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Schrick-Noah_CG-Network_Theory.aux
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Schrick-Noah_CG-Network_Theory.tex
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@ -96,7 +96,7 @@ The work conducted in this approach utilized three compliance graphs, with their
HIPAA & 2321 & 8063 & 0.150 \\ \hline HIPAA & 2321 & 8063 & 0.150 \\ \hline
PCI DSS & 61 & 163 & 4.381 \\ \hline PCI DSS & 61 & 163 & 4.381 \\ \hline
\end{tabular} \end{tabular}
\caption{Network Structural Properties for the Three Example Networks Used} \caption{Network Structural Properties (Nodes, Edges, and Connectivity Percentage) for the Three Example Networks}
\label{table:networks} \label{table:networks}
\end{table} \end{table}
@ -182,21 +182,21 @@ Following the properties of dominance, a dominator tree can be derived. In a dom
\includegraphics[scale=0.6]{"./images/dom_net_unshaded.png"} \includegraphics[scale=0.6]{"./images/dom_net_unshaded.png"}
\centering \centering
\vspace{.2truein} \centerline{} \vspace{.2truein} \centerline{}
\caption{Example Network for Illustrating Dominance}\label{fig:domNet} \caption{Example ``Base" Network for Illustrating Dominance. \\\hspace{\textwidth}An arbitrary DAG that has not yet undergone transformation.}\label{fig:domNet}
\end{figure} \end{figure}
\begin{figure}[htp] \begin{figure}[htp]
\includegraphics[scale=0.6]{"./images/dom_tree_unshaded.png"} \includegraphics[scale=0.6]{"./images/dom_tree_unshaded.png"}
\centering \centering
\vspace{.2truein} \centerline{} \vspace{.2truein} \centerline{}
\caption{Dominant Tree Derived from the Network Displayed in Figure \ref{fig:domNet}}\label{fig:domTree} \caption{Dominant Tree Derived from the Network Displayed in Figure \ref{fig:domNet}. \\\hspace{\textwidth}Node 1 dominates Node 2, and Node 2 possesses immediate dominance over Nodes 3, 4, 5, and 6.}\label{fig:domTree}
\end{figure} \end{figure}
Dominant trees do alter the structure of compliance graphs, and lead 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}. Dominant trees do alter the structure of compliance graphs, and lead 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} \section{Results and Result Analysis} \label{sec:results}
In this section, only results for the car network are displayed for brevity. These results can be seen in Tables \ref{table:car-deg} through \ref{table:car-betweenness}. The results for the HIPAA and PCI DSS networks can be found in the Supplementary Material included with this work. In this section, only results for the automobile network are displayed for brevity. These results can be seen in Figures \ref{fig:car-deg} through \ref{fig:car-betweenness}. The tabled results for the HIPAA and PCI DSS networks can be found in the Supplementary Material included with this work. Additional tabled results for the automobile network are also displayed in the Supplementary Material.
\begin{table}[] \begin{table}[]
\centering \centering
@ -231,144 +231,53 @@ In this section, only results for the car network are displayed for brevity. The
\label{table:car-deg} \label{table:car-deg}
\end{table} \end{table}
\begin{table}[] \begin{figure}[htp]
\includegraphics[width=\linewidth]{"./images/car-degree.png"}
\centering \centering
\setlength\tabcolsep{4pt} \vspace{.2truein} \centerline{}
\begin{tabular}{|cc|cc|cc|} \caption{Importance Distribution for Degree Centrality for the Automobile Network. \\\hspace{\textwidth} The ``Base" figure represents the distribution of the original network. \\\hspace{\textwidth} The Transitive Closure and Dominant Tree figures illustrate the distribution difference in node rankings after the base network underwent a transformation.}
\hline \label{fig:car-deg}
\end{figure}
\multicolumn{2}{|c|}{\textbf{Base}} \begin{figure}[htp]
& \multicolumn{2}{|c|}{\textbf{Transitive Closure}} \includegraphics[width=\linewidth]{"./images/car-katz.png"}
& \multicolumn{2}{|c|}{\textbf{Dominant Tree}}
\\ \hline
\multicolumn{1}{|c|}{\textbf{Node}} & \textbf{\% Value} & \multicolumn{1}{c|}{\textbf{Node}} & \textbf{\% Value} & \multicolumn{1}{c|}{\textbf{Node}} & \textbf{\% Value} \\ \hline
\multicolumn{1}{|c|}{314} & 0.13 & \multicolumn{1}{c|}{0} & 9.09 & \multicolumn{1}{c|}{1} & 4.88 \\ \hline
\multicolumn{1}{|c|}{377} & 0.10 & \multicolumn{1}{c|}{1} & 8.26 & \multicolumn{1}{c|}{3} & 3.47 \\ \hline
