SILVA Monotone Graph Equilibrium¶
This lab derives the monotone operator parameterization, its forward-backward
step, the normalized graph operator, and the corresponding SILVA family. It
checks an exact graph-elliptic dataset, trains a compact node field, and tests
node relabeling. The mechanism follows monotone implicit graph networks [47]
and remains inside the canonical silva_monotone_graph_equilibrium family.
Numbered literature: [1], [4], [47]. Each number opens the complete citation and its primary external source.
from pathlib import Path
import importlib.util
import subprocess
import sys
REPO_URL = "https://github.com/jseluis/silva-networks.git"
def find_local_silva_root():
candidates = [Path.cwd(), Path("/content/silva-networks")]
root = Path.cwd()
while root != root.parent:
candidates.append(root)
root = root.parent
for candidate in candidates:
if (candidate / "src" / "silva_networks").exists():
return candidate
return None
root = find_local_silva_root()
if root is not None:
sys.path.insert(0, str(root / "src"))
elif importlib.util.find_spec("silva_networks") is None:
subprocess.check_call([sys.executable, "-m", "pip", "install", f"git+{REPO_URL}"])
root = Path.cwd()
else:
root = Path.cwd()
import torch
import matplotlib.pyplot as plt
plt.rcParams.update({"figure.dpi": 300, "savefig.dpi": 300})
from silva_networks import (
SILVAMonotoneGraphEquilibrium,
SolverConfig,
make_monotone_chain_dataset,
normalized_laplacian_field,
)
torch.manual_seed(210)
<torch._C.Generator at 0x10c478ed0>
1. Graph Operator and Shape Contract¶
For node state $Z\in\mathbb R^{N\times d}$, define
$$ G=\frac12\left(I-D^{-1/2}AD^{-1/2}\right), \qquad GZ\in\mathbb R^{N\times d}. $$
The factor $1/2$ places the normalized-Laplacian spectrum in $[0,1]$. The package accepts a directed edge list; bidirectional edges represent an undirected graph.
data = make_monotone_chain_dataset(nodes=12, channels=1, diffusion=0.6, seed=21)
graph_field = normalized_laplacian_field(data.target, data.edge_index)
equation_error = data.equation_residual().abs().max()
assert data.source.shape == data.target.shape == (12, 1)
assert graph_field.shape == data.target.shape
assert equation_error < 1e-6
print("edges:", data.edge_index.shape[1])
print("maximum graph-equation residual:", float(equation_error))
edges: 22 maximum graph-equation residual: 1.1920928955078125e-07
2. Monotone Channel Parameterization¶
The channel operator is not an unconstrained matrix. It is formed as
$$ W=(1-m)I-CC^T+F-F^T, \qquad m>0. $$
Because the skew term vanishes in the symmetric part,
$$ I-\frac{W+W^T}{2}=mI+CC^T\succeq mI. $$
The smallest eigenvalue is therefore a directly testable certificate. This is
the stability constraint represented by
SILVAMonotoneGraphTransition.monotonicity_certificate().
3. Forward-Backward Step as a SILVA Transition¶
With source $B(X)$ and proximal activation, one operator-splitting step is
$$ Z^{k+1}=\operatorname{prox}_{\alpha f} \left((1-\alpha)Z^k+\alpha(WGZ^k+B(X))\right). $$
In SILVA, $B(X)$ is the source branch, $WGZ$ is the graph-local branch, and the proximal map is the output nonlinearity. Reusing this transition until convergence produces one implicit graph point rather than an explicit stack.
model = SILVAMonotoneGraphEquilibrium(
in_dim=1,
state_dim=6,
out_dim=1,
margin=0.15,
step_size=0.7,
config=SolverConfig(solver="picard", max_iter=25, tol=1e-6),
)
initial = model(data.source, data.edge_index, return_result=True)
assert initial.output.shape == data.target.shape
assert initial.monotonicity_certificate >= 0.15 - 1e-6
print("certificate:", float(initial.monotonicity_certificate))
print("equilibrium residual:", initial.solver_result.residual)
certificate: 0.15033301711082458 equilibrium residual: 0.005010434426367283
4. Tiny Equation-Supervised Task¶
The target solves
$$ (I+\nu G)u=s. $$
The loss below teaches the readout and equilibrium transition to approximate that solution. This is a deterministic small-scale reproduction of the graph mechanism, not a citation-network benchmark.
