from neuralogic.core.constructs.metadata import Metadata
from neuralogic.core.constructs.function import Transformation, Aggregation
from neuralogic.core.constructs.factories import R, V
from neuralogic.nn.module.module import Module
[docs]
class APPNPConv(Module):
r"""
Approximate Personalized Propagation of Neural Predictions layer from
`"Predict then Propagate: Graph Neural Networks meet Personalized PageRank" <https://arxiv.org/abs/1810.05997>`_.
Which can be expressed as:
.. math::
\mathbf{x}^{0}_i = \mathbf{x}_i
.. math::
\mathbf{x}^{k}_i = \alpha \cdot \mathbf{x}^0_i + (1 - \alpha) \cdot
{agg}_{j \in \mathcal{N}(i)}(\mathbf{x}^{k - 1}_j)
.. math::
\mathbf{x}^{\prime}_i = act(\mathbf{x}^{K}_i)
Where *act* is an activation function and *agg* aggregation function.
The first part of the second equation that is ":math:`\alpha \cdot \mathbf{x}^0_i`" is expressed
in the logic form as:
.. code-block:: logtalk
R.<output_name>__<k>(V.I) <= R.<feature_name>(V.I)[<alpha>].fixed()
The second part of the second equation that is
":math:`(1 - \alpha) \cdot {agg}_{j \in \mathcal{N}(i)}(\mathbf{x}^{k - 1}_j)`" is expressed as:
.. code-block:: logtalk
R.<output_name>__<k>(V.I) <= (R.<output_name>__<k-1>(V.J)[1 - <alpha>].fixed(), R.<edge_name>(V.J, V.I))
Examples
--------
The whole computation of this module
(parametrized as :code:`APPNPConv("h1", "h0", "_edge", 3, 0.1, Transformation.SIGMOID)`) is as follows:
.. code:: logtalk
metadata = Metadata(transformation=Transformation.IDENTITY, aggregation=Aggregation.SUM)
(R.h1__1(V.I) <= R.h0(V.I)[0.1].fixed()) | metadata
(R.h1__1(V.I) <= (R.h0(V.J)[0.9].fixed(), R._edge(V.J, V.I))) | metadata
R.h1__1/1 [Transformation.IDENTITY]
(R.h1__2(V.I) <= <0.1> R.h0(V.I)) | metadata
(R.h1__2(V.I) <= (<0.9> R.h1__1(V.J), R._edge(V.J, V.I))) | metadata
R.h1__2/1 [Transformation.IDENTITY]
(R.h1(V.I) <= <0.1> R.h0(V.I)) | metadata
(R.h1(V.I) <= (<0.9> R.h1__2(V.J), R._edge(V.J, V.I))) | metadata
R.h1 / 1 [Transformation.SIGMOID]
Parameters
----------
output_name : str
Output (head) predicate name of the module.
feature_name : str
Feature predicate name to get features from.
edge_name : str
Edge predicate name to use for neighborhood relations.
k : int
Number of iterations
alpha : float
Teleport probability
activation : Transformation
Activation function of the output.
Default: ``Transformation.IDENTITY``
aggregation : Aggregation
Aggregation function of nodes' neighbors.
Default: ``Aggregation.SUM``
"""
def __init__(
self,
output_name: str,
feature_name: str,
edge_name: str,
k: int,
alpha: float,
activation: Transformation = Transformation.IDENTITY,
aggregation: Aggregation = Aggregation.SUM,
):
self.output_name = output_name
self.feature_name = feature_name
self.edge_name = edge_name
self.alpha = alpha
self.k = k
self.aggregation = aggregation
self.activation = activation
def __call__(self):
head = R.get(self.output_name)(V.I)
metadata = Metadata(transformation=Transformation.IDENTITY, aggregation=self.aggregation)
edge = R.get(self.edge_name)
feature = R.get(self.feature_name)
rules = []
for k in range(1, self.k):
k_head = R.get(f"{self.output_name}__{k}")(V.I)
rules.append((k_head <= feature(V.I)[self.alpha].fixed()) | metadata)
if k == 1:
rules.append((k_head <= (feature(V.J)[1 - self.alpha].fixed(), edge(V.J, V.I))) | metadata)
else:
rules.append(
(k_head <= (R.get(f"{self.output_name}__{k - 1}")(V.J)[1 - self.alpha].fixed(), edge(V.J, V.I)))
| metadata
)
rules.append(R.get(f"{self.output_name}__{k}") / 1 | Metadata(transformation=Transformation.IDENTITY))
if self.k == 1:
output_rule = head <= (feature(V.J)[1 - self.alpha].fixed(), edge(V.J, V.I))
else:
output_rule = head <= (
R.get(f"{self.output_name}__{self.k - 1}")(V.J)[1 - self.alpha].fixed(),
edge(V.J, V.I),
)
return [
*rules,
(head <= feature(V.I)[self.alpha].fixed()) | metadata,
output_rule | metadata,
R.get(self.output_name) / 1 | Metadata(transformation=self.activation),
]