\multicolumn{1}{|c|}{346} & 0.10 & \multicolumn{1}{c|}{3} & 7.39 & \multicolumn{1}{c|}{7} & 0.60 \\ \hline
\multicolumn{1}{|c|}{376} & 0.10 & \multicolumn{1}{c|}{7} & 6.34 & \multicolumn{1}{c|}{0} & 0.52 \\ \hline
\multicolumn{1}{|c|}{374} & 0.10 & \multicolumn{1}{c|}{15} & 5.27 & \multicolumn{1}{c|}{42} & 0.45 \\ \hline
\multicolumn{1}{|c|}{378} & 0.10 & \multicolumn{1}{c|}{27} & 4.37 & \multicolumn{1}{c|}{314} & 0.15 \\ \hline
\multicolumn{1}{|c|}{380} & 0.10 & \multicolumn{1}{c|}{42} & 3.61 & \multicolumn{1}{c|}{27} & 0.08 \\ \hline
\multicolumn{1}{|c|}{381} & 0.10 & \multicolumn{1}{c|}{60} & 2.71 & \multicolumn{1}{c|}{250} & 0.05 \\ \hline
\multicolumn{1}{|c|}{382} & 0.10 & \multicolumn{1}{c|}{87} & 2.02 & \multicolumn{1}{c|}{15} & 0.04 \\ \hline
\multicolumn{1}{|c|}{262} & 0.10 & \multicolumn{1}{c|}{130} & 1.48 & \multicolumn{1}{c|}{187} & 0.04 \\ \hline
\multicolumn{1}{|c|}{370} & 0.10 & \multicolumn{1}{c|}{2} & 1.31 & \multicolumn{1}{c|}{130} & 0.04 \\ \hline
\multicolumn{1}{|c|}{379} & 0.10 & \multicolumn{1}{c|}{4} & 1.19 & \multicolumn{1}{c|}{87} & 0.04 \\ \hline
\multicolumn{1}{|c|}{418} & 0.08 & \multicolumn{1}{c|}{5} & 1.19 & \multicolumn{1}{c|}{60} & 0.04 \\ \hline
\multicolumn{1}{|c|}{459} & 0.08 & \multicolumn{1}{c|}{6} & 1.19 & \multicolumn{1}{c|}{2} & 0.04 \\ \hline
\multicolumn{1}{|c|}{467} & 0.08 & \multicolumn{1}{c|}{187} & 1.08 & \multicolumn{1}{c|}{4} & 0.04 \\ \hline
\end{tabular}
\caption{Top 15 Nodes with Katz Centrality for the Automobile Network. \\\hspace{\textwidth} Base columns represent the top ranked nodes and their percentage of importance held relative to the entire network. \\\hspace{\textwidth}Transitive Closure and Dominant Tree columns illustrate the difference in node rankings after the base network underwent a transformation. \\\hspace{\textwidth}}
\label{table:car-katz}
\end{table}
\begin{table}[]
\centering \centering
\setlength\tabcolsep{4pt} \vspace{.2truein} \centerline{}
\begin{tabular}{|cc|cc|cc|} \caption{Importance Distribution for Katz Centrality for the Automobile Network. \\\hspace{\textwidth} The ``Base" figure represents the distribution of the original network. \\\hspace{\textwidth} The Transitive Closure and Dominant Tree figures illustrate the distribution difference in node rankings after the base network underwent a transformation.}
\hline \label{fig:car-katz}
\end{figure}
\multicolumn{2}{|c|}{\textbf{Base}} \begin{figure}[htp]
& \multicolumn{2}{|c|}{\textbf{Transitive Closure}} \includegraphics[width=\linewidth]{"./images/car-pagerank.png"}
& \multicolumn{2}{|c|}{\textbf{Dominant Tree}}
\\ \hline
\multicolumn{1}{|c|}{\textbf{Node}} & \textbf{\% Value} & \multicolumn{1}{c|}{\textbf{Node}} & \textbf{\% Value} & \multicolumn{1}{c|}{\textbf{Node}} & \textbf{\% Value} \\ \hline
\multicolumn{1}{|c|}{314} & 0.25 & \multicolumn{1}{c|}{0} & 0.65 & \multicolumn{1}{c|}{1} & 37.54 \\ \hline
\multicolumn{1}{|c|}{346} & 0.19 & \multicolumn{1}{c|}{1} & 0.39 & \multicolumn{1}{c|}{0} & 35.05 \\ \hline
\multicolumn{1}{|c|}{362} & 0.19 & \multicolumn{1}{c|}{3} & 0.65 & \multicolumn{1}{c|}{3} & 17.53 \\ \hline
\multicolumn{1}{|c|}{370} & 0.19 & \multicolumn{1}{c|}{7} & 0.65 & \multicolumn{1}{c|}{7} & 2.54 \\ \hline
\multicolumn{1}{|c|}{374} & 0.19 & \multicolumn{1}{c|}{15} & 0.64 & \multicolumn{1}{c|}{15} & 1.88 \\ \hline
\multicolumn{1}{|c|}{376} & 0.19 & \multicolumn{1}{c|}{27} & 0.64 & \multicolumn{1}{c|}{27} & 1.88 \\ \hline
\multicolumn{1}{|c|}{377} & 0.19 & \multicolumn{1}{c|}{42} & 0.64 & \multicolumn{1}{c|}{42} & 1.88 \\ \hline
\multicolumn{1}{|c|}{378} & 0.19 & \multicolumn{1}{c|}{60} & 0.62 & \multicolumn{1}{c|}{187} & 0.53 \\ \hline
\multicolumn{1}{|c|}{379} & 0.19 & \multicolumn{1}{c|}{87} & 0.60 & \multicolumn{1}{c|}{250} & 0.51 \\ \hline
\multicolumn{1}{|c|}{380} & 0.19 & \multicolumn{1}{c|}{130} & 0.59 & \multicolumn{1}{c|}{314} & 0.50 \\ \hline
\multicolumn{1}{|c|}{381} & 0.19 & \multicolumn{1}{c|}{187} & 0.57 & \multicolumn{1}{c|}{60} & 0.05 \\ \hline
\multicolumn{1}{|c|}{382} & 0.19 & \multicolumn{1}{c|}{250} & 0.55 & \multicolumn{1}{c|}{86} & 0.05 \\ \hline
\multicolumn{1}{|c|}{398} & 0.14 & \multicolumn{1}{c|}{314} & 0.54 & \multicolumn{1}{c|}{130} & 0.05 \\ \hline
\multicolumn{1}{|c|}{406} & 0.14 & \multicolumn{1}{c|}{2} & 0.33 & \multicolumn{1}{c|}{2} & 0.00 \\ \hline
