Dominant Tree

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 \bibdata{Bibliography}
 \bibcite{phillips_graph-based_1998}{1}
 \bibcite{schneier_modeling_1999}{2}
@@ -76,10 +76,11 @@
 \bibcite{PMID:30064421}{13}
 \bibcite{Katz}{14}
 \bibcite{K_Path_Edge}{15}
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 \bibcite{PageRank}{16}
 \bibcite{Adapted_PageRank}{17}
 \bibcite{li_combining_2019}{18}
 \bibcite{zeng_cyber_2017}{19}
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Report/Schrick-Noah_CS-7863_Final-Report.bbl

@@ -91,4 +91,10 @@ M.~Li, P.~Hawrylak, and J.~Hale, ``Combining {OpenCL} and {MPI} to support
 K.~Zeng, ``Cyber {Attack} {Analysis} {Based} on {Markov} {Process} {Model},''
   2017.
 
+\bibitem{dominance}
+R.~T. Prosser, ``Applications of boolean matrices to the analysis of flow
+  diagrams,'' in {\em Papers Presented at the December 1-3, 1959, Eastern Joint
+  IRE-AIEE-ACM Computer Conference}, IRE-AIEE-ACM '59 (Eastern), (New York, NY,
+  USA), p.~133–138, Association for Computing Machinery, 1959.
+
 \end{thebibliography}

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Report/Schrick-Noah_CS-7863_Final-Report.tex

@@ -138,15 +138,25 @@ Section \ref{sec:networks}, and this transitive closure was then analyzed throug
 
 \section{Dominant Tree}
 \subsection{Introduction and Application}
-Dominance, as initially introduced by the author of \cite{dominance} in terms of flow, is defined as a node that is in every path to another node. For instance, if a node \textit{i} is a destination node, and every path to \textit{i} from a source node includes node \textit{j}, then node \textit{j} is said to dominate node \textit{i}. Figure \ref{fig:pre-DTree} \footnote{Image origin can be located at: https://commons.wikimedia.org/wiki/File:Dominator$\_$control$\_$flow$\_$graph.svg, and this image has been released into the public domain for use for any purpose, unless such conditions are required by law.} displays an example starting network. With node 1 being the source node, it is evident that node 2 immediately dominates nodes 3, 4, 5, and 6, since all messages from node 1 must pass through node 2. By definition, each node must also dominate itself, so node 2 also dominates node 2. 
 
-\begin{figure}[htp]
+Dominance, as initially introduced by the author of \cite{dominance} in terms of flow, is defined as a node that is in every path to another node. For instance, if a node \textit{i} is a destination node, and every path to \textit{i} from a source node includes node \textit{j}, then node \textit{j} is said to dominate node \textit{i}. Figure 2 displays an example starting network. With node 1 being the source node, it is evident that node 2 immediately dominates nodes 3, 4, 5, and 6, since all messages from node 1 must pass through node 2. By definition, each node must also dominate itself, so node 2 also dominates node 2.
+
+Following the properties of dominance, a dominator tree can be derived. In a dominator tree, each node has children that it immediately dominates. Immediate dominance is referred to nodes that strictly dominate a given node, but do not strictly dominate any other node that may strictly dominate a node. Figure 3 displays the dominant tree of the network seen in Figure 2.
+
+\begin{figure}[htp] \label{fig:preDtree}
     \includegraphics[width=\linewidth]{"./images/pre-Dtree.png"}
     \vspace{.2truein} \centerline{}
-        \caption{Example Network for Illustrating Dominance}
+        \caption[]{Example Network for Illustrating Dominance \footnote{Image origin can be located at: https://commons.wikimedia.org/wiki/File:Dominator$\_$control$\_$flow$\_$graph.svg, and this image has been released into the public domain for use for any purpose, unless such conditions are required by law.}}
-        \label{fig:TC}
 \end{figure}
 
+\begin{figure}[htp] 
+    \includegraphics[width=\linewidth]{"./images/post-Dtree.png"}
+    \vspace{.2truein} \centerline{}
+        \caption[]{Dominant Tree Derived from the Network Displayed in Figure 2 \footnote{Image origin can be located at: https://commons.wikimedia.org/wiki/File:Dominator$\_$tree.svg, and this image has been released into the public domain for use for any purpose, unless such conditions are required by law.}}
+        \label{fig:post-Dtree}
+\end{figure}
+
+Dominant trees do alter the structure of compliance graphs, and leads to leaf nodes and branches that do not exist in the original network. As a result, some nodes that have directed edges to other nodes may be moved to a position where the edge no longer points to the original nodes. However, in dominant trees, all node parents dominate their children. In this format, the information flow is guided predominantly by the upstream nodes, and all parents in the dominant tree exist as upstream nodes in the original compliance graph. While some downstream nodes may be altered, the importance of nodes can be reexamined in the dominant tree to see how importance differs when information flow is refined. To this end, dominant trees were identified for all networks described in Section \ref{sec:networks}, and these dominant trees were then analyzed through the five centrality methods discussed in Section \ref{sec:centralities}. Results and a discussion of the results can be seen in Section \ref{sec:results}.
 
 \section{Results and Result Analysis} \label{sec:results}
 

Report/Schrick-Noah_CS-7863_Final-Report.toc

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-\contentsline {section}{\numberline {7}Results and Result Analysis}{9}{}%
+\contentsline {section}{\numberline {7}Results and Result Analysis}{12}{}%
-\contentsline {section}{\numberline {8}Conclusions and Future Work}{9}{}%
+\contentsline {section}{\numberline {8}Conclusions and Future Work}{12}{}%
-\contentsline {section}{Bibliography}{11}{}%
+\contentsline {section}{Bibliography}{13}{}%