Reconstruction of higher-order complex networks under non-Markovian dynamics

Uncovering underlying topological structures is fundamental to understanding complex systems, but it presents a massive combinatorial challenge that is further complicated when spreading processes exhibit non-Markovian, memory-dependent characteristics and involve the reconstruction of higher-order structures. To simultaneously address both the structural and dynamical complexities, we propose a likelihood inference framework for the full-order reconstruction of simplicial complexes from general non-Markovian binary time series. We introduce a target-specific exposure clock and a Weibull memory kernel to explicitly model nonlinear aging effects in higher-order contagion. Methodologically, our central contribution is a relaxation strategy that overcomes the non-convex optimization difficulties typical of memory-dependent models, yielding a globally concave log-likelihood landscape. This formulation enables sparse optimization via fast iterative shrinkage-thresholding algorithm to separate true simplices from correlated spurious candidates, untangling the complex temporal-topological coupling. Numerical experiments on synthetic and empirical networks demonstrate that the framework adapts to different memory effects and aging regimes, achieving robust reconstruction beyond the regime covered by conventional Markovian models, while also demonstrating its applicability when the true structural orders are unknown.