optimizer = torch.optim.Adam(model.parameters(), lr=2e-2)
losses = []
for _ in range(12):
optimizer.zero_grad()
prediction = model(data.source, data.edge_index)
loss = torch.nn.functional.mse_loss(prediction, data.target)
loss.backward()
optimizer.step()
losses.append(float(loss.detach()))
trained = model(data.source, data.edge_index, return_result=True)
print("initial/final task loss:", losses[0], losses[-1])
print("final equilibrium residual:", trained.solver_result.residual)
initial/final task loss: 0.5128161311149597 0.09578704833984375 final equilibrium residual: 0.0008736563613638282
figure, axes = plt.subplots(1, 2, figsize=(7.2, 2.7))
axes[0].plot(losses, marker="o", markersize=2)
axes[0].set(xlabel="optimization step", ylabel="MSE", yscale="log")
axes[1].plot(data.target[:, 0], label="exact", linewidth=2)
axes[1].plot(trained.output.detach()[:, 0], "--", label="SILVA")
axes[1].set(xlabel="node", ylabel="field")
axes[1].legend()
figure.tight_layout()
plt.show()
5. Node Relabeling¶
For permutation matrix $P$, a graph equilibrium must satisfy
$$ F(PX,PEP^T)=PF(X,E). $$
The edge list must be relabeled with the nodes. This is different from permuting features while leaving graph topology unchanged.
permutation = torch.tensor([7, 1, 10, 3, 5, 9, 0, 11, 2, 8, 4, 6])
inverse = torch.empty_like(permutation)
inverse[permutation] = torch.arange(permutation.numel())
permuted_edges = inverse[data.edge_index]
with torch.no_grad():
reference = model(data.source, data.edge_index)
relabeled = model(data.source[permutation], permuted_edges)
equivariance_error = (relabeled - reference[permutation]).abs().max()
assert equivariance_error < 1e-5
print("relabeling error:", float(equivariance_error))
relabeling error: 0.0
6. What to Report¶
Record the graph normalization, directed-edge convention, margin $m$, proximal map, forward-backward step size, solver, fixed-point residual, and task metric. The monotonicity certificate diagnoses the parameterization; it does not replace the numerical residual or downstream accuracy.
From 21 Silva Monotone Graph Equilibrium to a Custom SILVA Family¶
The construction in this notebook can be separated into the universal conditioned-equilibrium contract
$$ z_0=I_\eta(x),\qquad z^\star=T_\theta(z^\star,x),\qquad \widehat y=Q_\psi(z^\star). $$
For this topic:
| Part | Concrete interpretation |
|---|---|
| Equilibrium state | one latent vector per node or entity |
| Condition | node features, edges, edge attributes, and graph batches |
| Repeated computation | a source-injected graph message or monotone graph transition |
| Required invariants | node relabeling equivariance, graph boundaries, and state shape |
| Replaceable components | input projection, message field, global field, transition, pooling, and head |
The initializer and source path are evaluated outside or alongside the root solve. Only the state-preserving transition is repeated. Replacing an internal architecture does not change this equation, provided the transition still maps the same state space into itself.
Replace the Monotone Transition and Certificate¶
model = SILVAMonotoneGraphEquilibrium(
in_dim=input_dim,
state_dim=width,
out_dim=output_dim,
transition=my_monotone_transition,
certificate=my_monotonicity_certificate,
readout=my_readout,
config=solver_config,
)
The transition has signature (state, inputs, edge_index, edge_weight) and
must preserve node count and state width. A custom certificate remains separate
from numerical convergence diagnostics.
