Abstract: We propose the expected graph Hilbert transform (EGHT), a robust extension of the Hilbert transform, for graph signals under topological uncertainty. By modeling the shift operator as a random variable and using its expected value as a stable spectral reference, the EGHT applies a π/2 phase shift to non-direct current (DC) graph Fourier modes, enabling the construction of analytic signals on uncertain graphs. The definitions of EGHT for undirected and directed graph ensembles are given, along with key properties such as linearity, anti-involution, and orthogonality; moreover, the expected graph analytic signal (EGAS) for node-wise modulation analysis is introduced. Experiments show that the EGHT outperforms conventional graph Hilbert transforms (GHTs) on noisy realizations, recovering coherent oscillations, producing smooth phase fields, and resisting topological perturbations. As such, it enables reliable time–frequency analysis in stochastic networks.

面向不确定图拓扑时频分析的期望图希尔伯特变换

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