Expected graph Hilbert transform for time–frequency analysis of uncertain graph topologies

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.

Keywords

Graph signal processing (GSP) / Hilbert transform / Analytic signal / Probabilistic graphs / Expected shift operator / Uncertain topology

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The Authors. Published by Zhejiang University Press Co., Ltd.