import torch as silva_extension_torch
from torch import nn as silva_extension_nn
from silva_networks import (
SILVAConditionedEquilibrium,
SILVAZeroInitializer,
SolverConfig,
validate_silva_transition,
)
class NotebookExtensionTransition(silva_extension_nn.Module):
def __init__(self, condition_dim=2, state_dim=3):
super().__init__()
self.source = silva_extension_nn.Linear(condition_dim, state_dim)
self.state_field = silva_extension_nn.Sequential(
silva_extension_nn.Linear(state_dim, 2 * state_dim),
silva_extension_nn.Tanh(),
silva_extension_nn.Linear(2 * state_dim, state_dim),
)
def forward(self, state, condition):
return silva_extension_torch.tanh(
self.source(condition) + 0.15 * self.state_field(state)
)
silva_extension_torch.manual_seed(610)
notebook_condition = silva_extension_torch.linspace(-1.0, 1.0, 8).reshape(4, 2)
notebook_state0 = silva_extension_torch.zeros(4, 3)
notebook_transition = NotebookExtensionTransition()
notebook_report = validate_silva_transition(
notebook_transition,
notebook_state0,
notebook_condition,
)
assert notebook_report.valid
with silva_extension_torch.no_grad():
notebook_reference_step = silva_extension_torch.tanh(
notebook_transition.source(notebook_condition)
+ 0.15 * notebook_transition.state_field(notebook_state0)
)
silva_extension_torch.testing.assert_close(
notebook_transition(notebook_state0, notebook_condition),
notebook_reference_step,
)
notebook_custom_model = SILVAConditionedEquilibrium(
notebook_transition,
SILVAZeroInitializer(3),
readout=silva_extension_nn.Linear(3, 1),
config=SolverConfig(
solver="picard",
max_iter=40,
tol=1e-7,
backward_mode="implicit",
backward_solver="gmres",
anderson_batch_dims=1,
),
)
notebook_custom_result = notebook_custom_model(
notebook_condition,
return_result=True,
)
assert notebook_custom_result.output.shape == (4, 1)
assert notebook_custom_result.solver_result.residual < 1e-5
notebook_custom_result.output.square().mean().backward()
assert all(
parameter.grad is not None and silva_extension_torch.isfinite(parameter.grad).all()
for parameter in notebook_custom_model.parameters()
)
print("custom transition:", notebook_report)
print("equilibrium residual:", notebook_custom_result.solver_result.residual)
custom transition: SILVATransitionReport(state_shape=(4, 3), output_shape=(4, 3), preserves_shape=True, preserves_device=True, preserves_dtype=True, finite=True, differentiable=True, parameter_count=54) equilibrium residual: 5.960464477539063e-08
Numerical Equivalence, Compact Reproduction, and Scale¶
Before training, compare one packaged transition with an independently written update:
$$ e_{\mathrm{step}} =\frac{\|T_\theta(z,x)-T_{\mathrm{ref}}(z,x)\|_2} {\|T_{\mathrm{ref}}(z,x)\|_2+\varepsilon}. $$
After solving, report the fixed-point residual separately:
$$ e_{\mathrm{fp}} =\frac{\|T_\theta(z^\star,x)-z^\star\|_2} {\|z^\star\|_2+\varepsilon}. $$
For this notebook, a compact reproduction must declare and assert node/graph error, physical graph residual, and fixed-point residual. A full experiment must additionally record the source dataset version and split, preprocessing, architecture widths, solver and optimizer schedules, random seeds, baseline configuration, checkpoints, and every deviation from the cited protocol.
The principal scaling axes are node count, edge count, feature width, and number of graphs. Increase one axis at a time, retain the compact deterministic case as a regression test, and record task error, domain-specific residual, forward residual, backward linear residual, memory use, and runtime independently.
Extension Exercises¶
- Replace one component from this notebook while preserving its state and domain invariants.
- Write the replacement first as an independent reference function, then as a module, and assert one-step equivalence.
- Compare two solver configurations on the identical trained transition.
- Add a compact baseline and a predeclared metric threshold.
- Create a full-scale configuration without weakening the compact tests.
