  The clockwork mechanism for gravity introduces a tower of massive graviton
modes, "clockwork gravitons," with a very compressed mass spectrum, whose
interaction strengths are much stronger than that of massless gravitons. In
this work, we compute the lowest order contributions of the clockwork gravitons
to the anomalous magnetic moment, $g-2$, of muon in the context of extra
dimensional model with a five dimensional Planck mass, $M_5$. We find that the
total contributions are rather insensitive to the detailed model parameters,
and determined mostly by the value of $M_5$. In order to account for the
current muon $g-2$ anomaly, $M_5$ should be around $0.2~{\rm TeV}$, and the
size of the extra dimension has to be quite large, $l_5 \gtrsim 10^{-7}\,$m.
For $M_5\gtrsim1~{\rm TeV}$, the clockwork graviton contributions are too small
to explain the current muon $g-2$ anomaly. We also compare the clockwork
graviton contributions with other extra dimension models such as
Randall-Sundrum models or large extra dimension models. We find that the
leading contributions in the small curvature limit are universal, but the
cutoff-independent subleading contributions vary for different background
geometries and the clockwork geometry gives the smallest subleading
contributions.
