21-EFT.WP.Metrology.Sync v1.0 | Chapter 11 — Distributed Graph Synchronization & Topology Optimization

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Chapter 11 — Distributed Graph Synchronization & Topology Optimization

One-line objective: On a distributed network G=(V,E), jointly model measurement, servo, and routing, and—via weighted-Laplacian optimization plus consensus algorithms—minimize end-to-end synchronization error subject to availability and cost constraints.

I. Scope & Objects

1. Applies to
   • Multi-GM, multi-domain, multi-link distributed timing networks (mixed PTP/NTP/SyncE/White Rabbit).
   • Scenarios requiring topology, edge weights, and redundancy optimization under throughput/cost/reliability constraints.
2. Inputs
   • Nodes V, candidate links E, link quality q_path(e), asymmetry estimates asym_e, measurement noise sigma_e, capacity/cost cost_e.
   • Reference set R ⊂ V (GM candidates) with scores score_r.
   • Arrival-time data: per-path dual-form T_arr and delta_form, time axes tau_mono/ts.
   • Servo & business constraints: slew_max, tol_sync, SLO targets.
3. Outputs
   • Optimal subgraph E* ⊂ E with weights w_e*; anchoring strategy (single-root / multi-root); parameters for consensus/distributed estimation.
   • Error-bound evaluation and panel metrics: trace(L_w^+), lambda_2(L_w), R_eff(i,j), and P95/P99 sync errors.
   • Manifests & audit: manifest.sync.graph.*.

II. Terms & Variables

• Graphs & matrices: G=(V,E), weighted Laplacian L_w, pseudoinverse L_w^+ (Moore–Penrose), algebraic connectivity lambda_2(L_w).
• States & measurements: node offsets x_i, edge measurement z_e, bias/asymmetry b_e, noise n_e ~ N(0, sigma_e^2).
• Weights & quality: w_e = 1 / sigma_e^2 or a function w(q_path, sigma, asym).
• Error gauges: R_eff(i,j) (effective resistance), cov(x) ≈ L_w^+ (after anchoring).
• Constraints & budgets: cost(E') = ∑_{e∈E'} cost_e ≤ B, deg(i) ∈ [d_min, d_max].
• Units: unit(x)="s", unit(sigma)="s", check_dim(expr) must pass.

III. Axioms P611- **

• P611-1 (Consistent anchoring): At least one anchor r ∈ R is fixed or strongly penalized to make L_w invertible on the anchored subspace.
• P611-2 (Explicit measurement model): Edge measurements follow a differencing model with explicit measure; asymmetry enters as b_e in estimation and correction.
• P611-3 (Dual-form arrivals): For any path, record both arrival forms and enforce delta_form ≤ tol_Tarr.
• P611-4 (Spectral–variance concordance): Network error lower bounds use L_w^+ and R_eff; topology objectives must reflect those bounds.
• P611-5 (Minimum disturbance): Topology reconfiguration and weight updates proceed gradually, maintaining tau_mono continuity with slew_max limits.
• P611-6 (Multi-root anti-backflow): Multi-GM anchoring via a multi-root spanning forest; forbid bias backflow and uncalibrated bi-directional injection.

IV. Minimal Equations S611- **

• S611-1 (Edge measurement model)
  z_e = ( x_i - x_j ) + b_e + n_e, e = (i,j)
  w_e = f( sigma_e, q_path(e), |asym_e| )
• S611-2 (Weighted least-squares)
  x_hat = arg min_x ( ∑_{e∈E} w_e * ( z_e - ( x_i - x_j ) - b_e )^2 )
  subject to anchor(x_r = 0, r ∈ R)
  Covariance approximation: cov(x_hat) ≈ L_w^+
• S611-3 (Pairwise error lower bound)
  var( x_i - x_j ) ≥ R_eff(i,j) with R_eff(i,j) = e_{ij}^T L_w^+ e_{ij}
• S611-4 (Spectral connectivity & convergence)
  Consensus convergence rate ∝ lambda_2(L_w); mixing time T_mix ≈ O( 1 / lambda_2(L_w) )
• S611-5 (Topology selection)
  Under budget B, minimize the variance surrogate:
  E_obj(E') = trace( L_w(E')^+ )
  minimize E_obj(E') subject to cost(E') ≤ B, connectivity, degree bounds
• S611-6 (Multi-root anchoring penalty)
  With anchor penalty A = diag(a_i), a_i > 0 for anchors:
  x_hat = arg min_x ( x^T L_w x + (x - x_ref)^T A (x - x_ref) )