\multicolumn{1}{|c|}{410} & 0.14 & \multicolumn{1}{c|}{4} & 0.33 & \multicolumn{1}{c|}{4} & 0.00 \\ \hline
\end{tabular}
\caption{Top 15 Nodes with K-path Edge Centrality for the Automobile Network. \\\hspace{\textwidth} Base columns represent the top ranked nodes and their percentage of importance held relative to the entire network. \\\hspace{\textwidth}Transitive Closure and Dominant Tree columns illustrate the difference in node rankings after the base network underwent a transformation. \\\hspace{\textwidth}}
\label{table:car-kpe}
\end{table}
\begin{table}[]
\centering \centering
\setlength\tabcolsep{4pt} \vspace{.2truein} \centerline{}
\begin{tabular}{|cc|cc|cc|} \caption{Importance Distribution for Adapted PageRank Centrality for the Automobile Network. \\\hspace{\textwidth} The ``Base" figure represents the distribution of the original network. \\\hspace{\textwidth} The Transitive Closure and Dominant Tree figures illustrate the distribution difference in node rankings after the base network underwent a transformation.}
\hline \label{fig:car-pr}
\end{figure}
\multicolumn{2}{|c|}{\textbf{Base}} \begin{figure}[htp]
& \multicolumn{2}{|c|}{\textbf{Transitive Closure}} \includegraphics[width=\linewidth]{"./images/car-kpe.png"}
& \multicolumn{2}{|c|}{\textbf{Dominant Tree}}
\\ \hline
\multicolumn{1}{|c|}{\textbf{Node}} & \textbf{\% Value} & \multicolumn{1}{c|}{\textbf{Node}} & \textbf{\% Value} & \multicolumn{1}{c|}{\textbf{Node}} & \textbf{\% Value} \\ \hline
\multicolumn{1}{|c|}{2490} & 8.27 & \multicolumn{1}{c|}{2490} & 19.92 & \multicolumn{1}{c|}{314} & 0.17 \\ \hline
\multicolumn{1}{|c|}{1004} & 1.06 & \multicolumn{1}{c|}{2479} & 1.58 & \multicolumn{1}{c|}{250} & 0.15 \\ \hline
\multicolumn{1}{|c|}{1467} & 0.97 & \multicolumn{1}{c|}{2480} & 1.58 & \multicolumn{1}{c|}{187} & 0.13 \\ \hline
\multicolumn{1}{|c|}{2479} & 0.95 & \multicolumn{1}{c|}{2481} & 1.58 & \multicolumn{1}{c|}{130} & 0.10 \\ \hline
\multicolumn{1}{|c|}{2480} & 0.95 & \multicolumn{1}{c|}{2482} & 1.58 & \multicolumn{1}{c|}{42} & 0.10 \\ \hline
\multicolumn{1}{|c|}{2481} & 0.95 & \multicolumn{1}{c|}{2483} & 1.58 & \multicolumn{1}{c|}{87} & 0.07 \\ \hline
\multicolumn{1}{|c|}{2482} & 0.95 & \multicolumn{1}{c|}{2484} & 1.4 & \multicolumn{1}{c|}{27} & 0.07 \\ \hline
\multicolumn{1}{|c|}{2483} & 0.95 & \multicolumn{1}{c|}{2485} & 1.4 & \multicolumn{1}{c|}{1} & 0.07 \\ \hline
\multicolumn{1}{|c|}{667} & 0.92 & \multicolumn{1}{c|}{2486} & 1.39 & \multicolumn{1}{c|}{378} & 0.04 \\ \hline
\multicolumn{1}{|c|}{2484} & 0.83 & \multicolumn{1}{c|}{2487} & 1.39 & \multicolumn{1}{c|}{379} & 0.04 \\ \hline
\multicolumn{1}{|c|}{2485} & 0.83 & \multicolumn{1}{c|}{2488} & 1.39 & \multicolumn{1}{c|}{380} & 0.04 \\ \hline
\multicolumn{1}{|c|}{2486} & 0.83 & \multicolumn{1}{c|}{2489} & 1.39 & \multicolumn{1}{c|}{381} & 0.04 \\ \hline
\multicolumn{1}{|c|}{2487} & 0.83 & \multicolumn{1}{c|}{2424} & 0.29 & \multicolumn{1}{c|}{382} & 0.04 \\ \hline
\multicolumn{1}{|c|}{2488} & 0.83 & \multicolumn{1}{c|}{2425} & 0.29 & \multicolumn{1}{c|}{470} & 0.04 \\ \hline
\multicolumn{1}{|c|}{2489} & 0.83 & \multicolumn{1}{c|}{2426} & 0.29 & \multicolumn{1}{c|}{471} & 0.04 \\ \hline
\end{tabular}
\caption{Top 15 Nodes with PageRank Centrality for the Automobile Network. \\\hspace{\textwidth} Base columns represent the top ranked nodes and their percentage of importance held relative to the entire network. \\\hspace{\textwidth}Transitive Closure and Dominant Tree columns illustrate the difference in node rankings after the base network underwent a transformation. \\\hspace{\textwidth}}
\label{table:car-APC}
\end{table}
\begin{table}[]
\centering \centering
\setlength\tabcolsep{4pt} \vspace{.2truein} \centerline{}
\begin{tabular}{|cc|cc|cc|} \caption{Importance Distribution for K-Path Edge Centrality for the Automobile Network. \\\hspace{\textwidth} The ``Base" figure represents the distribution of the original network. \\\hspace{\textwidth} The Transitive Closure and Dominant Tree figures illustrate the distribution difference in node rankings after the base network underwent a transformation.}
\hline \label{fig:car-kpe}
\end{figure}
\multicolumn{2}{|c|}{\textbf{Base}} \begin{figure}[htp]
& \multicolumn{2}{|c|}{\textbf{Transitive Closure}} \includegraphics[width=\linewidth]{"./images/car-betweenness.png"}
& \multicolumn{2}{|c|}{\textbf{Dominant Tree}} \centering
