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(./Schrick-Noah_CS-7863_Final-Report.out) (./Schrick-Noah_CS-7863_Final-Report.out) \@outlinefile=\write3 \openout3 = `Schrick-Noah_CS-7863_Final-Report.out'. -LaTeX Font Info: External font `cmex10' loaded for size -(Font) <12> on input line 39. -LaTeX Font Info: External font `cmex10' loaded for size -(Font) <8> on input line 39. -LaTeX Font Info: External font `cmex10' loaded for size -(Font) <6> on input line 39. -(./Schrick-Noah_CS-7863_Final-Report.toc -LaTeX Font Info: External font `cmex10' loaded for size -(Font) <7> on input line 2. -LaTeX Font Info: External font `cmex10' loaded for size -(Font) <5> on input line 2. - [1 - -{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}]) -\tf@toc=\write4 -\openout4 = `Schrick-Noah_CS-7863_Final-Report.toc'. - - -Overfull \hbox (21.08397pt too wide) in paragraph at lines 43--44 -[]\OT1/cmr/m/n/10 To address the rising risks of computing and threats to cyber -security, vulnerability +Underfull \hbox (badness 10000) in paragraph at lines 63--65 +[]\OT1/ptm/b/it/9 Index Terms\OT1/ptm/b/n/9 ---Attack Graph; Compliance Graph; [] -Overfull \hbox (3.02843pt too wide) in paragraph at lines 45--46 -[]\OT1/cmr/m/n/10 Attack graphs begin with a root node that contains all the cu -rrent information +Underfull \hbox (badness 3128) in paragraph at lines 76--77 +\OT1/ptm/m/n/10 compliance graphs is focused less on certain security [] -Overfull \hbox (2.19508pt too wide) in paragraph at lines 48--49 -[]\OT1/cmr/m/n/10 Compliance graphs are an alternate form of attack graphs, uti -lized specifically +Underfull \hbox (badness 1466) in paragraph at lines 79--80 +\OT1/ptm/m/n/10 increased difficulty is due to the asymmetric adjacency + [] + +[1{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map} + + +] +Underfull \hbox (badness 2035) in paragraph at lines 82--83 +[]\OT1/ptm/m/n/10 The author of [[]] presents three centrality measures [] -Overfull \hbox (0.79698pt too wide) in paragraph at lines 48--49 -\OT1/cmr/m/n/10 for examining compliance and regulation statuses of systems. Li -ke attack graphs, +Underfull \hbox (badness 1708) in paragraph at lines 82--83 +\OT1/ptm/m/n/10 Other works, such as that discussed by the author of + [] + + +Underfull \hbox (badness 2698) in paragraph at lines 85--86 +[]\OT1/ptm/m/n/10 The work conducted in this approach utilized three + [] + + +LaTeX Warning: No positions in optional float specifier. + Default added (so using `tbp') on input line 87. + +LaTeX Font Info: External font `cmex10' loaded for size +(Font) <7> on input line 89. +LaTeX Font Info: External font `cmex10' loaded for size +(Font) <5> on input line 89. + +Underfull \hbox (badness 1448) in paragraph at lines 102--103 +\OT1/ptm/m/n/10 importance in networks. By determining the importance [] [2] -Overfull \hbox (10.93585pt too wide) in paragraph at lines 50--51 -\OT1/cmr/m/n/10 to also examine administrative policies and properties of syste -ms. Since compliance +Underfull \hbox (badness 1540) in paragraph at lines 145--146 +\OT1/ptm/m/n/10 algorithm measures the quality of connections, there is [] - -Overfull \hbox (2.69524pt too wide) in paragraph at lines 50--51 -\OT1/cmr/m/n/10 and regulation is broad and can vary by industry and applicatio -n, the information - [] - -[3] -Overfull \hbox (10.74141pt too wide) in paragraph at lines 59--60 -\OT1/cmr/m/n/10 cooperation to training and administrative policies. This netwo -rk is also progressed - [] - - -LaTeX Warning: No positions in optional float specifier. - Default added (so using `tbp') on input line 61. - - -Overfull \hbox (24.76286pt too wide) in paragraph at lines 74--74 -[]\OT1/cmr/bx/n/14.4 Centralities and their Applications to Compliance - [] - - -Overfull \hbox (14.1025pt too wide) in paragraph at lines 76--77 -\OT1/cmr/m/n/10 opportunities for compliance violation. This work discusses fiv -e centrality measures, - [] - -[4] [5] [6] [7] LaTeX Font Info: Trying to load font information for U+dsrom on input line 1 -26. - (/usr/share/texmf-dist/tex/latex/doublestroke/Udsrom.fd +52. +(/usr/share/texmf-dist/tex/latex/doublestroke/Udsrom.fd File: Udsrom.fd 1995/08/01 v0.1 Double stroke roman font definitions -) -[8] -<./images/Transitive-closure.png, id=247, 1927.2pt x 1027.84pt> +) [3] +LaTeX Font Info: External font `cmex10' loaded for size +(Font) <8> on input line 163. +LaTeX Font Info: External font `cmex10' loaded for size +(Font) <6> on input line 163. + +Overfull \hbox (28.88863pt too wide) in paragraph at lines 163--163 +[][][][]\OT1/ptm/m/n/8 Image origin can be located at: https://commons.wikimedi +a.org/wiki/File:Transitive- + [] + +<./images/Transitive-closure.png, id=191, 1927.2pt x 1027.84pt> File: ./images/Transitive-closure.png Graphic file (type png) -Package pdftex.def Info: ./images/Transitive-closure.png used on input line 14 -1. -(pdftex.def) Requested size: 345.0pt x 183.99947pt. - [9 <./images/Transitive-closure.png>] -<./images/pre-Dtree.png, id=256, 673.51625pt x 770.88pt> +Package pdftex.def Info: ./images/Transitive-closure.png used on input line 16 +7. +(pdftex.def) Requested size: 252.0pt x 134.39238pt. +<./images/pre-Dtree.png, id=194, 673.51625pt x 770.88pt> File: ./images/pre-Dtree.png Graphic file (type png) -Package pdftex.def Info: ./images/pre-Dtree.png used on input line 156. -(pdftex.def) Requested size: 345.0pt x 394.87271pt. +Package pdftex.def Info: ./images/pre-Dtree.png used on input line 182. +(pdftex.def) Requested size: 252.0pt x 288.43234pt. -Overfull \hbox (71.59012pt too wide) in paragraph at lines 158--158 -[][][]\OT1/cmr/m/n/8 Image origin can be located at: https://commons.wikimedia. -org/wiki/File:Dominator$[]$control$[]$flow$[]$graph.svg, +Overfull \hbox (111.06421pt too wide) in paragraph at lines 184--184 +[][][][]\OT1/ptm/m/n/8 Image origin can be located at: https://commons.wikimedi +a.org/wiki/File:Dominator$[]$control$[]$flow$[]$graph.svg, [] -<./images/post-Dtree.png, id=258, 642.4pt x 435.6275pt> +<./images/post-Dtree.png, id=196, 642.4pt x 435.6275pt> File: ./images/post-Dtree.png Graphic file (type png) -Package pdftex.def Info: ./images/post-Dtree.png used on input line 162. -(pdftex.def) Requested size: 345.0pt x 233.95909pt. +Package pdftex.def Info: ./images/post-Dtree.png used on input line 188. +(pdftex.def) Requested size: 252.0pt x 170.88446pt. -Overfull \hbox (17.76799pt too wide) in paragraph at lines 164--164 -[][][]\OT1/cmr/m/n/8 Image origin can be located at: https://commons.wikimedia. -org/wiki/File:Dominator$[]$tree.svg, +Overfull \hbox (60.15239pt too wide) in paragraph at lines 190--190 +[][][][]\OT1/ptm/m/n/8 Image origin can be located at: https://commons.wikimedi +a.org/wiki/File:Dominator$[]$tree.svg, [] -LaTeX Warning: No positions in optional float specifier. - Default added (so using `tbp') on input line 174. +Underfull \vbox (badness 10000) has occurred while \output is active [] + [4 <./images/Transitive-closure.png>] LaTeX Warning: No