`)
method: str, optional
The method used to calculate the homophily,
either :obj:`"edge"` (first formula), :obj:`"node"` (second
formula) or :obj:`"edge_insensitive"` (third formula).
(default: :obj:`"edge"`)
Examples:
>>> edge_index = tlx.convert_to_tensor([[0, 1, 2, 3],
... [1, 2, 0, 4]])
>>> y = tlx.convert_to_tensor([0, 0, 0, 0, 1])
>>> # Edge homophily ratio
>>> homophily(edge_index, y, method='edge')
0.75
>>> # Node homophily ratio
>>> homophily(edge_index, y, method='node')
0.6000000238418579
>>> # Class insensitive edge homophily ratio
>>> homophily(edge_index, y, method='edge_insensitive')
0.19999998807907104
"""
assert method in {'edge', 'node', 'edge_insensitive'}
if tlx.is_tensor(edge_index):
edge_index = tlx.convert_to_numpy(edge_index)
if tlx.is_tensor(y):
y = tlx.convert_to_numpy(y)
y = np.squeeze(y, axis=-1) if y.ndim > 1 else y
row, col = edge_index[0], edge_index[1]
if method == 'edge':
out = np.zeros(row.shape[0])
out[y[row] == y[col]] = 1.
if batch is None:
return float(out.mean())
else:
out = tlx.convert_to_tensor(out, dtype=tlx.float32)
col = tlx.convert_to_tensor(col)
batch = tlx.convert_to_tensor(batch)
return unsorted_segment_mean(out, tlx.gather(batch, col))
elif method == 'node':
out = np.zeros(row.shape[0])
out[y[row] == y[col]] = 1.
out = unsorted_segment_mean(tlx.convert_to_tensor(out, dtype=tlx.float32), tlx.convert_to_tensor(col))
print(out)
if batch is None:
return float(tlx.reduce_mean(out))
else:
return unsorted_segment_mean(out, batch)
elif method == 'edge_insensitive':
assert y.ndim == 1
num_classes = int(y.max()) + 1
assert num_classes >= 2
if batch is None:
batch = np.zeros_like(y)
elif tlx.is_tensor(batch):
batch = tlx.convert_to_numpy(batch)
num_nodes = tlx.convert_to_numpy(degree(batch))
num_graphs = num_nodes.size
batch = num_classes * batch + y
h = tlx.convert_to_numpy(homophily(edge_index, y, batch, method='edge'))
h = np.reshape(h, [num_graphs, num_classes])
counts = np.bincount(batch, minlength=num_classes * num_graphs)
counts = np.reshape(counts, [num_graphs, num_classes])
proportions = counts / np.reshape(num_nodes, [-1, 1])
out = np.clip((h - proportions), a_min=0, a_max=None).sum(axis=-1)
out /= num_classes - 1
return out if out.size > 1 else float(out)
else:
raise NotImplementedError