Spectral projectors of temporally integrated functional brain networks

Abstract

Functional-connectivity matrices mix pairwise association strength, spectral weighting, and eigenspace orientation, making it difficult to determine which components carry stable information about network organization. We develop a spectral-geometric framework that uses rank-k orthogonal projector matrices as subject-level representations of functional operators. The projector preserves eigenspace orientation in measurement coordinates while removing relative eigen-value weighting. We further compare two noncommuting temporal constructions--projecting a temporally integrated operator and averaging projectors estimated from local operators-- and examine how the resulting representations depend on temporal scale and retained rank. We demonstrate the framework in resting-state fNIRS recordings from 99 four-month-old infants exposed to monolingual or bilingual language environments. Pearson correlation favored temporal integration before projection and exhibited a localized intermediate-rank regime at shorter scales, whereas multitaper coherence favored projection before integration and showed a broader regime at longer scales. Reverse engineering of the Pearson representation showed that the intermediate-rank regime was carried primarily by angular relations among channel embeddings, whereas a secondary near-full-rank regime corresponded to a distinct bottom-of-spectrum channel-occupancy carrier. A fully nested two-rank analysis repeatedly retained both regimes. These results show that temporally integrated functional-network organization depends jointly on operator definition, temporal support, projection order, and retained rank. The framework provides an interpretable way to separate complementary spectral carriers of network structure, with group labels used only for cross-fitted validation rather than for representation construction. Key PointsO_LIRank-k spectral projectors provide an interpretable, basis-invariant representation of functional-network eigenspace geometry in the original measurement coordinates. C_LIO_LITemporal integration and spectral projection are noncommuting operations, and their relative informativeness depends on the functional operator and temporal scale. C_LIO_LIDistinct retained-rank regimes reveal complementary geometric carriers of functional-network organization, including relational dominant-subspace structure and bottom-of-spectrum channel occupancy. C_LI