  Understanding the geodesic properties of fractal aggregates is essential, as
their thermal and mechanical properties are characterized by their geodesics.
In this study, we investigate the root mean square (RMS) of the geodesic
between two constituent particles within fractal aggregates prepared by
ballistic cluster-cluster aggregation (BCCA), diffusion-limited aggregation
(DLA), and growing self-avoiding walk (GSAW) processes in two- and
three-dimensional spaces. We find that the dependence of the RMS of the
geodesic on the number of constituent particles is given by the following
equation: $N \approx k_{\rm g} {D_{\rm RMS}}^{d_{\rm g}}$, where $N$ is the
number of constituent particles and $D_{\rm RMS}$ is the RMS of the geodesic.
We numerically obtain the prefactor $k_{\rm g}$ and exponent $d_{\rm g}$ for
these fractal aggregates. We name the exponent ``the geodesic dimension'', and
it is compared with the fractal dimension. Our findings show that the
difference between fractal and geodesic dimensions varies significantly
depending on the preparation procedure for fractals.