\\ \hline \vspace{.2truein} \centerline{}
\caption{Importance Distribution for Betweenness Centrality for the Automobile Network. \\\hspace{\textwidth} The ``Base" figure represents the distribution of the original network. \\\hspace{\textwidth} The Transitive Closure and Dominant Tree figures illustrate the distribution difference in node rankings after the base network underwent a transformation.}
\label{fig:car-betweenness}
\end{figure}
\multicolumn{1}{|c|}{\textbf{Node}} & \textbf{\% Value} & \multicolumn{1}{c|}{\textbf{Node}} & \textbf{\% Value} & \multicolumn{1}{c|}{\textbf{Node}} & \textbf{\% Value} \\ \hline
\multicolumn{1}{|c|}{42} & 0.43 & \multicolumn{1}{c|}{0} & 0 & \multicolumn{1}{c|}{1} & 24.71 \\ \hline
\multicolumn{1}{|c|}{27} & 0.40 & \multicolumn{1}{c|}{1} & 0 & \multicolumn{1}{c|}{3} & 24.68 \\ \hline
\multicolumn{1}{|c|}{60} & 0.39 & \multicolumn{1}{c|}{2} & 0 & \multicolumn{1}{c|}{7} & 9.20 \\ \hline
\multicolumn{1}{|c|}{87} & 0.36 & \multicolumn{1}{c|}{3} & 0 & \multicolumn{1}{c|}{42} & 9.00 \\ \hline
\multicolumn{1}{|c|}{15} & 0.36 & \multicolumn{1}{c|}{4} & 0 & \multicolumn{1}{c|}{27} & 7.55 \\ \hline
\multicolumn{1}{|c|}{130} & 0.33 & \multicolumn{1}{c|}{5} & 0 & \multicolumn{1}{c|}{15} & 6.08 \\ \hline
\multicolumn{1}{|c|}{7} & 0.31 & \multicolumn{1}{c|}{6} & 0 & \multicolumn{1}{c|}{314} & 3.69 \\ \hline
\multicolumn{1}{|c|}{187} & 0.29 & \multicolumn{1}{c|}{7} & 0 & \multicolumn{1}{c|}{250} & 3.49 \\ \hline
\multicolumn{1}{|c|}{50} & 0.28 & \multicolumn{1}{c|}{8} & 0 & \multicolumn{1}{c|}{187} & 3.28 \\ \hline
\multicolumn{1}{|c|}{70} & 0.28 & \multicolumn{1}{c|}{9} & 0 & \multicolumn{1}{c|}{130} & 3.04 \\ \hline
\multicolumn{1}{|c|}{104} & 0.27 & \multicolumn{1}{c|}{10} & 0 & \multicolumn{1}{c|}{87} & 2.78 \\ \hline
\multicolumn{1}{|c|}{156} & 0.26 & \multicolumn{1}{c|}{11} & 0 & \multicolumn{1}{c|}{60} & 2.50 \\ \hline
\multicolumn{1}{|c|}{1467} & 0.25 & \multicolumn{1}{c|}{12} & 0 & \multicolumn{1}{c|}{0} & 0.00 \\ \hline
\multicolumn{1}{|c|}{250} & 0.25 & \multicolumn{1}{c|}{13} & 0 & \multicolumn{1}{c|}{2} & 0.00 \\ \hline
\multicolumn{1}{|c|}{115} & 0.25 & \multicolumn{1}{c|}{14} & 0 & \multicolumn{1}{c|}{4} & 0.00 \\ \hline
\end{tabular}
\caption{Top 15 Nodes with Betweenness Centrality for the Automobile Network. \\\hspace{\textwidth} Base columns represent the top ranked nodes and their percentage of importance held relative to the entire network. \\\hspace{\textwidth}Transitive Closure and Dominant Tree columns illustrate the difference in node rankings after the base network underwent a transformation. \\\hspace{\textwidth}}
\label{table:car-betweenness}
\end{table}
\subsection{Result Analysis} \subsection{Result Analysis}
When viewing the results of the car networks, unsurprisingly, each centrality method ranks nodes in a different order. These differences in rankings can be used based on additional metrics, such as severity, cost, or disturbance of systems, to identify correction schemes best suited for a given environment. However, degree centrality and K-path edge centrality rankings for the top 15 were identical for the car network. This also extends to the HIPAA network, as seen in the included results in the Supplementary Material, but does not extend to the PCI DSS network. The value for \textit{k} in K-path edge centrality was set to 3. With a relatively small \textit{k} value in comparison to the overall size of the car and HIPAA networks, coupled with the high degree count of the top 15 nodes ranked with degree centrality, it is likely that the high degree count correlates to the K-path edge centrality scoring. This reasoning extends to the PCI DSS network, where the network is substantially smaller and there is a greater percentage of connectivity. When viewing the results of the car networks, unsurprisingly, each centrality method ranks nodes in a different order. These differences in rankings can be used based on additional metrics, such as severity, cost, or disturbance of systems, to identify correction schemes best suited for a given environment. However, degree centrality and K-path edge centrality rankings for the top 15 were identical for the car network. This also extends to the HIPAA network, as seen in the included results in the Supplementary Material, but does not extend to the PCI DSS network. The value for \textit{k} in K-path edge centrality was set to 3. With a relatively small \textit{k} value in comparison to the overall size of the car and HIPAA networks, coupled with the high degree count of the top 15 nodes ranked with degree centrality, it is likely that the high degree count correlates to the K-path edge centrality scoring. This reasoning extends to the PCI DSS network, where the network is substantially smaller and there is a greater percentage of connectivity.
