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Ana-Maria Constantin added clearpage_section_textbf_Chapter_5__.tex
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\section{\textbf{Chapter 5)\\{Discussion}}}
\subsection{\textit{Minimum Spanning Trees - Diagnosis}}
We analyze the images obtained in Chapter 4) and perform tree diagnostics on the MSTs corresponding to the 10 found filaments by \cite{Zucker_2015}. By analyzing these MSTs and deciding what are the parameters which describe them and which set them apart from other MSTs in the image, we want to be able to automatically distinguish minimum spanning trees of CO molecular clouds which match the location of bones of the galaxy. Ultimately, we have the goal of performing tree diagnostics on the entire minimum spanning forest that results from running Prim’s algorithm on the Peretto & Fuller catalog, such that we identify potential areas of interest for the exploration of additional bones of the Milky Way. Once we identify potential regions of interest in the catalog and additional bone candidates, we can proceed with performing data analytics in these areas by using previous methods as well, such as the visual method in l-b-v space suggested by \cite{Zucker_2015}. The MST approach is thus valuable since it can give us a better intuition with regards to locations on the sky where we are most likely to find additional bones, and eventually map the entire skeleton of the galaxy.
For purposes of performing tree diagnostics, we suggest using the parameters below in order to describe the shape of a given tree and its likelihood for identifying a bone of the galaxy. We motivate choosing these parameters not only based on common properties of already identified filaments, but also based on the study of social networking graphs and of community detection algorithms.
\begin{itemize}
\item \textbf{Number of nodes} – total number of vertices in a tree, or in our case total number of clouds connected inside a filament
\item \textbf{Tree Size} – total number of edges in a tree, always equal to N-1, where N is the total number of nodes
\item \textbf{Tree Total Length} – sum of weights of all edges inside a given tree
\item \textbf{Longest Path Between Two Vertices} – sum of weights along the longest path connecting two nodes inside a given tree
\item \textbf{Average Degree} – average connectivity degree of nodes inside a tree. In graph theory, the degree of a node is the total number of edges incident on that node.
\item \textbf{Average Clustering Coefficient} – measuring the degree to which nodes inside a graph are clustering together. Always zero for the case of a spanning tree (inside which there are no cycles)
\item \textbf{Density of Tree} – the density of undirected graphs is calculated using the formula
ρ= 2m/(n ( n-1) ), where m is the number of edges, and n is the number of nodes.
\item \textbf{Length on x-axis }– spanning degree in longitude, calculated using∆x=x_(max )- x_min, where x is the longitude coordinate of a given node inside the tree.
\item \textbf{Length on y-axis} – spanning degree in latitude, calculated using ∆y=y_(max )- y_min, where y is the latitude coordinate of a given node inside the tree.
\item \textbf{Length Ratio} - ∆x/∆y, showing how features are more elongated on the x-axis than on the y-axis
\item \textbf{Average Inclination Angle }– average value of the slope calculated for each tree edge
\end{itemize}
