\subsection{MASS DISTRIBUTIONS}
To better understand the origins of the GMAs' structures, we studied their mass distributions to find the parameters of the powerlaw fit to the data. The theoretical cumulative mass function, $N(>M)$, representing the number of clouds more massive than a given mass, $M$, is expressed by Eq.~\ref{massdist} where $\beta$ is a normalization constant, $\alpha$ is the index, $M_{\mathrm{max}}$ is the maximum cloud mass, and $N_{\mathrm{max}}$ is the number of clouds near that mass \citep{Rosolowsky_2005}.

\begin{equation}
 N(>M) = \frac{\beta M_{\odot}}{\alpha +1} \left(\frac{M}{M_{\odot}}\right)^\alpha = \frac{N_\mathrm{max}}{M_\mathrm{max}} \left[ \left( \frac{M}{M_\mathrm{max}} \right) ^{\alpha +1}-1 \right] .
\label{massdist}
\end{equation}

The index $\alpha$ is of particular interest as it illustrates how the mass is distributed across the range of masses and suggests what drives fragmentation. For pure gravitational fragmentation, we expect an index of -2, but turbulence and galactic dynamics alter that value.

Created using the python package {\sc powerlaw}, Fig.~\ref{powerlaw} shows the powerlaw and truncated powerlaw fits for each of our populations, where the powerlaw fit has a distribution function in the form of Eq.~\ref{powerlawdist}, and the truncated powerlaw uses an exponential cutoff as in Eq.~\ref{truncpowerlawdist}, with index $\lambda$. Note that we have manually set the minimum mass included in the fits to $10^7~M_{\odot}$ based on the noise in the data sets and the CPROPS algorithm. 

\begin{equation}
p(x) = x^{\alpha}
\label{powerlawdist}
\end{equation}

\begin{equation}
p(x) = x^{\alpha}\mathrm{e}^{-\lambda x}
\label{truncpowerlawdist}
\end{equation}