Comparing the transitive closure format of compliance graphs, the associated centrality rankings greatly vary from their original compliance graph rankings. As expected however, the root or a leaf node has the highest centrality value. Since the root node can reach all nodes, and a leaf node can be reached by all nodes, these two nodes are expectedly ranked high. What is unexpected, however, is that the top 15 rankings are not comprised of the most upstream 15 nodes or the 15 most downstream nodes. While rankings do tend to be higher for more upstream for K-path edge, Katz, and degree centralities, node IDs in the 100s, 200s, and 300s (``midstream" nodes) all make appearances. Betweenness centrality for the transitive closure representation yielded no valuable insight, since shortest paths to a node from any given node is reachable in 1 step. Comparing the transitive closure format of compliance graphs, the associated centrality rankings greatly vary from their original compliance graph rankings. As expected, the root or a leaf node has the highest centrality value. Since the root node can reach all nodes, and a leaf node can be reached by all nodes, these two nodes are expectedly ranked high. What is unexpected, however, is that the highest rankings are not comprised of the most upstream nodes or the most downstream nodes. While rankings do tend to be higher for more upstream for K-path edge (Figure \ref{fig:car-kpe}), Katz (Figure \ref{fig:car-katz}), and degree centralities (Figure \ref{fig:car-deg}), node IDs in the 100s, 200s, and 300s (``midstream" nodes) all make appearances in the upper rankings. Betweenness centrality (Figure \ref{fig:car-betweenness}) for the transitive closure representation yielded no valuable insight, since shortest paths to a node from any given node is reachable in 1 step.
For the dominant tree representation, it was initially hypothesized that nodes ranked highly in the original compliance graph's betweenness centrality or Katz centrality measures would closely relate to the dominant tree results. However, the dominant tree rankings also vary greatly from the original compliance graph's rankings. Even nodes that saw no appearances in the top 15 of the base compliance graph or transitive closure representation made appearances in the dominant tree results. Since the dominant tree format does favor the upstream nodes due to a lesser reordering effect caused by dominance, the PageRank ordering were not predominantly downstream nodes, but mostly node IDs in the 300s. For the dominant tree representation, it was initially hypothesized that nodes ranked highly in the original compliance graph's betweenness centrality or Katz centrality measures would closely relate to the dominant tree results. However, the dominant tree rankings also vary greatly from the original compliance graph's rankings. Even nodes that saw no appearances in the top rankings of the base compliance graph or transitive closure representation made appearances in the dominant tree results. Since the dominant tree format does favor the upstream nodes due to a lesser reordering effect caused by dominance, the PageRank ordering (Figure \ref{fig:car-pr}) were not predominantly downstream nodes, but mostly node IDs in the 300s, with all nodes having near identical importance. Apart from this PageRank ordering, the dominant tree representation yielded results where almost all importance was cumulated into a few nodes. As opposed to the base network and transitive closure networks which had many nodes with noticeable importance, the dominant tree representation had only few nodes with importance. This is due to the structure change from dominance computations, and this is validated by the upstream nodes having the importance in the transformed network, with the root node possessing the maximum importance value.
