From 9eacf9eb2073bc18c9ed9ea73635248c7696adbf Mon Sep 17 00:00:00 2001 From: noah Date: Mon, 2 May 2022 19:02:57 -0500 Subject: [PATCH] Degree, Betweenness, and beginning work of Katz Centrality --- .~lock.Formatted_Results.ods# | 2 +- Formatted_Results.ods | Bin 20828 -> 23888 bytes Report/Bibliography.bib | 23 ++++++++ Report/Schrick-Noah_CS-7863_Final-Report.aux | 6 +- Report/Schrick-Noah_CS-7863_Final-Report.bbl | 4 ++ Report/Schrick-Noah_CS-7863_Final-Report.blg | 56 +++++++++---------- Report/Schrick-Noah_CS-7863_Final-Report.log | 34 +++++++---- Report/Schrick-Noah_CS-7863_Final-Report.pdf | Bin 116347 -> 161981 bytes Report/Schrick-Noah_CS-7863_Final-Report.tex | 18 +++++- Report/Schrick-Noah_CS-7863_Final-Report.toc | 2 +- Schrick-Noah_CG-Analysis.R | 33 +++++++++++ centralities.R | 7 ++- 12 files changed, 139 insertions(+), 46 deletions(-) diff --git a/.~lock.Formatted_Results.ods# b/.~lock.Formatted_Results.ods# index 8fd9f05..73d0e93 100644 --- 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Available: [] + +Underfull \hbox (badness 1571) in paragraph at lines 62--64 +[]\OT1/cmr/m/n/10 L. Katz, ``A new status index derived from sociometric analys +is,'' + [] + ) [7 ] (./Schrick-Noah_CS-7863_Final-Report.aux) ) Here is how much of TeX's memory you used: - 5373 strings out of 478238 - 96520 string characters out of 5850456 - 398294 words of memory out of 5000000 - 23550 multiletter control sequences out of 15000+600000 + 5376 strings out of 478238 + 96547 string characters out of 5850456 + 398315 words of memory out of 5000000 + 23553 multiletter control sequences out of 15000+600000 473860 words of font info for 43 fonts, out of 8000000 for 9000 1141 hyphenation exceptions out of 8191 67i,8n,77p,1807b,289s stack positions out of 5000i,500n,10000p,200000b,80000s -Output written on Schrick-Noah_CS-7863_Final-Report.pdf (7 pages, 116347 bytes) +ts/cm/cmbx12.pfb> +Output written on Schrick-Noah_CS-7863_Final-Report.pdf (7 pages, 161981 bytes) . 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zLIT^xzWL~J$Y)?{z=(yX?fVM>2UjhEBnl>^2nO0ew21;oC-hG%IX-jC40sn8XdjI* zAyiqSNlb-RG`g$Hb^}!inJ~yEJ5($}^14P`EQC_7W<)IJ^i@5XSjh7qz$%gWGj!?C zX>`n3RL)p*5GBfBBz}6t+lsDo=ISq6wXw$~a+NH6z4dvWTS|R({xUEsATfEWw xgQf;l1-P+?!Rc#-PoQ%wT=@U{ZCzZAoLxPg&CFq#nAo_O*kQ=X#1zC~{tJTU*RTKp diff --git a/Report/Schrick-Noah_CS-7863_Final-Report.tex b/Report/Schrick-Noah_CS-7863_Final-Report.tex index d9a316d..d135728 100644 --- a/Report/Schrick-Noah_CS-7863_Final-Report.tex +++ b/Report/Schrick-Noah_CS-7863_Final-Report.tex @@ -65,9 +65,25 @@ The work conducted in this approach utilized three compliance graphs, with their \subsection{Introduction} The author of \cite{PMID:30064421} provides a survey of centrality measures, and discusses how various centrality measures have been implemented and brought forth in order to determine node importance in networks. By determining the importance of nodes, various conclusions can be drawn regarding the network. In the case of compliance graphs, conclusions can be drawn regarding the prioritization of patching or correction schemes. If one node is known to lead to the creation of many other nodes, it may be said that a patch is imperative to prevent further opportunities for compliance violation. This work discusses five centrality measures, and discusses their application to compliance graphs. \subsection{Degree} -Degree