Multi-sequential motif profile in multilayer visibility graphs

The concept of sequential visibility graph motifs, typical subgraphs with distinctive frequency features in time series-derived visibility graphs, has been well established, and relevant analytical theories are mainly limited to single-sequence horizontal visibility graphs (HVGs) and natural visibility graphs (NVGs). Here we extend the theoretical framework of sequential visibility graph motifs to multi-sequence and multilayer visibility graphs (ML-VGs). The developed theory yields exact analytical results for both deterministic dynamical systems and stochastic Markov processes with smooth invariant measures and continuous marginal distributions, regardless of whether the variables are defined on bounded or unbounded intervals. Benefiting from the inherent sparsity of visibility graph adjacency matrices and the inverse distance law of edge formation, our method supports linear-time numerical extraction of multi-sequential motif profiles for real-world multivariate time series. We further conduct comprehensive comparisons between single-layer and multilayer motif characteristics, and verify the discriminative ability and robustness of the proposed multi-sequential motif profile under different marginal distributions, chaotic dynamics and noisy conditions.