\section{Conclusions and Future Work} \section{Conclusions and Future Work}
\subsection{Conclusions} \subsection{Conclusions}

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**Schrick-Noah_Supplemental.tex **Schrick-Noah_Supplemental
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** Conference Paper ** ** Conference Paper **
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@ -40,10 +40,153 @@
Tulsa, USA \\ Tulsa, USA \\
peter-hawrylak@utulsa.edu peter-hawrylak@utulsa.edu
} }
\and
\IEEEauthorblockN{Brett A. McKinney}
\IEEEauthorblockA{
\textit{Tandy School of Computer Science} \\
\textit{The University of Tulsa}\\
Tulsa, USA \\
brett-mckinney@utulsa.edu
}
} }
\maketitle \maketitle
\begin{table}[]
\centering
\setlength\tabcolsep{4pt}
\begin{tabular}{|cc|cc|cc|}
\hline
\multicolumn{2}{|c|}{\textbf{Base}}
& \multicolumn{2}{|c|}{\textbf{Transitive Closure}}
& \multicolumn{2}{|c|}{\textbf{Dominant Tree}}
\\ \hline
\multicolumn{1}{|c|}{\textbf{Node}} & \textbf{\% Value} & \multicolumn{1}{c|}{\textbf{Node}} & \textbf{\% Value} & \multicolumn{1}{c|}{\textbf{Node}} & \textbf{\% Value} \\ \hline
\multicolumn{1}{|c|}{314} & 0.13 & \multicolumn{1}{c|}{0} & 9.09 & \multicolumn{1}{c|}{1} & 4.88 \\ \hline
\multicolumn{1}{|c|}{377} & 0.10 & \multicolumn{1}{c|}{1} & 8.26 & \multicolumn{1}{c|}{3} & 3.47 \\ \hline
\multicolumn{1}{|c|}{346} & 0.10 & \multicolumn{1}{c|}{3} & 7.39 & \multicolumn{1}{c|}{7} & 0.60 \\ \hline
\multicolumn{1}{|c|}{376} & 0.10 & \multicolumn{1}{c|}{7} & 6.34 & \multicolumn{1}{c|}{0} & 0.52 \\ \hline
\multicolumn{1}{|c|}{374} & 0.10 & \multicolumn{1}{c|}{15} & 5.27 & \multicolumn{1}{c|}{42} & 0.45 \\ \hline
\multicolumn{1}{|c|}{378} & 0.10 & \multicolumn{1}{c|}{27} & 4.37 & \multicolumn{1}{c|}{314} & 0.15 \\ \hline
\multicolumn{1}{|c|}{380} & 0.10 & \multicolumn{1}{c|}{42} & 3.61 & \multicolumn{1}{c|}{27} & 0.08 \\ \hline
\multicolumn{1}{|c|}{381} & 0.10 & \multicolumn{1}{c|}{60} & 2.71 & \multicolumn{1}{c|}{250} & 0.05 \\ \hline
\multicolumn{1}{|c|}{382} & 0.10 & \multicolumn{1}{c|}{87} & 2.02 & \multicolumn{1}{c|}{15} & 0.04 \\ \hline
\multicolumn{1}{|c|}{262} & 0.10 & \multicolumn{1}{c|}{130} & 1.48 & \multicolumn{1}{c|}{187} & 0.04 \\ \hline
\multicolumn{1}{|c|}{370} & 0.10 & \multicolumn{1}{c|}{2} & 1.31 & \multicolumn{1}{c|}{130} & 0.04 \\ \hline
\multicolumn{1}{|c|}{379} & 0.10 & \multicolumn{1}{c|}{4} & 1.19 & \multicolumn{1}{c|}{87} & 0.04 \\ \hline
\multicolumn{1}{|c|}{418} & 0.08 & \multicolumn{1}{c|}{5} & 1.19 & \multicolumn{1}{c|}{60} & 0.04 \\ \hline
\multicolumn{1}{|c|}{459} & 0.08 & \multicolumn{1}{c|}{6} & 1.19 & \multicolumn{1}{c|}{2} & 0.04 \\ \hline
\multicolumn{1}{|c|}{467} & 0.08 & \multicolumn{1}{c|}{187} & 1.08 & \multicolumn{1}{c|}{4} & 0.04 \\ \hline
\end{tabular}
\caption{Top 15 Nodes with Katz Centrality for the Automobile Network. \\\hspace{\textwidth} Base columns represent the top ranked nodes and their percentage of importance held relative to the entire network. \\\hspace{\textwidth}Transitive Closure and Dominant Tree columns illustrate the difference in node rankings after the base network underwent a transformation. \\\hspace{\textwidth}}
\label{table:car-katz}
\end{table}
\begin{table}[]
\centering
\setlength\tabcolsep{4pt}
\begin{tabular}{|cc|cc|cc|}
\hline
\multicolumn{2}{|c|}{\textbf{Base}}
& \multicolumn{2}{|c|}{\textbf{Transitive Closure}}
& \multicolumn{2}{|c|}{\textbf{Dominant Tree}}
\\ \hline
\multicolumn{1}{|c|}{\textbf{Node}} & \textbf{\% Value} & \multicolumn{1}{c|}{\textbf{Node}} & \textbf{\% Value} & \multicolumn{1}{c|}{\textbf{Node}} & \textbf{\% Value} \\ \hline
