Specify message direction, normalization and nonlinear placement

For this lesson use A[v,u]=1 when u→v, so a receiving destination occupies a matrix row. This is the transpose of the source-row convention in the storage lesson; name that change deliberately. Multiplying A by a node-feature matrix sums incoming source features. Dividing each nonempty row by its incoming degree yields a mean. For an empty neighborhood, choose an explicit convention such as a zero aggregate or a separate self transformation; division by zero is not an implementation detail to ignore.

A sum retains information about neighborhood multiplicity, while a mean removes a uniform scaling from duplicated identical neighbors. Neither is universally better. Edge weights, relation types and parallel edges can each change the aggregate. GraphSAGE’s mean variant with self concatenation differs from a single mean that includes the self feature. Verify the exact architecture rather than treating every mean-based layer as the same operator. A self contribution can be separately parameterized, included in normalization or absent altogether.

The familiar GCN operator for an undirected graph adds self-loops and uses symmetric normalization on both ends of each edge. Each message coefficient depends on the augmented degrees of the source and destination. A destination-only mean instead uses one row-degree divisor. They coincide only in special cases and should not be substituted casually. Directed graphs require a deliberate extension or another normalization rule; the standard undirected derivation does not uniquely choose the correct directed operator.

Graph attention layers compute learned coefficients and normalize them over a specified destination neighborhood, often including self-loops. Those coefficients depend on the current features, unlike fixed degree coefficients. Sampling a smaller neighborhood changes the attention softmax denominator and generally changes its value; reusing the full-graph formula without examining that denominator is not an unbiasedness proof. Likewise, degrees computed on a sampled subgraph differ from full-graph degrees, so normalization metadata must identify which graph and self-loop policy produced it.

Use a tiny asymmetric graph with different source features to test direction, and an irregular undirected graph to test normalization. Compare per-node messages before nonlinearities, then the destination update. Maintain an explicit edge map when graph transformations reorder records. A final accuracy metric can conceal swapped directions or duplicated self contributions, particularly on a dataset where node features alone are predictive. Record whether bias and activation are applied to each message or once to the accumulated destination value.

The network lab isolates that last distinction. Reset and set both x=1 and w=1; predict σ(0.5)≈0.62246 for the fixed bias -0.5. Separately Reset and set both x=2 and w=1; predict σ(1.5)≈0.81757. Applying this unit to two identical scalar messages then summing gives about 1.24492, different from summing the inputs first and applying it once. The lab is one sigmoid unit, not a graph layer, and does not calculate GCN degrees or attention weights.