  We propose Hilbert transform (HT) and analytic signal (AS) construction for
signals over graphs. This is motivated by the popularity of HT, AS, and
modulation analysis in conventional signal processing, and the observation that
complementary insight is often obtained by viewing conventional signals in the
graph setting. Our definitions of HT and AS use a conjugate-symmetry-like
property exhibited by the graph Fourier transform (GFT). We show that a real
graph signal (GS) can be represented using smaller number of GFT coefficients
than the signal length. We show that the graph HT (GHT) and graph AS (GAS)
operations are linear and shift-invariant over graphs. Using the GAS, we define
the amplitude, phase, and frequency modulations for a graph signal (GS).
Further, we use convex optimization to develop an alternative definition of
envelope for a GS. We illustrate the proposed concepts by showing applications
to synthesized and real-world signals. For example, we show that the GHT is
suitable for anomaly detection/analysis over networks and that GAS reveals
complementary information in speech signals.
