Citation-Aware Reporting

Use this example after a model runs and before writing a methods section. The goal is to report exactly what the package used: SILVA-specific components, solver choices, operator families, diagnostics, and dataset sources.

Example Configuration

from silva_networks import SILVAGraphPresetNetwork

model = SILVAGraphPresetNetwork(
    in_dim=num_features,
    hidden_dim=[64, 48],
    out_dim=num_classes,
    task="node",
    graph_mode="GAT",
    attention_mode="simple",
    stack_alphas=[0.5, 0.2],
    solver="anderson",
    max_iter=20,
)

This configuration uses:

Choice	Meaning
`SILVAGraphPresetNetwork`	SILVA structured equilibrium graph preset
`graph_mode="GAT"`	graph-attention local branch
`attention_mode="simple"`	gated mean-field global branch
`stack_alphas=[0.5, 0.2]`	two equilibrium layers with different damping
`solver="anderson"`	Anderson-accelerated fixed-point solve

Citation Checklist

For this configuration, cite:

SILVA paper/package [1] [2] for the structured interaction field and implementation.
Deep Equilibrium Models [4] for the fixed-point layer view.
Graph Attention Networks [16] for the local graph-attention operator.
Attention Is All You Need [29] if discussing the attention score form.
Deep Sets [18] if discussing permutation-invariant graph/set pooling.
Anderson 1965 [10] and Walker-Ni 2011 [11] for Anderson acceleration.
Dataset source, such as the UCI repository [42], for the dataset used in the experiment.

If you add stability_report, hutchinson_jacobian_norm, or jacobian_regularization_loss, also cite Hutchinson trace estimation and Jacobian-regularized DEQs [14] [6].

Methods Sentence

The model used SILVA Networks as structured equilibrium graph layers with a
GAT-style local branch, a gated mean-field global branch, and two damped
equilibrium stages. Fixed points were solved with Anderson acceleration, and
solver residuals were reported together with local Jacobian diagnostics.

Then attach the citations selected above. The full package-wide mapping is in Research Citation Audit.

Report Table

Report item	Value to include
package version	installed `silva-networks` version or commit
case family	graph/node, graph-level, vision, molecular, or custom
solver	Picard, Anderson, Broyden, or GMRES adjoint
damping	each `alpha` value
iteration budget	`max_iter` and tolerance
local branch	graph mean, GAT, kNN, channel kNN, custom
global branch	mean, gated mean, top-k attention, channel attention, custom
diagnostics	residuals, energy trace, spectral radius, Jacobian norm
dataset citation	dataset source and preprocessing choices

The exact article, software, solver, graph, attention, and dataset links are in Paper and References. Report tensor layouts with the table above when the state is not an ordinary (batch, features) matrix.

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 the tensor solved to equilibrium, the condition is the observed input or source tensor, and the repeated map is the state-preserving transition evaluated by the root solver.

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 a complete machine-readable source and verification record. The invariants that must remain true are shape, device, dtype, finiteness, and differentiability.

Run the Complete Example

python examples/reproduction_registry.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.

silva_fno_deq paper-adaptation compact-verified
silva_monotone_graph_equilibrium paper-adaptation compact-verified
silva_physics_informed_equilibrium paper-adaptation compact-verified
diffusion_equilibrium paper-adaptation compact-verified
transition report SILVATransitionReport(state_shape=(5, 4), output_shape=(5, 4), preserves_shape=True, preserves_device=True, preserves_dtype=True, finite=True, differentiable=True, parameter_count=28)
equilibrium residual 1.095007249318769e-07

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: citation-aware-reporting
  state: the tensor solved to equilibrium
  condition: the observed input or source tensor
  repeated_transition: the state-preserving transition evaluated by the root solver
  invariant_checks: shape, device, dtype, finiteness, and differentiability
  compact_evidence: a complete machine-readable source and verification record
  scale_axes: state width, batch size, and data volume
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 a source-conforming multi-seed study with archived configuration and receipts. Increase only one of state width, batch size, and data volume 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
Which sources and citation fields have been audited?	Research Citation Audit
Which measured outputs can be reported?	Results
Where is the complete bibliography?	Paper and References