centrality is a trivial, localized measure of node importance based on the number of edges that a node has. In an undirected graph, the degree centrality is predicated solely on the number of edges. However, in the case of a directed graph, a distinction is drawn with a degree centrality oriented on the number of edges coming into a node, and another measure focused on the number of edges leaving a node. Both of these cases provide useful information for compliance graphs. When a node has a large number of other nodes it points to, this node may be prioritized since it creates further room for violation. When a node has a large number of edges pointing to it, this node may be prioritized since the probability that systems may enter this state is higher due to the increased number of ways that a system could lead to this state. +Degree centrality is a trivial, localized measure of node importance based on the number of edges that a node has. In an undirected graph, the degree centrality is predicated solely on the number of edges. However, in the case of a directed graph, a distinction is drawn with a degree centrality oriented on the number of edges coming into a node, and another measure focused on the number of edges leaving a node. Both of these cases provide useful information for compliance graphs. When a node has a large number of other nodes it points to, this node may be prioritized since it creates further opportunity for violation. When a node has a large number of edges pointing to it, this node may be prioritized since the probability that systems may enter this state is higher due to the increased number of ways that a system could lead to this state. \subsection{Betweenness} +Betweenness centrality ranks node importance based on its ability to transfer information flow in a network. For all pairs of nodes in a network, a shortest path is determined. A node that is in this shortest path is considered to have importance. The total betweenness centrality is based on the number of shortest paths that pass through a given node. For compliance graphs, the shortest paths are useful to identify the quickest way that systems may fall out of compliance. By prioritizing the nodes that fall in the highest number of shortest paths, correction schemes can be employed to prolong or prevent systems from falling out of compliance. + +Betweenness centrality is given in Equation \ref{eq:between}, where \textit{i} and \textit{j} are two different, individual nodes in the network, $\sigma_{ij}$ is the total number of shortest paths from \textit{i} to \textit{j}, and $\sigma _{ij}(v)$ is the number of shortest paths that include a node \textit{v}. + +\begin{equation} +\sum_{s \neq v \neq t} \frac{\sigma_{ij}(v)}{\sigma_{ij}} +\label{eq:between} +\end{equation} + \subsection{Katz} +Katz centrality was first introduced by the author of \cite{Katz}, and measures the importance of nodes through all paths in a network, and is not limited to solely the shortest path between any two given nodes. The original work by the author defines Katz as seen in Equation \ref{eq:Katz}. + +\begin{equation} +C_{\mathrm {Katz} }(i)=\sum _{k=1}^{\infty }\sum _{j=1}^{n}\alpha ^{k}(A^{k})_{ji} +\label{eq:Katz} +\end{equation} + \subsection{K-Path Edge} \subsection{Adapted Page Rank} diff --git a/Report/Schrick-Noah_CS-7863_Final-Report.toc b/Report/Schrick-Noah_CS-7863_Final-Report.toc