The complete authoring protocol is documented in Extending SILVA.
notebook_reproduction_record = {
"notebook": '21_silva_monotone_graph_equilibrium.ipynb',
"state": 'one latent vector per node or entity',
"condition": 'node features, edges, edge attributes, and graph batches',
"transition": 'a source-injected graph message or monotone graph transition',
"invariants": 'node relabeling equivariance, graph boundaries, and state shape',
"compact_metric": 'node/graph error, physical graph residual, and fixed-point residual',
"scale_axis": 'node count, edge count, feature width, and number of graphs',
}
assert all(notebook_reproduction_record.values())
notebook_reproduction_record
{'notebook': '21_silva_monotone_graph_equilibrium.ipynb',
'state': 'one latent vector per node or entity',
'condition': 'node features, edges, edge attributes, and graph batches',
'transition': 'a source-injected graph message or monotone graph transition',
'invariants': 'node relabeling equivariance, graph boundaries, and state shape',
'compact_metric': 'node/graph error, physical graph residual, and fixed-point residual',
'scale_axis': 'node count, edge count, feature width, and number of graphs'}
Worked Convergence and Sensitivity Study¶
The preceding example demonstrates one configured solve. This additional study changes the graph propagation factor while keeping the source fixed, so solver effort and implicit sensitivity can be read separately from task behavior. Locally, one eigendirection of a nonlinear transition can be represented by
$$ z_{k+1} = \rho z_k + u, \qquad 0 \leq \rho < 1. $$
Its equilibrium is
$$ z^\star = \frac{u}{1-\rho}. $$
Subtracting the fixed-point equation from the iteration gives the exact error recursion
$$ e_{k+1} = \rho e_k, \qquad |e_k| = \rho^k |e_0|. $$
For a requested absolute tolerance $\tau$, the idealized iteration estimate is
$$ k \geq \frac{\log(\tau/|e_0|)}{\log \rho}. $$
The same factor controls sensitivity. Differentiating the equilibrium with respect to the source gives
$$ \frac{\partial z^\star}{\partial u} =\frac{1}{1-\rho}. $$
Thus a transition can remain contractive while becoming expensive and highly sensitive as $\rho$ approaches one. The table and figure below measure this effect rather than merely stating it. They provide a reference envelope for the notebook's actual state, one latent vector per node or entity, and its repeated map, a source-injected graph message or monotone graph transition. The scalar study does not replace the domain model; it supplies a result whose convergence rate and derivative are known exactly, so the same reporting code can be trusted before it is applied to the larger transition.
import math as silva_deepening_math
import torch as silva_deepening_torch
silva_deepening_rates = (0.20, 0.45, 0.70, 0.85)
silva_deepening_source = 0.35
silva_deepening_tolerance = 1e-8
silva_deepening_histories = {}
silva_deepening_rows = []
for silva_deepening_rho in silva_deepening_rates:
silva_deepening_state = silva_deepening_torch.tensor(0.0)
silva_deepening_exact = silva_deepening_source / (1.0 - silva_deepening_rho)
silva_deepening_history = []
for silva_deepening_iteration in range(1, 241):
silva_deepening_next = (
silva_deepening_rho * silva_deepening_state + silva_deepening_source
)
silva_deepening_residual = abs(
float(silva_deepening_next - silva_deepening_state)
)
silva_deepening_history.append(silva_deepening_residual)
silva_deepening_state = silva_deepening_next
if silva_deepening_residual < silva_deepening_tolerance:
break
silva_deepening_u = silva_deepening_torch.tensor(
silva_deepening_source, requires_grad=True
)
silva_deepening_solution = silva_deepening_u / (1.0 - silva_deepening_rho)
silva_deepening_solution.backward()
silva_deepening_expected_sensitivity = 1.0 / (1.0 - silva_deepening_rho)
silva_deepening_gradient_error = abs(
float(silva_deepening_u.grad) - silva_deepening_expected_sensitivity
)
silva_deepening_histories[silva_deepening_rho] = silva_deepening_history
silva_deepening_rows.append(
(
silva_deepening_rho,
silva_deepening_iteration,
silva_deepening_history[-1],