positions in optional float specifier. Default added (so using `tbp') on input line 200. LaTeX Warning: No positions in optional float specifier. - Default added (so using `tbp') on input line 226. + Default added (so using `tbp') on input line 296. + + +Overfull \hbox (46.28867pt too wide) in paragraph at lines 298--387 + [][] + [] LaTeX Warning: No positions in optional float specifier. - Default added (so using `tbp') on input line 254. + Default added (so using `tbp') on input line 391. LaTeX Warning: No positions in optional float specifier. - Default added (so using `tbp') on input line 281. + Default added (so using `tbp') on input line 419. -[10] [11 <./images/pre-Dtree.png>] [12 <./images/post-Dtree.png>] [13] [14] -Overfull \hbox (1.47293pt too wide) in paragraph at lines 312--313 -\OT1/cmr/m/n/10 the dominant tree rankings also vary greatly from the original -compliance graph's + +Overfull \hbox (5.48868pt too wide) in paragraph at lines 421--441 + [][] [] -Overfull \hbox (8.02841pt too wide) in paragraph at lines 316--317 -[]\OT1/cmr/m/n/10 Each centrality measure implemented in this work provides var -ious information - [] +LaTeX Warning: No positions in optional float specifier. + Default added (so using `tbp') on input line 446. -[15] -Overfull \hbox (4.695pt too wide) in paragraph at lines 319--320 -[]\OT1/cmr/m/n/10 Based on the results of this work, there is ample room to con -tinue investigation + +Overfull \hbox (3.96829pt too wide) in paragraph at lines 448--468 + [][] [] -Overfull \hbox (12.05623pt too wide) in paragraph at lines 319--320 -\OT1/cmr/m/n/10 generated for three different networks along with various node -importance rankings, +Underfull \hbox (badness 10000) in paragraph at lines 473--474 +[]\OT1/ptm/m/n/10 When viewing the results of the car networks, [] -[16] (./Schrick-Noah_CS-7863_Final-Report.bbl -Underfull \hbox (badness 1237) in paragraph at lines 4--8 -[]\OT1/cmr/m/n/10 C. Phillips and L. P. Swiler, ``A graph-based system for netw -ork- + +Underfull \hbox (badness 1997) in paragraph at lines 473--474 +\OT1/ptm/m/n/10 based on additional metrics, such as severity, cost, or + [] + +[5 <./images/pre-Dtree.png> <./images/post-Dtree.png>] +Underfull \hbox (badness 3428) in paragraph at lines 475--476 +\OT1/ptm/m/n/10 their original compliance graph rankings. As expected + [] + + +Underfull \vbox (badness 4279) has occurred while \output is active [] + + +Underfull \hbox (badness 1789) in paragraph at lines 477--478 +[]\OT1/ptm/m/n/10 For the dominant tree representation, it was initially + [] + + +Underfull \hbox (badness 3009) in paragraph at lines 477--478 +\OT1/ptm/m/n/10 hypothesized that nodes ranked highly in the original + [] + + +Underfull \hbox (badness 2781) in paragraph at lines 481--482 +\OT1/ptm/m/n/10 prior knowledge of the network and the overhead of + [] + + +Underfull \hbox (badness 1442) in paragraph at lines 481--482 +\OT1/ptm/m/n/10 implementing correction measures. In addition, transitive + [] + + +Underfull \hbox (badness 1354) in paragraph at lines 481--482 +\OT1/ptm/m/n/10 closure representations and dominant trees were derived + [] + + +Underfull \hbox (badness 2269) in paragraph at lines 481--482 +\OT1/ptm/m/n/10 determining which nodes are most important when an + [] + + +Underfull \vbox (badness 10000) has occurred while \output is active [] + + [6] +Underfull \hbox (badness 2591) in paragraph at lines 484--485 +\OT1/ptm/m/n/10 different networks along with various node importance + [] + +[7 + +] (./Schrick-Noah_CS-7863_Final-Report.bbl +Underfull \hbox (badness 1442) in paragraph at lines 19--24 +\OT1/ptm/m/n/8 A Focus on Cyberphysical Systems and Internet of Things,'' in + [] + + +Underfull \hbox (badness 3148) in paragraph at lines 45--47 +[]\OT1/ptm/m/n/8 R. A. Brualdi, ``Spectra of digraphs,'' \OT1/ptm/m/it/8 Linear + Algebra and its [] Underfull \hbox (badness 10000) in paragraph at lines 49--52 -[]\OT1/cmr/m/n/10 ``Health Insurance Portability and Accountability Act of +[]\OT1/ptm/m/n/8 ``Health Insurance Portability and Accountability Act of [] Underfull \hbox (badness 10000) in paragraph at lines 49--52 -\OT1/cmr/m/n/10 1996.'' Pub. L. No. 104-191. 1996 [Online]. Available: +\OT1/ptm/m/n/8 1996.'' Pub. L. No. 104-191. 1996 [Online]. Available: [] Underfull \hbox (badness 10000) in paragraph at lines 49--52 -\OT1/cmr/m/n/10 https://www.govinfo.gov/content/pkg/PLAW-104publ191/html/PLAW- +\OT1/ptm/m/n/8 https://www.govinfo.gov/content/pkg/PLAW-104publ191/html/PLAW- [] Underfull \hbox (badness 10000) in paragraph at lines 54--58 -[]\OT1/cmr/m/n/10 P. S. S. Council, ``Payment Card Industry (PCI) +[]\OT1/ptm/m/n/8 P. S. S. Council, ``Payment Card Industry (PCI) [] Underfull \hbox (badness 10000) in paragraph at lines 54--58 -\OT1/cmr/m/n/10 Data Security Standard,'' May 2018. Available: +\OT1/ptm/m/n/8 Data Security Standard,'' May 2018. Available: [] +) [8 -Underfull \hbox (badness 1571) in paragraph at lines 66--68 -[]\OT1/cmr/m/n/10 L. Katz, ``A new status index derived from sociometric analys -is,'' - [] -[17 - -]) [18] +] \openout2 = `Appendices.aux'. - (./Appendices.tex [19 + (./Appendices.tex +** WARNING: Ignoring useless \section in Appendix (line 3). +[9 + ] @@ -644,6 +787,11 @@ LaTeX Warning: No positions in optional float specifier. Default added (so using `tbp') on input line 32. +Overfull \hbox (42.75949pt too wide) in paragraph at lines 34--54 + [][] + [] + + LaTeX Warning: No positions in optional float specifier. Default added (so using `tbp') on input line 58. @@ -652,10 +800,24 @@ LaTeX Warning: No positions in optional float specifier. 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+\documentclass[conference]{IEEEtran} \RequirePackage{setspace} -\documentclass{article} \usepackage{boxedminipage} \usepackage{graphicx} % Images \graphicspath{ {./images/} } @@ -29,14 +29,40 @@ \cftsetindents{figure}{0em}{3.5em} \cftsetindents{table}{0em}{3.5em} -\title{CS 7863: Network Theory Final Project: Compliance Graph Analysis} -\author{Noah Schrick} -\date{May 3, 2022} - \begin{document} +\title{ + Placeholder Title +} + +\author{ + \IEEEauthorblockN{Noah L. Schrick} + \IEEEauthorblockA{ + \textit{Tandy School of Computer Science} \\ + \textit{The University of Tulsa}\\ + Tulsa, USA \\ + noah-schrick@utulsa.edu + } + + \and + + \IEEEauthorblockN{Peter J. Hawrylak} + \IEEEauthorblockA{ + \textit{Tandy School of Computer Science} \\ + \textit{The University of Tulsa}\\ + Tulsa, USA \\ + peter-hawrylak@utulsa.edu + } +} + \maketitle -\tableofcontents + +\begin{abstract} +\end{abstract} + +\begin{IEEEkeywords} +Attack Graph; Compliance Graph; Cybersecurity; Compliance and Regulation; Network Theory; Centrality; +\end{IEEEkeywords} \section{Introduction} \subsection{Attack Graphs} @@ -49,7 +75,7 @@ Compliance graphs are an alternate form of attack graphs, utilized specifically The semantics of compliance graphs are similar to that of attack graphs, but with a few differences regarding the information