\multicolumn{1}{|c|}{314} & 0.25 & \multicolumn{1}{c|}{0} & 0.65 & \multicolumn{1}{c|}{1} & 37.54 \\ \hline
\multicolumn{1}{|c|}{346} & 0.19 & \multicolumn{1}{c|}{1} & 0.39 & \multicolumn{1}{c|}{0} & 35.05 \\ \hline
\multicolumn{1}{|c|}{362} & 0.19 & \multicolumn{1}{c|}{3} & 0.65 & \multicolumn{1}{c|}{3} & 17.53 \\ \hline
\multicolumn{1}{|c|}{370} & 0.19 & \multicolumn{1}{c|}{7} & 0.65 & \multicolumn{1}{c|}{7} & 2.54 \\ \hline
\multicolumn{1}{|c|}{374} & 0.19 & \multicolumn{1}{c|}{15} & 0.64 & \multicolumn{1}{c|}{15} & 1.88 \\ \hline
\multicolumn{1}{|c|}{376} & 0.19 & \multicolumn{1}{c|}{27} & 0.64 & \multicolumn{1}{c|}{27} & 1.88 \\ \hline
\multicolumn{1}{|c|}{377} & 0.19 & \multicolumn{1}{c|}{42} & 0.64 & \multicolumn{1}{c|}{42} & 1.88 \\ \hline
\multicolumn{1}{|c|}{378} & 0.19 & \multicolumn{1}{c|}{60} & 0.62 & \multicolumn{1}{c|}{187} & 0.53 \\ \hline
\multicolumn{1}{|c|}{379} & 0.19 & \multicolumn{1}{c|}{87} & 0.60 & \multicolumn{1}{c|}{250} & 0.51 \\ \hline
\multicolumn{1}{|c|}{380} & 0.19 & \multicolumn{1}{c|}{130} & 0.59 & \multicolumn{1}{c|}{314} & 0.50 \\ \hline
\multicolumn{1}{|c|}{381} & 0.19 & \multicolumn{1}{c|}{187} & 0.57 & \multicolumn{1}{c|}{60} & 0.05 \\ \hline
\multicolumn{1}{|c|}{382} & 0.19 & \multicolumn{1}{c|}{250} & 0.55 & \multicolumn{1}{c|}{86} & 0.05 \\ \hline
\multicolumn{1}{|c|}{398} & 0.14 & \multicolumn{1}{c|}{314} & 0.54 & \multicolumn{1}{c|}{130} & 0.05 \\ \hline
\multicolumn{1}{|c|}{406} & 0.14 & \multicolumn{1}{c|}{2} & 0.33 & \multicolumn{1}{c|}{2} & 0.00 \\ \hline
\multicolumn{1}{|c|}{410} & 0.14 & \multicolumn{1}{c|}{4} & 0.33 & \multicolumn{1}{c|}{4} & 0.00 \\ \hline
\end{tabular}
\caption{Top 15 Nodes with K-path Edge Centrality for the Automobile Network. \\\hspace{\textwidth} Base columns represent the top ranked nodes and their percentage of importance held relative to the entire network. \\\hspace{\textwidth}Transitive Closure and Dominant Tree columns illustrate the difference in node rankings after the base network underwent a transformation. \\\hspace{\textwidth}}
\label{table:car-kpe}
\end{table}
\begin{table}[]
\centering
\setlength\tabcolsep{4pt}
\begin{tabular}{|cc|cc|cc|}
\hline
\multicolumn{2}{|c|}{\textbf{Base}}
& \multicolumn{2}{|c|}{\textbf{Transitive Closure}}
& \multicolumn{2}{|c|}{\textbf{Dominant Tree}}
\\ \hline
\multicolumn{1}{|c|}{\textbf{Node}} & \textbf{\% Value} & \multicolumn{1}{c|}{\textbf{Node}} & \textbf{\% Value} & \multicolumn{1}{c|}{\textbf{Node}} & \textbf{\% Value} \\ \hline
\multicolumn{1}{|c|}{2490} & 8.27 & \multicolumn{1}{c|}{2490} & 19.92 & \multicolumn{1}{c|}{314} & 0.17 \\ \hline
\multicolumn{1}{|c|}{1004} & 1.06 & \multicolumn{1}{c|}{2479} & 1.58 & \multicolumn{1}{c|}{250} & 0.15 \\ \hline
\multicolumn{1}{|c|}{1467} & 0.97 & \multicolumn{1}{c|}{2480} & 1.58 & \multicolumn{1}{c|}{187} & 0.13 \\ \hline
\multicolumn{1}{|c|}{2479} & 0.95 & \multicolumn{1}{c|}{2481} & 1.58 & \multicolumn{1}{c|}{130} & 0.10 \\ \hline
\multicolumn{1}{|c|}{2480} & 0.95 & \multicolumn{1}{c|}{2482} & 1.58 & \multicolumn{1}{c|}{42} & 0.10 \\ \hline
\multicolumn{1}{|c|}{2481} & 0.95 & \multicolumn{1}{c|}{2483} & 1.58 & \multicolumn{1}{c|}{87} & 0.07 \\ \hline
\multicolumn{1}{|c|}{2482} & 0.95 & \multicolumn{1}{c|}{2484} & 1.4 & \multicolumn{1}{c|}{27} & 0.07 \\ \hline
\multicolumn{1}{|c|}{2483} & 0.95 & \multicolumn{1}{c|}{2485} & 1.4 & \multicolumn{1}{c|}{1} & 0.07 \\ \hline
\multicolumn{1}{|c|}{667} & 0.92 & \multicolumn{1}{c|}{2486} & 1.39 & \multicolumn{1}{c|}{378} & 0.04 \\ \hline
\multicolumn{1}{|c|}{2484} & 0.83 & \multicolumn{1}{c|}{2487} & 1.39 & \multicolumn{1}{c|}{379} & 0.04 \\ \hline