V. Flow M60-11 (Graph Synchronization & Topology Optimization)

• Topology readiness
  Build the candidate graph G=(V,E); collect q_path(e), sigma_e, asym_e, cost_e; run check_dim and delta_form audits.
• Edge-weight calibration
  Normalize w_e = g(sigma_e, q_path, asym_e); drop edges with sub-threshold q_path.
• Anchor planning
  Choose single- or multi-root strategy; form a multi-root spanning forest; configure anchor penalties a_i.
• Distributed estimation
  Select algorithm: consensus, ADMM, distributed Kalman; distribute steps and bandwidth limits; run on tau_mono.
• Error audit
  Estimate cov(x_hat) ≈ L_w^+; compute R_eff(i,j), trace(L_w^+), lambda_2(L_w); compare with tol_sync & SLOs.
• Topology optimization
  Solve min trace(L_w^+) s.t. cost(E') ≤ B (greedy/submodular approximation/MILP); satisfy degree, redundancy, loop-avoidance constraints.
• Progressive deployment
  Update w_e and activate/deactivate edges with minimal disturbance; bound slew rate; record manifest.sync.graph.update.
• Continuous monitoring
  Refresh q_path, sigma_e, asym_e online; apply re-optimization thresholds and cool-downs; rollback on anomalies; switch redundancy if needed.

VI. Contracts & Assertions

• C60-111 (Connectivity & anchoring): G is connected per domain and has a valid anchor set R.
• C60-112 (Dual-form alignment): All selected paths satisfy delta_form ≤ tol_Tarr.
• C60-113 (Spectral floor): lambda_2(L_w) ≥ lambda_min; below threshold triggers reconstruction.
• C60-114 (Variance ceiling): trace(L_w^+) ≤ tau_trace_max and max_{i,j} R_eff(i,j) ≤ R_max.
• C60-115 (Budget & degree): cost(E') ≤ B and deg(i) ∈ [d_min, d_max].
• C60-116 (Progressive updates): Per-step edge-weight change |Δw_e| ≤ dw_max; slew ≤ slew_max.

VII. Implementation Bindings I60-11*

• build_graph(measures) -> G, attrs
• calibrate_edge_weights(attrs) -> {w_e}
• select_anchors(candidates, policy) -> R, A
• distributed_sync(G, w, A, algo, steps) -> {x_hat, stats}
• compute_graph_metrics(G, w) -> {L_w, L_w^+, lambda2, R_eff_max, traceL+}
• optimize_topology(G, w, cost, budget, constraints) -> {E*, w*}
• progressive_deploy(E→E*, w→w*) -> plan
• emit_graph_manifest(G, metrics, plan) -> manifest.sync.graph
• Invariants: units/dimensions validated; tau_mono continuity; delta_form persisted; updates are rollback-safe.

VIII. Cross-References

• Asymmetry modeling & correction: Chapter 7.
• Servo & filter selection: Chapter 6.
• Oscillator noise → tdev propagation: EFT.WP.Metrology.TimeBase v1.0 Chapter 7.
• Cleaning contracts & manifest signing: Methods.Cleaning v1.0 Chapter 10.
• Streaming & backpressure: this volume Chapter 12/11 (per ToC).

IX. Quality & Risk Control

1. SLIs: lambda_2(L_w), trace(L_w^+), R_eff_max, sync_error_p95/p99, convergence_time_p95, time_in_compliance, reopt_count.
2. Risk actions:
   • If lambda_2 nears threshold → add edges or retune weights.
   • If R_eff_max breaches → enable backup links or increase GM density.
   • Frequent re-optimizations → raise thresholds & cool-down; inspect q_path jitter and routing policy.
   • delta_form breach → disable affected paths and trigger arrival-time revalidation.

Summary