The classic GCN model employs both of the symmetric normalization and self-loop approach. Specifically,
$\mathbf{h}_{u}^{(k)}=\sigma(\mathbf{W}^{(k)}\sum_{v\in \mathcal{N}(u)\cap\{u\}} \frac {\mathbf{h}_v}{\sqrt{|\mathcal{N}(u)\mathcal{N}(v)|}})$


Normalized Aggregation in Graph Convolutional Network (GCN)

Information from very high degree nodes might not be useful, so researchers has suggested the following normalization:

$\mathbf{m}_{\mathcal{N}(u)}=\sum_{v\in \mathcal{N}(u)} \frac {\mathbf{h}_v}{\sqrt{|\mathcal{N}(u)\mathcal{N}(v)|}}$

University of Notre Dame

The most basic neighborhood aggregator takes a sum of a node’s neighbors. One drawback of this is that it’s highly sensitive to the node degrees. So there’s several ways of normalizing the aggregator.

Although it solves the degree sensitivity problem, it also caused loss of information. Indeed, it has been proven that an normalized GNN is less powerful than an not normalized GNN.


Neighborhood Normalization

Hamilton, W.L. Graph Representation Learning. *Synthesis Lectures on Artificial Intelligence and Machine Learning*. 2020, 14(3), 1–159. https://doi.org/10.2200/S01045ED1V01Y202009AIM046

Graph Representation Learning by William Hamilton

The simplist approach to  overcome node degree sensitivity is to take average:
$\mathbf{m}_{\mathcal{N}(u)}=\frac{\sum_{v\in \mathcal{N}(u)}\mathbf{h}_v}{|\mathcal{N}(u)|}$


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