index a9e4728..dd94fa2 100644 --- a/Report/Schrick-Noah_CS-7863_Final-Report.toc +++ b/Report/Schrick-Noah_CS-7863_Final-Report.toc @@ -8,7 +8,7 @@ \contentsline {section}{\numberline {4}Centralities}{4}{}% \contentsline {subsection}{\numberline {4.1}Introduction}{4}{}% \contentsline {subsection}{\numberline {4.2}Degree}{5}{}% -\contentsline {subsection}{\numberline {4.3}Betweenness}{6}{}% +\contentsline {subsection}{\numberline {4.3}Betweenness}{5}{}% \contentsline {subsection}{\numberline {4.4}Katz}{6}{}% \contentsline {subsection}{\numberline {4.5}K-Path Edge}{6}{}% \contentsline {subsection}{\numberline {4.6}Adapted Page Rank}{6}{}% diff --git a/Schrick-Noah_CG-Analysis.R b/Schrick-Noah_CG-Analysis.R index c4e04fa..dd3a4cd 100644 --- a/Schrick-Noah_CG-Analysis.R +++ b/Schrick-Noah_CG-Analysis.R @@ -45,9 +45,25 @@ base_centralities[[3,1]] <- pci.deg %>% sort(decreasing = T) #### Katz car.katz <- katz.cent(car) +nodes <- car.katz %>% order(decreasing=T) +nodes <- head(nodes, 15)-1 +vals <- car.katz %>% sort(decreasing=T) +vals <- head(vals, 15) base_centralities[[1,2]] <- car.katz[rowSums(apply(car.katz,2,is.nan))==0,] %>% sort(decreasing = T) + base_centralities[[2,2]] <- katz.cent(hipaa) %>% sort(decreasing = T) +hipaa.katz <- katz.cent(hipaa) +nodes <- hipaa.katz %>% order(decreasing=T) +nodes <- head(nodes, 15)-1 +vals <- hipaa.katz %>% sort(decreasing=T) +vals <- head(vals, 15) + base_centralities[[3,2]] <- katz.cent(pci) %>% sort(decreasing = T) +pci.katz <- katz.cent(pci) +nodes <- pci.katz %>% order(decreasing=T) +nodes <- head(nodes, 15)-1 +vals <- pci.katz %>% sort(decreasing=T) +vals <- head(vals, 15) ### Page Rank base_centralities[[1,3]] <- page.rank(car)$vector %>% sort(decreasing = T) @@ -159,8 +175,25 @@ tc_centralities[[3,1]] <- pci.tc.deg %>% sort(decreasing = T) #### Katz car.tc.katz <- katz.cent(car.tc) tc_centralities[[1,2]] <- car.tc.katz[rowSums(apply(car.tc.katz,2,is.nan))==0,] %>% sort(decreasing = T) +car.tc.katz <- katz.cent(car.tc) +nodes <- car.tc.katz %>% order(decreasing=T) +nodes <- head(nodes, 15)-1 +vals <- car.tc.katz %>% sort(decreasing=T) +vals <- head(vals, 15) + tc_centralities[[2,2]] <- katz.cent(hipaa.tc) %>% sort(decreasing = T) +hipaa.tc.katz <- katz.cent(hipaa.tc) +nodes <- hipaa.tc.katz %>% order(decreasing=T) +nodes <- head(nodes, 15)-1 +vals <- hipaa.tc.katz %>% sort(decreasing=T) +vals <- head(vals, 15) + tc_centralities[[3,2]] <- katz.cent(pci.tc) %>% sort(decreasing = T) +pci.tc.katz <- katz.cent(pci.tc) +nodes <- pci.tc.katz %>% order(decreasing=T) +nodes <- head(nodes, 15)-1 +vals <- pci.tc.katz %>% sort(decreasing=T) +vals <- head(vals, 15) ### Page Rank tc_centralities[[1,3]] <- page.rank(car.tc)$vector %>% sort(decreasing = T) diff --git a/centralities.R b/centralities.R index 067d14c..a09e90e 100644 --- a/centralities.R +++ b/centralities.R @@ -6,7 +6,10 @@ katz.cent <- function(A, alpha=NULL, beta=NULL){ #NULL sets the default value lam.dom <- eigen(A)$values[1] #dom eigenvec if (is.null(alpha)){ - alpha <- 0.9 * (1/lam.dom) #Set alpha to 90% of max allowed + if(lam.dom == 0) + alpha = 0.1 + else + alpha <- 0.9 * (1/lam.dom) #Set alpha to 90% of max allowed if (is.complex(alpha)){ alpha <- Re(alpha) } @@ -19,6 +22,6 @@ katz.cent <- function(A, alpha=NULL, beta=NULL){ #NULL sets the default value #Katz scores scores <- solve(diag(n) - alpha*A,beta) - + return(scores) }