Tracking Dynamic Simplicial Complexes via Constrained State-Space Estimation

1. INTRODUCTION

• Simplicial complexes model group interactions while coupling simplex levels through inclusion, but existing inference methods predominantly assume static topologies.Source: §1 · p. 1
• Because higher-order interactions evolve over time, dynamic SC inference must recover changing structures across simplex levels while satisfying SC constraints.Source: §1 · p. 1
• The authors identify no existing method that jointly tracks edges and triangles in a time-varying SC; prior methods learn triangles while filtering signals [7] or infer higher-order cells from known lower-order structure.Source: §1 · p. 1
• Building on dynamic SC generation models, the proposed closure-aware Markov transition preserves inclusion in its conditional mean and captures simplex dependencies in process covariance.Source: §1 · p. 1
• Observation-derived virtual measurements use common SC-learning signal priors to avoid known input-output pairs, while expected simplex counts are incorporated into the measurement model.Source: §1 · p. 1
• Nonlinear pseudo-measurements encode SC constraints in correction, with EKF, IEKF, and Laplace approximation progressively incorporating richer constraint information into covariance updates.Source: §1 · p. 1

Problem Statement

• A simplicial complex of dimension D contains simplices of dimension at most D and is closed under inclusion; this work restricts attention to dimension two.Nodes, edges, and triangles are respectively 0-, 1-, and 2-simplices.Source: §2 · p. 1
• Following, binary selection vectors represent active edges and triangles among all candidate edges and triangles.The node and candidate-simplex sets remain fixed over time while simplex selections may change at each time instant.Source: §2 · p. 1
• Each active triangle must have all three boundary edges active, expressed as the constraint [s2]t ≤ [s1]e for every boundary edge e of triangle t.The inclusion constraint is imposed at every time step.Source: §2 · p. 1
• The goal is to sequentially infer time-varying topology from nodal and candidate-edge observations, given a signal–topology relationship and subject to inclusion constraints.Edge signals are assumed available over the full candidate edge set; otherwise they may be constructed from nodal signals using a permutation-invariant fusion function, as in.Source: §2 · pp. 1–2

• At each simplex level, inactive simplices are born with probability β_i and active simplices survive with probability ρ_i, with edges and intrinsic triangle proposals sampled independently conditioned on s_k.Source: §3 · p. 2
• A triangle is active only if all its boundary edges are present, so each triangle proposal is gated by its newly sampled boundary.Source: §3 · p. 2
• Unlike latent on-off gating, an inactive triangle reactivates only through birth probability β_2 when its boundary returns, rather than becoming active automatically; dynamic clique complexes fill every induced clique.Source: §3 · p. 2

4.1. Transition Equation

• The transition uses a continuous heteroscedastic Gaussian surrogate whose mean preserves inclusion, while its covariance captures cross-level and shared-boundary dependencies.The surrogate approximates Bernoulli transitions by matching conditional means and covariances, evaluated at continuous states.Source: §4.1 · pp. 2–3

4.2. Measurement Equation

• Virtual measurements encode node- and edge-signal smoothness alongside edge and triangle cardinalities, relating observed signals to the complex topology.Cardinality measurements prevent the trivial all-zero solution for zero-valued smoothness measurements and can introduce sparsity under structural-equation models.Source: §4.2 · p. 3

4.3. Constraint-Aware Measurement Update

• Nonlinear inclusion and box pseudo-measurements let constraints inform state and covariance: EKF linearizes once, IEKF uses MAP and Gauss–Newton updates, and Laplace adds full local curvature.Unlike a constrained MAP update that retains an unconstrained covariance, the methods incorporate constraint information progressively into covariance updates.Source: §4.3 · pp. 3–4

5.1. Simulated Experiments

• The simulated setup uses 15-node Erdős–Rényi seed complexes, half of feasible triangles activated, and signals generated by Laplacian-filtering random inputs.The rollout uses β1 = β2 = 0.05 and ρ1 = ρ2 = 0.98; dynamics-informed methods receive true transition probabilities, and all methods receive true simplex counts.Source: §5.1 · p. 4
• In simulation, Laplace generally performs best, followed by IEKF; both improve triangle recovery and relative error over CM, while EKF sometimes falls below CM.Edge-support F1 differences are small, and RLS performs worst because it ignores the topology-transition model.Source: §5.1 · p. 4

5.2. Real Experiments

• On real topology, all estimators outperform RLS on both metrics at all but two time steps, where confidently wrong calibrated simplex counts hurt them.Differences among the proposed estimators are marginal on relative error; their consistent margin over RLS and F1 scores show tracking under the observation model. The test uses one classroom of 25 students over four hours, with synthetic smooth nodal and low-curl edge signals; parameters learned on day 1 are frozen for day 2.Source: §5.2 · p. 4

• The framework jointly tracks edges and triangles using a closure-aware birth-survival model for transition means and covariances, and virtual measurements that relate latent topology to observed signals and encode structural constraints.Source: §6 · p. 4
• EKF, IEKF, and Laplace-approximation tracking results demonstrate the value of topology-aware dynamics and constraint-informed covariance updates, particularly in simulations.Source: §6 · p. 4
• Future work will examine observability and identifiability under limited measurements and develop scalable algorithms for larger candidate complexes.Source: §6 · p. 4