from torch_geometric.typing import Adj
import torch
from torch import Tensor
from torch_sparse import SparseTensor
from torch_scatter import scatter_mean
@torch.jit._overload
def homophily(adj_t, y, method):
# type: (SparseTensor, Tensor) -> float
pass
@torch.jit._overload
def homophily(edge_index, y, method):
# type: (Tensor, Tensor) -> float
pass
[docs]def homophily(edge_index: Adj, y: Tensor, 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::
\text{homophily} = \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::
\text{homophily} = \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*.
Args:
edge_index (Tensor or SparseTensor): The graph connectivity.
y (Tensor): The labels.
method (str, optional): The method used to calculate the homophily,
either :obj:`"edge"` (first formula) or :obj:`"node"`
(second formula). (default: :obj:`"edge"`)
"""
assert method in ['edge', 'node']
y = y.squeeze(-1) if y.dim() > 1 else y
if isinstance(edge_index, SparseTensor):
col, row, _ = edge_index.coo()
else:
row, col = edge_index
if method == 'edge':
return int((y[row] == y[col]).sum()) / row.size(0)
else:
out = torch.zeros_like(row, dtype=float)
out[y[row] == y[col]] = 1.
out = scatter_mean(out, col, 0, dim_size=y.size(0))
return float(out.mean())