abs(float(silva_deepening_state) - silva_deepening_exact),
float(silva_deepening_u.grad),
silva_deepening_gradient_error,
)
)
print('graph propagation factor')
print("rho | iterations | final residual | exact-state error | sensitivity | gradient error")
for silva_deepening_row in silva_deepening_rows:
print(
f"{silva_deepening_row[0]:.2f} | {silva_deepening_row[1]:3d} | "
f"{silva_deepening_row[2]:.3e} | {silva_deepening_row[3]:.3e} | "
f"{silva_deepening_row[4]:.4f} | {silva_deepening_row[5]:.3e}"
)
assert all(row[2] < silva_deepening_tolerance for row in silva_deepening_rows)
assert all(row[3] < 1e-6 for row in silva_deepening_rows)
assert all(row[5] < 1e-6 for row in silva_deepening_rows)
graph propagation factor rho | iterations | final residual | exact-state error | sensitivity | gradient error 0.20 | 12 | 0.000e+00 | 5.551e-17 | 1.2500 | 0.000e+00 0.45 | 23 | 0.000e+00 | 1.084e-08 | 1.8182 | 6.502e-08 0.70 | 45 | 0.000e+00 | 1.589e-07 | 3.3333 | 7.947e-08 0.85 | 93 | 0.000e+00 | 5.563e-07 | 6.6667 | 1.589e-07
import matplotlib.pyplot as silva_deepening_plt
silva_deepening_plt.rcParams.update({"figure.dpi": 300, "savefig.dpi": 300})
silva_deepening_figure, silva_deepening_axes = silva_deepening_plt.subplots(
1, 2, figsize=(8.6, 3.2)
)
for silva_deepening_rho, silva_deepening_history in silva_deepening_histories.items():
silva_deepening_axes[0].semilogy(
range(1, len(silva_deepening_history) + 1),
silva_deepening_history,
marker="o",
markersize=2,
linewidth=1.2,
label=f"rho={silva_deepening_rho:.2f}",
)
silva_deepening_axes[0].axhline(
silva_deepening_tolerance, color="black", linestyle="--", linewidth=0.9
)
silva_deepening_axes[0].set_xlabel("iteration")
silva_deepening_axes[0].set_ylabel("absolute residual")
silva_deepening_axes[0].set_title("Residual trajectories")
silva_deepening_axes[0].legend(fontsize=7)
silva_deepening_axes[1].plot(
[row[0] for row in silva_deepening_rows],
[row[1] for row in silva_deepening_rows],
marker="o",
label="iterations",
)
silva_deepening_sensitivity_axis = silva_deepening_axes[1].twinx()
silva_deepening_sensitivity_axis.plot(
[row[0] for row in silva_deepening_rows],
[row[4] for row in silva_deepening_rows],
color="tab:red",
marker="s",
label="sensitivity",
)
silva_deepening_axes[1].set_xlabel('graph propagation factor')
silva_deepening_axes[1].set_ylabel("iterations")
silva_deepening_sensitivity_axis.set_ylabel("implicit sensitivity", color="tab:red")
silva_deepening_axes[1].set_title("Cost and sensitivity")
silva_deepening_figure.tight_layout()
silva_deepening_plt.show()
Reading and Extending the Result¶
The measured residual curves become flatter as the graph propagation factor increases. The iteration count and the exact sensitivity rise together, but they answer different questions: iterations measure numerical work, while sensitivity describes how strongly the equilibrium reacts to the source. The gradient-error column verifies the differentiation path against the analytic derivative.
Apply the same separation to this notebook's full model:
| Report | Notebook-specific interpretation |
|---|---|
| Task evidence | node/graph error, physical graph residual, and fixed-point residual |
| Forward residual | Re-evaluate the complete transition at the returned state |
| Empirical rate | Compare consecutive residuals only after the transient regime |
| Backward residual | Record the linear-adjoint stopping value independently |
| Sensitivity | Perturb one declared source field while preserving all other inputs |
| Structural checks | node relabeling equivariance, graph boundaries, and state shape |
| Scale sweep | Change one of node count, edge count, feature width, and number of graphs at a time |
A richer experiment should now repeat the sweep with at least two forward solvers, two tolerances, and multiple seeds. Keep model parameters and data identical when comparing solvers. Then change one architecture or data-scale axis, retain the compact analytic study as a regression test, and report task quality, residuals, iterations, runtime, memory, gradient norms, and failed convergence cases together.