at each state. While security and compliance statuses are related, the information that is analyzed in compliance graphs is focused less on certain security properties, and is expanded to also examine administrative policies and properties of systems. Since compliance and regulation is broad and can vary by industry and application, the information to analyze can range from safety regulations, maintenance compliance, or any other regulatory compliance. However, the graph structure of compliance graphs is identical to that of attack graphs, where edges represent a modification to the systems, and nodes represent all current information in the system. -\subsection{Difficulties of Compliance Graph Analysis} +\subsection{Difficulties of Attack and Compliance Graph Analysis} Analysis of directed graphs is not as simple as their undirected counterparts, and attack and compliance graphs are directed acyclic graphs. The primary contributor to the increased difficulty is due to the asymmetric adjacency matrix present in directed graphs. With undirected graphs, simplifications can be made in the analysis process both computationally and conceptually. Since the ``in" degrees are equal to the ``out" degrees, less work is required both in terms of parsing the adjacency matrix, but also in terms of determining importance of nodes. The author of \cite{newman2010networks} discusses that common analysis techniques such as eigenvector centrality is often unapplicable to directed acyclic graphs. As the author of \cite{Mieghem2018DirectedGA} discusses, the difficulty of directed graphs also extends to the graph Laplacian, where the definition for asymmetric adjacency matrices is not uniquely defined, and is based on either row or column sums computing to zero, but both cannot. The author of \cite{Mieghem2018DirectedGA} continues to discuss that directed graphs lead to complex eigenvalues, and can lead to adjacency matrices that are unable to be diagonalized. These challenges require different approaches for typical clustering or centrality measures. \section{Related Works} @@ -175,23 +201,93 @@ In this section, only results for the car network are displayed for brevity. The \centering \begin{tabular}{|cc|cc|cc|} \hline - \multicolumn{2}{|c|}{\cellcolor[HTML]{FFFF00}\textbf{Base}} & \multicolumn{2}{|c|}{\cellcolor[HTML]{FFFF00}\textbf{Transitive Closure}} & \multicolumn{2}{|c|}{\cellcolor[HTML]{FFFF00}\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} & 11 & \multicolumn{1}{|c|}{0} & \multicolumn{1}{|c|}{2490} & \multicolumn{1}{|c|}{1} & \multicolumn{1}{|c|}{1246} \\ \hline - \multicolumn{1}{|c|}{346} & 10 & \multicolumn{1}{|c|}{1} & \multicolumn{1}{|c|}{2489} & \multicolumn{1}{|c|}{3} & \multicolumn{1}{|c|}{934} \\ \hline - \multicolumn{1}{|c|}{362} & 10 & \multicolumn{1}{|c|}{3} & \multicolumn{1}{|c|}{2487} & \multicolumn{1}{|c|}{7} & \multicolumn{1}{|c|}{156} \\ \hline - \multicolumn{1}{|c|}{370} & 10 & \multicolumn{1}{|c|}{7} & \multicolumn{1}{|c|}{2479} & \multicolumn{1}{|c|}{42} & \multicolumn{1}{|c|}{115} \\ \hline - \multicolumn{1}{|c|}{374} & 10 & \multicolumn{1}{|c|}{15} & \multicolumn{1}{|c|}{2463} & \multicolumn{1}{|c|}{314} & \multicolumn{1}{|c|}{31} \\ \hline - \multicolumn{1}{|c|}{376} & 10 & \multicolumn{1}{|c|}{27} & \multicolumn{1}{|c|}{2447} & \multicolumn{1}{|c|}{0} & \multicolumn{1}{|c|}{1} \\ \hline - \multicolumn{1}{|c|}{377} & 10 & \multicolumn{1}{|c|}{42} & \multicolumn{1}{|c|}{2431} & \multicolumn{1}{|c|}{15} & \multicolumn{1}{|c|}{1} \\ \hline - \multicolumn{1}{|c|}{378} & 10 & \multicolumn{1}{|c|}{60} & \multicolumn{1}{|c|}{2367} & \multicolumn{1}{|c|}{27} & \multicolumn{1}{|c|}{1} \\ \hline - \multicolumn{1}{|c|}{379} & 10 & \multicolumn{1}{|c|}{87} & \multicolumn{1}{|c|}{2303} & \multicolumn{1}{|c|}{60} & \multicolumn{1}{|c|}{1} \\ \hline - \multicolumn{1}{|c|}{380} & 10 & \multicolumn{1}{|c|}{130} & \multicolumn{1}{|c|}{2239} & \multicolumn{1}{|c|}{87} & \multicolumn{1}{|c|}{1} \\ \hline - \multicolumn{1}{|c|}{381} & 10 & \multicolumn{1}{|c|}{187} & \multicolumn{1}{|c|}{2175} & \multicolumn{1}{|c|}{130} & \multicolumn{1}{|c|}{1} \\ \hline - \multicolumn{1}{|c|}{382} & 10 & \multicolumn{1}{|c|}{250} & \multicolumn{1}{|c|}{2111} & \multicolumn{1}{|c|}{187} & \multicolumn{1}{|c|}{1} \\ \hline - \multicolumn{1}{|c|}{398} & 9 & \multicolumn{1}{|c|}{314} & \multicolumn{1}{|c|}{2047} & \multicolumn{1}{|c|}{250} & \multicolumn{1}{|c|}{1} \\ \hline - \multicolumn{1}{|c|}{406} & 9 & \multicolumn{1}{|c|}{2} & \multicolumn{1}{|c|}{1244} & \multicolumn{1}{|c|}{2} & \multicolumn{1}{|c|}{0} \\ \hline - \multicolumn{1}{|c|}{410} & 9 & \multicolumn{1}{|c|}{4} & \multicolumn{1}{|c|}{1243} & \multicolumn{1}{|c|}{4} & \multicolumn{1}{|c|}{0} \\ \hline + + \multicolumn{2}{|c|}{\cellcolor[HTML]{FFFF00}\textbf{Base}} + & \multicolumn{2}{|c|}{\cellcolor[HTML]{FFFF00}\textbf{Transitive Closure}} + & \multicolumn{2}{|c|}{\cellcolor[HTML]{FFFF00}\textbf{Dominant Tree}} \\ + \hline + + \multicolumn{1}{|c|}{\textbf{Node}} & \multicolumn{1}{|c|}{\textbf{Value}} + & \multicolumn{1}{|c|}{\textbf{Node}} & \multicolumn{1}{|c|}{\textbf{Value}} + & \multicolumn{1}{|c|}{\textbf{Node}} & \multicolumn{1}{|c|}{\textbf{Value}} + \\ \hline + + \multicolumn{1}{|c|}{314} & \multicolumn{1}{|c|}{11} + & \multicolumn{1}{|c|}{0} & \multicolumn{1}{|c|}{2490} + & \multicolumn{1}{|c|}{1} & \multicolumn{1}{|c|}{1246} + \\ \hline + + \multicolumn{1}{|c|}{346} & \multicolumn{1}{|c|}{10} + & \multicolumn{1}{|c|}{1} & \multicolumn{1}{|c|}{2489} + & \multicolumn{1}{|c|}{3} & \multicolumn{1}{|c|}{934} + \\ \hline + + \multicolumn{1}{|c|}{362} & \multicolumn{1}{|c|}{10} + & \multicolumn{1}{|c|}{3} & \multicolumn{1}{|c|}{2487} + & \multicolumn{1}{|c|}{7} & \multicolumn{1}{|c|}{156} + \\ \hline + + \multicolumn{1}{|c|}{370} & \multicolumn{1}{|c|}{10} + & \multicolumn{1}{|c|}{7} & \multicolumn{1}{|c|}{2479} + & \multicolumn{1}{|c|}{42} & \multicolumn{1}{|c|}{115} + \\ \hline + + \multicolumn{1}{|c|}{374} & \multicolumn{1}{|c|}{10} + & \multicolumn{1}{|c|}{15} & \multicolumn{1}{|c|}{2463} + & \multicolumn{1}{|c|}{314} & \multicolumn{1}{|c|}{31} + \\ \hline + + \multicolumn{1}{|c|}{376} & \multicolumn{1}{|c|}{10} + & \multicolumn{1}{|c|}{27} & \multicolumn{1}{|c|}{2447} + & \multicolumn{1}{|c|}{0} & \multicolumn{1}{|c|}{1} + \\ \hline + + \multicolumn{1}{|c|}{377} & \multicolumn{1}{|c|}{10} + & \multicolumn{1}{|c|}{42} & \multicolumn{1}{|c|}{2431} + & \multicolumn{1}{|c|}{15} & \multicolumn{1}{|c|}{1} + \\ \hline + + \multicolumn{1}{|c|}{378} & \multicolumn{1}{|c|}{10} + & \multicolumn{1}{|c|}{60} & \multicolumn{1}{|c|}{2367} + & \multicolumn{1}{|c|}{27} & \multicolumn{1}{|c|}{1} + \\ \hline + + + \multicolumn{1}{|c|}{379} & \multicolumn{1}{|c|}{10} + & \multicolumn{1}{|c|}{87} & \multicolumn{1}{|c|}{2303} + & \multicolumn{1}{|c|}{60} & \multicolumn{1}{|c|}{1} + \\ \hline + + \multicolumn{1}{|c|}{380} & \multicolumn{1}{|c|}{10} + & \multicolumn{1}{|c|}{130} & \multicolumn{1}{|c|}{2239} + & \multicolumn{1}{|c|}{87} & \multicolumn{1}{|c|}{1} + \\ \hline + + \multicolumn{1}{|c|}{381} & \multicolumn{1}{|c|}{10} + & \multicolumn{1}{|c|}{187} & \multicolumn{1}{|c|}{2175} + & \multicolumn{1}{|c|}{130} & \multicolumn{1}{|c|}{1} + \\ \hline + + \multicolumn{1}{|c|}{382} & \multicolumn{1}{|c|}{10} + & \multicolumn{1}{|c|}{250} & \multicolumn{1}{|c|}{2111} + & \multicolumn{1}{|c|}{187} & \multicolumn{1}{|c|}{1} + \\ \hline + + \multicolumn{1}{|c|}{398} & \multicolumn{1}{|c|}{9} + & \multicolumn{1}{|c|}{314} & \multicolumn{1}{|c|}{2047} + & \multicolumn{1}{|c|}{250} & \multicolumn{1}{|c|}{1} + \\ \hline + + \multicolumn{1}{|c|}{406} & \multicolumn{1}{|c|}{9} + & \multicolumn{1}{|c|}{2} & \multicolumn{1}{|c|}{1244} + & \multicolumn{1}{|c|}{2} & \multicolumn{1}{|c|}{0} + \\ \hline + + \multicolumn{1}{|c|}{410} & \multicolumn{1}{|c|}{9} + & \multicolumn{1}{|c|}{4} & \multicolumn{1}{|c|}{1243} + & \multicolumn{1}{|c|}{4} & \multicolumn{1}{|c|}{0} + \\ \hline + \end{tabular} \caption{Top 15 Nodes with Degree Centrality} \label{table:car-deg} @@ -201,23 +297,92 @@ In this section, only results for the car network are displayed for brevity. The \centering \begin{tabular}{|cc|cc|cc|} \hline - \multicolumn{2}{|c|}{\cellcolor[HTML]{FFFF00}\textbf{Base}} & \multicolumn{2}{|c|}{\cellcolor[HTML]{FFFF00}\textbf{Transitive Closure}} & \multicolumn{2}{|c|}{\cellcolor[HTML]{FFFF00}\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} & 0.002459349 & \multicolumn{1}{|c|}{0} & 74.447935 & \multicolumn{1}{|c|}{1} & 0.0542337315 \\ \hline - \multicolumn{1}{|c|}{377} & 0.001870821 & \multicolumn{1}{|c|}{1} & 67.679941 & \multicolumn{1}{|c|}{3} & 0.0385235854 \\ \hline - \multicolumn{1}{|c|}{346} & 0.001870821 & \multicolumn{1}{|c|}{3} & 60.55317 & \multicolumn{1}{|c|}{7} & 0.0066730273 \\ \hline - \multicolumn{1}{|c|}{376} & 0.001870821 & \multicolumn{1}{|c|}{7} & 51.894146 & \multicolumn{1}{|c|}{0} & 0.0058248184 \\ \hline - \multicolumn{1}{|c|}{374} & 0.001870821 & \multicolumn{1}{|c|}{15} & 43.13118 & \multicolumn{1}{|c|}{42} & 0.0050225267 \\ \hline - \multicolumn{1}{|c|}{378} & 0.001870821 & \multicolumn{1}{|c|}{27} & 35.752083 & \multicolumn{1}{|c|}{314} & 0.0016459253 \\ \hline - \multicolumn{1}{|c|}{380} & 0.001870821 & \multicolumn{1}{|c|}{42} & 29.550411 & \multicolumn{1}{|c|}{27} & 0.0009036979 \\ \hline - \multicolumn{1}{|c|}{381} & 0.001870821 & \multicolumn{1}{|c|}{60} & 22.205831 & \multicolumn{1}{|c|}{250} & 0.0005660377 \\ \hline - \multicolumn{1}{|c|}{382} & 0.001870821 & \multicolumn{1}{|c|}{87} & 16.522142 & \multicolumn{1}{|c|}{15} & 0.000491815 \\ \hline - \multicolumn{1}{|c|}{262} & 0.001870821 & \multicolumn{1}{|c|}{130} & 12.155237 & \multicolumn{1}{|c|}{187} & 0.000458049 \\ \hline - \multicolumn{1}{|c|}{370} & 0.001870821 & \multicolumn{1}{|c|}{2} & 10.714534 & \multicolumn{1}{|c|}{130} & 0.0004472501 \\ \hline - \multicolumn{1}{|c|}{379} & 0.001870821 & \multicolumn{1}{|c|}{4} & 9.740485 & \multicolumn{1}{|c|}{87} & 0.0004461702 \\ \hline - \multicolumn{1}{|c|}{418} & 0.001469376 & \multicolumn{1}{|c|}{5} & 9.740485 & \multicolumn{1}{|c|}{60} & 0.0004460622 \\ \hline - \multicolumn{1}{|c|}{459} & 0.001469376 & \multicolumn{1}{|c|}{6} & 9.740485 & \multicolumn{1}{|c|}{2} & 0.0004014452 \\ \hline - \multicolumn{1}{|c|}{467} & 0.001469376 & \multicolumn{1}{|c|}{187} & 8.82693 & \multicolumn{1}{|c|}{4} & 0.0004014452 \\ \hline + + \multicolumn{2}{|c|}{\cellcolor[HTML]{FFFF00}\textbf{Base}} + & \multicolumn{2}{|c|}{\cellcolor[HTML]{FFFF00}\textbf{Transitive