Graph SILVA

examples/graph_silva.py builds a small ring graph and applies SILVAGraphLayer.

python examples/graph_silva.py

The graph has eight entities and edges

\[ 0\to1,\;1\to2,\;\ldots,\;7\to0. \]

The layer solves

\[ z^\star = f_\theta(z^\star,x) = \Phi\{S_\theta(x)+L_\theta(z^\star,E)+G_\theta(z^\star)\}. \]

After solving, a linear head creates two logits per node:

z = layer(x, edge_index=edge_index)
loss = torch.nn.functional.cross_entropy(head(torch.tanh(z)), y)
loss.backward()

The printed state_shape confirms the hidden representation, loss confirms gradient flow, and spectral_radius gives a local stability diagnostic for the solved state.

The compact ring has state shape (8, 12). Inspect the fixed-point residual before using the task loss as evidence, and compare the spectral radius under the same damping used by the solver. Graph, attention, and message-passing sources are listed in Graphs, Attention, and Messages.

Complete Worked Study

The short construction above identifies the main API. A complete study must also distinguish the state equation, task objective, numerical residual, gradient path, and scale transfer. In this example, the equilibrium state is one latent vector per node or entity, the condition is node features, edges, edge attributes, and graph batches, and the repeated map is a source-injected graph message or monotone graph transition.

Derivation From Transition to Reported Result

The forward solve is defined by

\[ z^\star = T_\theta(z^\star,x). \]

The task output and task objective are separate from convergence:

\[ \widehat y = R_\phi(z^\star), \qquad \mathcal L_{\mathrm{task}}=\ell(\widehat y,y). \]

For a computed state \(z_K\), the normalized fixed-point residual is

\[ r_K = \frac{\lVert T_\theta(z_K,x)-z_K\rVert_2} {\lVert z_K\rVert_2+\varepsilon}. \]

A small task loss does not imply a small \(r_K\), and a small \(r_K\) does not establish task quality. Both belong in the result. For implicit training, the parameter sensitivity follows

\[ \frac{\mathrm d z^\star}{\mathrm d\theta} = \left(I-\partial_z T_\theta(z^\star,x)\right)^{-1} \partial_\theta T_\theta(z^\star,x). \]

This is why the example checks gradients in addition to forward convergence. The reader-facing evidence for this route is node-state shape, task loss, equilibrium residual, and gradients. The invariants that must remain true are node relabeling equivariance, graph boundaries, and state shape.

Run the Complete Example

python examples/graph_silva.py

Measured Compact Output

The following output was produced by the executable program in the current repository. Floating-point values may vary slightly across devices and library builds, while shapes, finite values, invariants, and declared tolerances must remain stable.

state_shape (8, 12)
loss 0.7801069021224976
residual 0.07725001126527786
spectral_radius 0.7778381109237671

Interpret the Output

Evidence	What it answers	What would require investigation
Tensor shapes	Did every source, state, branch, and readout preserve its declared contract?	A changed entity, channel, token, or spatial dimension
Task metric	Did the compact task execute and produce finite evidence?	Non-finite loss, a missing mask, or a metric computed on the wrong split
Fixed-point residual	Did the returned state satisfy the repeated transition to the requested tolerance?	A residual plateau, rising trajectory, or convergence flag inconsistent with the value
Iteration or trajectory data	How much numerical work was required?	Solver effort that grows sharply under a small input or resolution change
Gradient evidence	Can the loss reach every trainable component through the selected backward mode?	Missing, non-finite, or implausibly large gradients
Domain invariant	Did the method retain positivity, feasibility, boundary values, permutation behavior, or another structural requirement?	A task metric that looks acceptable while the structural contract fails

The compact output is a mechanism check, not a paper-scale benchmark claim. It shows that data enter the intended construction, the transition executes, the solver returns diagnostics, and differentiation reaches trainable parameters.

Add a Solver and Scale Sweep

The next run should hold model parameters and data fixed while changing one numerical control at a time. A complete experiment record can use this schema:

experiment:
  example: graph-silva
  state: one latent vector per node or entity
  condition: node features, edges, edge attributes, and graph batches
  repeated_transition: a source-injected graph message or monotone graph transition
  invariant_checks: node relabeling equivariance, graph boundaries, and state shape
  compact_evidence: node-state shape, task loss, equilibrium residual, and gradients
  scale_axes: node count, edge count, feature width, and number of graphs
solver_sweep:
  methods: [picard, anderson, broyden]
  tolerances: [1.0e-4, 1.0e-6, 1.0e-8]
  maximum_iterations: [25, 50, 100]
report:
  - task_metric
  - fixed_point_residual
  - backward_linear_residual
  - iterations
  - wall_time
  - peak_memory
  - gradient_norm

At full scale, move toward the complete graph split with sparse operators and task metrics. Increase only one of node count, edge count, feature width, and number of graphs at a time. Retain this compact run as a regression test, preserve the source split and preprocessing receipt, archive the resolved configuration and checkpoint, and report convergence failures rather than discarding them.

Where to Go Next

Question	Page
How is this graph transition derived branch by branch?	SILVA From Scratch
Which graph-layer arguments are public?	Layers API
How is graph pooling extended to molecules?	Molecules Example