A grid, before the network

Keep the relations in view

Suppose five buses in a small power-grid sketch each carry a feature. You want a model that can combine a bus's own information with information from connected buses. A table can store the features, but the table's row order does not say who is connected to whom. The graph supplies that missing relation.

We will use the shape of a grid to make message passing inspectable. The values are invented, dimensionless features. They are not voltage, power, fault probabilities, or calibrated stress readings. No electrical equation runs underneath the picture.

A graph is a set of nodes and a set of edges. Here the nodes are A, B, C, D and E. The five undirected edges are AB, AC, BC, CD and DE. Undirected means that our model permits messages in both directions along each listed connection. It says nothing about the direction of electrical flow.

A's neighbors are B and C. C's neighbors are A, B and D. E has only D. Move the circles around on the page and those neighbor sets stay the same: a graph's drawn position is different from its connectivity.

CL-001 · Gilmer et al., §2: attributed graph inputs

A number attached to a node

A feature is a recorded input attached to an entity. Write the input at node $i$ as $x_i$. In A–E order, our input is:

The superscript $\top$ indicates a column: five rows, one feature per node. A richer input could give each node a vector—an ordered list of several features. We start with a scalar, one number, so every operation fits on paper.

A hidden state is the representation carried between layers. We write it as $h_i^{(\ell)}$, where $\ell$ counts layers. At the start, set $h_i^{(0)}=x_i$. The network will change states while the input remains the original record.

Edges can also carry features. We assign each edge a synthetic gate $e_{ij}=1$. Later we will halve the C–D gate. This multiplier scales a model message; it is not a measured property of an electrical line.

CL-007 · Toy assumption: all five values, edge gates and the graph itself were chosen for this lesson.

A representation is a starting point

So far, we have a graph and features. We have not specified what to predict, what counts as a good prediction, or how to learn parameters. Those are separate modeling choices.

The immediate task is smaller: compute a new state for every node using the same local rule. If you can explain what C receives, you can reconstruct the main mechanism of a message-passing layer. Then we can ask how far a local change travels.

CL-014 · Model boundary: topology chooses communication paths. This toy contains no physical power-balance or conservation equation.