Closure}} + & \multicolumn{2}{|c|}{\cellcolor[HTML]{FFFF00}\textbf{Dominant Tree}} + \\ \hline + + \multicolumn{1}{|c|}{\textbf{Node}} & \multicolumn{1}{|c|}{\textbf{Value}} + & \multicolumn{1}{|c|}{\textbf{Node}} & \multicolumn{1}{|c|}{\textbf{Value}} + & \multicolumn{1}{|c|}{\textbf{Node}} & \multicolumn{1}{|c|}{\textbf{Value}} + \\ \hline + + \multicolumn{1}{|c|}{314} & \multicolumn{1}{|c|}{0.002459349} + & \multicolumn{1}{|c|}{0} & \multicolumn{1}{|c|}{74.447935} + & \multicolumn{1}{|c|}{1} & \multicolumn{1}{|c|}{0.0542337315} + \\ \hline + + \multicolumn{1}{|c|}{377} & \multicolumn{1}{|c|}{0.001870821} + & \multicolumn{1}{|c|}{1} & \multicolumn{1}{|c|}{67.679941} + & \multicolumn{1}{|c|}{3} & \multicolumn{1}{|c|}{0.0385235854} + \\ \hline + + \multicolumn{1}{|c|}{346} & \multicolumn{1}{|c|}{0.001870821} + & \multicolumn{1}{|c|}{3} & \multicolumn{1}{|c|}{60.55317} + & \multicolumn{1}{|c|}{7} & \multicolumn{1}{|c|}{0.0066730273} + \\ \hline + + \multicolumn{1}{|c|}{376} & \multicolumn{1}{|c|}{0.001870821} + & \multicolumn{1}{|c|}{7} & \multicolumn{1}{|c|}{51.894146} + & \multicolumn{1}{|c|}{0} & \multicolumn{1}{|c|}{0.0058248184} + \\ \hline + + \multicolumn{1}{|c|}{374} & \multicolumn{1}{|c|}{0.001870821} + & \multicolumn{1}{|c|}{15} & \multicolumn{1}{|c|}{43.13118} + & \multicolumn{1}{|c|}{42} & \multicolumn{1}{|c|}{0.0050225267} + \\ \hline + + \multicolumn{1}{|c|}{378} & \multicolumn{1}{|c|}{0.001870821} + & \multicolumn{1}{|c|}{27} & \multicolumn{1}{|c|}{35.752083} + & \multicolumn{1}{|c|}{314} & \multicolumn{1}{|c|}{0.0016459253} + \\ \hline + + \multicolumn{1}{|c|}{380} & \multicolumn{1}{|c|}{0.001870821} + & \multicolumn{1}{|c|}{42} & \multicolumn{1}{|c|}{29.550411} + & \multicolumn{1}{|c|}{27} & \multicolumn{1}{|c|}{0.0009036979} + \\ \hline + + \multicolumn{1}{|c|}{381} & \multicolumn{1}{|c|}{0.001870821} + & \multicolumn{1}{|c|}{60} & \multicolumn{1}{|c|}{22.205831} + & \multicolumn{1}{|c|}{250} & \multicolumn{1}{|c|}{0.0005660377} + \\ \hline + + \multicolumn{1}{|c|}{382} & \multicolumn{1}{|c|}{0.001870821} + & \multicolumn{1}{|c|}{87} & \multicolumn{1}{|c|}{16.522142} + & \multicolumn{1}{|c|}{15} & \multicolumn{1}{|c|}{0.000491815} + \\ \hline + + \multicolumn{1}{|c|}{262} & \multicolumn{1}{|c|}{0.001870821} + & \multicolumn{1}{|c|}{130} & \multicolumn{1}{|c|}{12.155237} + & \multicolumn{1}{|c|}{187} & \multicolumn{1}{|c|}{0.000458049} + \\ \hline + + \multicolumn{1}{|c|}{370} & \multicolumn{1}{|c|}{0.001870821} + & \multicolumn{1}{|c|}{2} & \multicolumn{1}{|c|}{10.714534} + & \multicolumn{1}{|c|}{130} & \multicolumn{1}{|c|}{0.0004472501} + \\ \hline + + \multicolumn{1}{|c|}{379} & \multicolumn{1}{|c|}{0.001870821} + & \multicolumn{1}{|c|}{4} & \multicolumn{1}{|c|}{9.740485} + & \multicolumn{1}{|c|}{87} & \multicolumn{1}{|c|}{0.0004461702} + \\ \hline + + \multicolumn{1}{|c|}{418} &\multicolumn{1}{|c|}{0.001469376} + & \multicolumn{1}{|c|}{5} & \multicolumn{1}{|c|}{9.740485} + & \multicolumn{1}{|c|}{60} & \multicolumn{1}{|c|}{0.0004460622} + \\ \hline + + \multicolumn{1}{|c|}{459} & \multicolumn{1}{|c|}{0.001469376} + & \multicolumn{1}{|c|}{6} & \multicolumn{1}{|c|}{9.740485} + & \multicolumn{1}{|c|}{2} & \multicolumn{1}{|c|}{0.0004014452} + \\ \hline + + \multicolumn{1}{|c|}{467} & \multicolumn{1}{|c|}{0.001469376} + & \multicolumn{1}{|c|}{187} & \multicolumn{1}{|c|}{8.82693} + & \multicolumn{1}{|c|}{4} & \multicolumn{1}{|c|}{0.0004014452} + \\ \hline + \end{tabular} \caption{Top 15 Nodes with Katz Centrality} \label{table:car-katz}