Structured Equilibria Example

The runnable source is examples/structured_equilibria.py. It exercises all six public families on deterministic compact data and prints the diagnostic that defines success for each mechanism.

Every example retains the same two-stage SILVA contract,

\[ z^\star=T_\theta(z^\star,x), \qquad \widehat y=Q_\psi(z^\star), \]

while changing the mathematical structure inside \(T_\theta\). The mechanisms follow monDEQ [75], pcDEQ [76], NEMON [77], EIGNN [78], MGNNI [79], and DeltaDEQ [80]. The reference list links each primary article and source repository separately.

python examples/structured_equilibria.py

The example reports:

the monotonicity certificate for the monotone-operator point;
the minimum positive-concave state value;
the weighted-infinity one-sided bound;
the EIGNN spectral denominator margin;
the MGNNI attention normalization;
the DeltaDEQ active fraction and exact full-map residual.

These are mechanism checks. For a benchmark study, replace the compact builder, source/readout modules, dimensions, and runtime configuration while retaining the same result object and diagnostics.

The dedicated notebooks add the construction paths that do not fit in this single smoke example: notebook 37 executes source-style pcDEQ projection, notebook 40 differentiates through a custom graph-conditioned MGNNI source, and notebook 41 compares source-aligned full-map training with delta-cached evaluation before exercising delta-forward implicit differentiation as a separately labeled SILVA extension.

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 certificates, positivity, one-sided bounds, scale weights, and cache activity. The invariants that must remain true are node relabeling equivariance, graph boundaries, and state shape.

Run the Complete Example

python examples/structured_equilibria.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.

monotone operator torch.Size([8, 2]) certificate 0.5005146265029907
positive concave torch.Size([8, 1]) minimum state 0.042785972356796265
non-Euclidean torch.Size([8, 2]) one-sided bound 0.04999999701976776
efficient infinite graph torch.Size([12, 1]) spectral margin 0.44062745571136475
multiscale graph torch.Size([12, 1]) attention sums tensor([1.0000, 1.0000, 1.0000], grad_fn=<SliceBackward0>)
delta equilibrium torch.Size([8, 1]) mean active fraction 0.2036637931034483 exact residual 0.0014585574390366673

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: structured-equilibria
  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: certificates, positivity, one-sided bounds, scale weights, and cache activity
  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 source benchmark associated with the selected structured family. 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 each diagnostic derived?	Structured Equilibrium Families
What can be replaced inside each model?	Structured Equilibria API
Which data and metrics reproduce the source studies?	Reproduction Registry