Spectral moment analysis of pinning synchronization in simplicial complexes

The eigenvalue spectrum plays a pivotal role in the analysis of network synchronization. Motivated by this observation, this paper proposes a spectral moment-based approach to investigate pinning synchronization in higher-order networks modeled by simplicial complexes. Focusing on three representative classes of simplicial complexes, namely globally coupled, nearest-neighbor coupled, and random structures, we derive analytical expressions for the first three expected spectral moments of the coupling-control matrix. For regular simplicial complexes constructed from globally coupled and nearest-neighbor coupled networks, the first three expected spectral moments are characterized in terms of structural properties, including the network size, first-order node degree, the number of 2-simplices incident to each node, and weighted closed walks associated with the higher-order topology, as well as coupling and pinning parameters such as the coupling strength ratio, control gain, and the number of pinned nodes. For random simplicial complexes, the expected spectral moments further depend on structural statistics including the mean and higher-order statistics of the first- and second-order degree sequences and triangle counts, together with the coupling and pinning parameters. Building on these moment expressions, a piecewise-linear reconstruction technique is employed to obtain a triangular approximation of the eigenvalue distribution of the coupling-control matrix. This approximation enables an efficient prediction of pinning synchronization behavior in large-scale higher-order networks without explicitly computing the full eigenvalue spectrum. Numerical simulations on the three representative simplicial complexes, with Lorenz and Chen oscillators as the intrinsic node dynamics, further validate the effectiveness of the proposed method in analyzing pinning synchronization in simplicial complexes.