03
How far can a change travel?
Connected does not mean immediately visible. Each local layer extends the dependency path.
Can E notice A?
Keep the graph and weights fixed. Raise only A's initial feature from 2 to 4. Before trying the lab, predict: can E's state change after one layer? After two?
In the first layer, B and C read A directly. A also retains part of its own change. D reads C's old state, which is still 1 in both runs. E reads D's old state, which is still 3. Neither D nor E can notice A yet.
In the second layer, D can receive C's changed state from layer one. E still reads D's unchanged layer-one state. The earliest path from A to E is A–C–D–E: three edges, hence three local layers.
CL-004 · Battaglia et al., §4.3 and Figure 7: local propagation without a global channel
Trace the two forward passes
The lab compares a baseline A=2 with your selected A value. Both runs use the same selected connection setting and the same weights. This isolates the effect of A's feature. Double rings and signed differences mark a numerical change; a connected node may still have zero difference at the current layer.
Follow the changed feature
Toy values · fixed a=b=½ · no training or electrical solver
Layer 1, A=4, C–D present. Within 1 hops of A: A, B, C. Observed nonzero differences: A, B, C. E has no path short enough to respond yet.
Selected-A run. Double ring = nonzero difference. Δ means selected minus baseline.
| Node | A=2 run | Selected A | Difference |
|---|---|---|---|
| A | 1.250 | 2.250 | +1.000 |
| B | 0.750 | 1.250 | +0.500 |
| C | 1.333 | 1.667 | +0.333 |
| D | 1.750 | 1.750 | 0.000 |
| E | 1.500 | 1.500 | 0.000 |
C at layer 1 reads layer 0. A: 4.000 × gate 1 = 4.000; B: 0.000 × gate 1 = 0.000; D: 3.000 × gate 1 = 3.000.
Message sum 7.000 ÷ degree 3 = 2.333. New C = ½ × own 1.000 + ½ × mean 2.333 = 1.667.
Isolate the edge intervention
This separate comparison holds your A input fixed and changes only C–D relative to the full graph.
| Node | Full C–D | Selected C–D | Difference |
|---|---|---|---|
| A | 2.250 | 2.250 | 0.000 |
| B | 1.250 | 1.250 | 0.000 |
| C | 1.667 | 1.667 | 0.000 |
| D | 1.750 | 1.750 | 0.000 |
| E | 1.500 | 1.500 | 0.000 |
The range is 0–4 layers and A is 0–6. Reset returns A=4, layer 1, full C–D. Computation uses full precision; display rounds to 3 decimals. Degree counts retained edges. Nothing here is an electrical quantity.
| Layer | Changed nodes | E difference |
|---|---|---|
| 0 | A | 0.000 |
| 1 | A, B, C | 0.000 |
| 2 | A, B, C, D | 0.000 |
| 3 | A, B, C, D, E | +0.042 |
At layer 1, A/B/C respond. At layer 2, D joins them. At layer 3, E first responds. Self retention keeps earlier effects present. These are deterministic outputs of the shared model.
Try A=4 at layers 0, 1, 2 and 3. Then return A to 2: every difference vanishes, even though paths still exist. The graph gives a set of possible dependencies. An actual change also requires a different input and a computation that responds to it.
For this positive linear toy and a nonzero change at A, every node within the selected number of hops responds. For a general learned GNN, zero weights, cancellation or a nonlinearity can suppress a reachable input. “Within k hops” is a dependency bound, not a promise that every reachable feature changes every output.
CL-010 · Derived for the stipulated positive self and neighbor coefficients. Displayed differences use a 10−12 threshold; computation uses full precision.
An edge edit changes the rule locally
Reset, then remove C–D. A's change can now reach only A, B and C, however many local layers you apply. The graph has split into two connected components. D and E still update their own states; they simply have no computational path from A.
Removing C–D also changes the neighbor count at its endpoints. At the original input, C now averages A=2 and B=0: the mean is 1, and the new C is 1. D now averages only E=0, so its new state is 3/2.
Now keep C–D but halve its gate. C still has three neighbors and D still has two; their messages across this edge are multiplied by 1/2. C's new value becomes 13/12 and D's becomes 13/8. A weaker message and a missing edge are different interventions in this model.
| C–D mode | Degree C | New C | New D |
|---|---|---|---|
| present | 3 | 1.333 | 1.750 |
| half | 3 | 1.083 | 1.625 |
| removed | 2 | 1.000 | 1.500 |
The lesson is not that a particular normalization is correct for a grid. It is that you must inspect the normalization actually used. Here a half gate attenuates a message without renormalizing the mean. It does not simulate an electrical line change.
CL-011 · Derived. The table holds A at its original value 2 to isolate the edge intervention. Later effects can spread from both edited endpoints.