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Schrick-Noah_CG-Network_Theory.tex
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\maketitle \maketitle
\begin{abstract} \begin{abstract}
Compliance graphs are generated graphs (or networks) that represent systems' compliance or regulation standings at present, with expected changes, or both. These graphs are generated as directed acyclic graphs (DAGs), and can be used to identify possible correction or mitigation schemes for environments necessitating compliance to mandates or regulations.
DAGs complicate the analysis process due to their underlying graph structures and asymmetry. This work presents network science centralities compatible with DAGs, and structure variations as a means to analyze three different example compliance graphs. Each centrality measure and structural change offers a unique importance ranking that can be used for prioritizing correction or mitigation schemes.
\end{abstract} \end{abstract}
\begin{IEEEkeywords} \begin{IEEEkeywords}
@ -169,7 +171,7 @@ Section \ref{sec:networks}, and this transitive closure was then analyzed throug
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\centering \centering
\vspace{.2truein} \centerline{} \vspace{.2truein} \centerline{}
\caption{Example of Transitive Closure} \caption{Transitive Closure Illustration}
\label{fig:TC} \label{fig:TC}
\end{figure} \end{figure}
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\section{Conclusions and Future Work} \section{Conclusions and Future Work}
\subsection{Conclusions} \subsection{Conclusions}
Each centrality measure implemented in this work provides various information that is useful for identifying correction schemes based on a network science approach. The results from the centrality methods differ, and each network can determine which rankings should be preferred based on prior knowledge of the network and the overhead of implementing correction measures. In addition, transitive closure representations and dominant trees were derived from the original compliance graphs, and unique rankings were identified. Transitive closure rankings are useful for determining which nodes are most important when an adversarial action can be considered to have infinite time and resources to perform changes to the original system. Dominant tree rankings are useful for determining which nodes are most important from an information flow perspective, where adversarial actions must pass though a series of nodes to reach any other node in the network. By applying correction schemes to the bottlenecks of the network, it may be possible to eliminate branches of the dominant tree entirely, leading to a removal of nodes in the original compliance graph. Each centrality measure implemented in this work provides various information that is useful for identifying correction schemes based on a network science approach. The results from the centrality methods differ, and each network can determine which rankings should be preferred based on prior knowledge of the network and the overhead of implementing correction measures. In addition, transitive closures and dominant trees were derived from the original compliance graphs, and unique rankings were identified. Transitive closure rankings are useful for determining which nodes are most important when an adversarial action can be considered to have infinite time and resources to perform changes to the original system. Dominant tree rankings are useful for determining which nodes are most important from an information flow perspective, where adversarial actions must pass though a series of nodes to reach any other node in the network. By applying correction schemes to the bottlenecks of the network, it may be possible to eliminate branches of the dominant tree entirely, leading to a removal of nodes in the original compliance graph.
\subsection{Future Work} \subsection{Future Work}
Based on the results of this work, there is ample room to continue investigation of centrality methods for compliance graphs. With three compliance graphs generated for three different networks along with various node importance rankings, it would be useful to artificially implement correction schemes based on the rankings to see their effects on the compliance graph. Likewise, using a user-defined data matrix in centrality methods like PageRank, further research could examine how node importance varies based on user-defined metrics. Edge weights could also be assigned to the original compliance graphs to represent the probability that a given change in the network could occur. Edge weights would be reflected in the adjacency matrices of the graphs, and centrality methods could be reexamined to determine node importance when state transition probabilities are given. Transitive closures and dominant trees derived from the compliance graphs present a new approach for examining compliance graphs. Further research can be conducted to determine the effects of correction schemes when employed on nodes ranked highly in their respective centrality measures in these formats. Based on the results of this work, there is ample room to continue investigation of centrality methods for compliance graphs. With three compliance graphs generated for three different networks along with various node importance rankings, it would be useful to artificially implement correction schemes based on the rankings to see their effects on the compliance graph. Likewise, using a user-defined data matrix in centrality methods like PageRank, further research could examine how node importance varies based on user-defined metrics. Edge weights could also be assigned to the original compliance graphs to represent the probability that a given change in the network could occur. Edge weights would be reflected in the adjacency matrices of the graphs, and centrality methods could be reexamined to determine node importance when state transition probabilities are given. Transitive closures and dominant trees derived from the compliance graphs present a new approach for examining compliance graphs. Further research can be conducted to determine the effects of correction schemes when employed on nodes ranked highly in their respective centrality measures in these formats.