\multicolumn{1}{|c|}{2485} & 0.83 & \multicolumn{1}{c|}{2488} & 1.39 & \multicolumn{1}{c|}{380} & 0.04 \\ \hline
\multicolumn{1}{|c|}{2486} & 0.83 & \multicolumn{1}{c|}{2489} & 1.39 & \multicolumn{1}{c|}{381} & 0.04 \\ \hline
\multicolumn{1}{|c|}{2487} & 0.83 & \multicolumn{1}{c|}{2424} & 0.29 & \multicolumn{1}{c|}{382} & 0.04 \\ \hline
\multicolumn{1}{|c|}{2488} & 0.83 & \multicolumn{1}{c|}{2425} & 0.29 & \multicolumn{1}{c|}{470} & 0.04 \\ \hline
\multicolumn{1}{|c|}{2489} & 0.83 & \multicolumn{1}{c|}{2426} & 0.29 & \multicolumn{1}{c|}{471} & 0.04 \\ \hline
\end{tabular}
\caption{Top 15 Nodes with PageRank Centrality for the Automobile Network. \\\hspace{\textwidth} Base columns represent the top ranked nodes and their percentage of importance held relative to the entire network. \\\hspace{\textwidth}Transitive Closure and Dominant Tree columns illustrate the difference in node rankings after the base network underwent a transformation. \\\hspace{\textwidth}}
\label{table:car-APC}
\end{table}
\begin{table}[]
\centering
\setlength\tabcolsep{4pt}
\begin{tabular}{|cc|cc|cc|}
\hline
\multicolumn{2}{|c|}{\textbf{Base}}
& \multicolumn{2}{|c|}{\textbf{Transitive Closure}}
& \multicolumn{2}{|c|}{\textbf{Dominant Tree}}
\\ \hline
\multicolumn{1}{|c|}{\textbf{Node}} & \textbf{\% Value} & \multicolumn{1}{c|}{\textbf{Node}} & \textbf{\% Value} & \multicolumn{1}{c|}{\textbf{Node}} & \textbf{\% Value} \\ \hline
\multicolumn{1}{|c|}{42} & 0.43 & \multicolumn{1}{c|}{0} & 0 & \multicolumn{1}{c|}{1} & 24.71 \\ \hline
\multicolumn{1}{|c|}{27} & 0.40 & \multicolumn{1}{c|}{1} & 0 & \multicolumn{1}{c|}{3} & 24.68 \\ \hline
\multicolumn{1}{|c|}{60} & 0.39 & \multicolumn{1}{c|}{2} & 0 & \multicolumn{1}{c|}{7} & 9.20 \\ \hline
\multicolumn{1}{|c|}{87} & 0.36 & \multicolumn{1}{c|}{3} & 0 & \multicolumn{1}{c|}{42} & 9.00 \\ \hline
\multicolumn{1}{|c|}{15} & 0.36 & \multicolumn{1}{c|}{4} & 0 & \multicolumn{1}{c|}{27} & 7.55 \\ \hline
\multicolumn{1}{|c|}{130} & 0.33 & \multicolumn{1}{c|}{5} & 0 & \multicolumn{1}{c|}{15} & 6.08 \\ \hline
\multicolumn{1}{|c|}{7} & 0.31 & \multicolumn{1}{c|}{6} & 0 & \multicolumn{1}{c|}{314} & 3.69 \\ \hline
\multicolumn{1}{|c|}{187} & 0.29 & \multicolumn{1}{c|}{7} & 0 & \multicolumn{1}{c|}{250} & 3.49 \\ \hline
\multicolumn{1}{|c|}{50} & 0.28 & \multicolumn{1}{c|}{8} & 0 & \multicolumn{1}{c|}{187} & 3.28 \\ \hline
\multicolumn{1}{|c|}{70} & 0.28 & \multicolumn{1}{c|}{9} & 0 & \multicolumn{1}{c|}{130} & 3.04 \\ \hline
\multicolumn{1}{|c|}{104} & 0.27 & \multicolumn{1}{c|}{10} & 0 & \multicolumn{1}{c|}{87} & 2.78 \\ \hline
\multicolumn{1}{|c|}{156} & 0.26 & \multicolumn{1}{c|}{11} & 0 & \multicolumn{1}{c|}{60} & 2.50 \\ \hline
\multicolumn{1}{|c|}{1467} & 0.25 & \multicolumn{1}{c|}{12} & 0 & \multicolumn{1}{c|}{0} & 0.00 \\ \hline
\multicolumn{1}{|c|}{250} & 0.25 & \multicolumn{1}{c|}{13} & 0 & \multicolumn{1}{c|}{2} & 0.00 \\ \hline
\multicolumn{1}{|c|}{115} & 0.25 & \multicolumn{1}{c|}{14} & 0 & \multicolumn{1}{c|}{4} & 0.00 \\ \hline
\end{tabular}
\caption{Top 15 Nodes with Betweenness Centrality for the Automobile Network. \\\hspace{\textwidth} Base columns represent the top ranked nodes and their percentage of importance held relative to the entire network. \\\hspace{\textwidth}Transitive Closure and Dominant Tree columns illustrate the difference in node rankings after the base network underwent a transformation. \\\hspace{\textwidth}}
\label{table:car-betweenness}
\end{table}
\begin{table}[] \begin{table}[]
\centering \centering
\setlength\tabcolsep{4pt} \setlength\tabcolsep{4pt}

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