  (abridged) The large-scale distribution of galaxies is generally analyzed
using the two-point correlation function. However, this statistic does not
capture the topology of the distribution, and it is necessary to resort to
higher order correlations to break degeneracies. We demonstrate that an
alternate approach using network analysis can discriminate between
topologically different distributions that have similar two-point correlations.
We investigate two galaxy point distributions, one produced by a cosmological
simulation and the other by a L\'evy walk. For the cosmological simulation, we
adopt the redshift $z = 0.58$ slice from Illustris (Vogelsberger et al. 2014A)
and select galaxies with stellar masses greater than $10^8$$M_\odot$. The two
point correlation function of these simulated galaxies follows a single
power-law, $\xi(r) \sim r^{-1.5}$. Then, we generate L\'evy walks matching the
correlation function and abundance with the simulated galaxies. We find that,
while the two simulated galaxy point distributions have the same abundance and
two point correlation function, their spatial distributions are very different;
most prominently, \emph{filamentary structures}, absent in L\'evy fractals. To
quantify these missing topologies, we adopt network analysis tools and measure
diameter, giant component, and transitivity from networks built by a
conventional friends-of-friends recipe with various linking lengths. Unlike the
abundance and two point correlation function, these network quantities reveal a
clear separation between the two simulated distributions; therefore, the galaxy
distribution simulated by Illustris is not a L\'evy fractal quantitatively. We
find that the described network quantities offer an efficient tool for
discriminating topologies and for comparing observed and theoretical
distributions.
