Degree, Betweenness, and beginning work of Katz Centrality

.~lock.Formatted_Results.ods#

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-,noah,NovaArchSys,02.05.2022 16:43,file:///home/noah/.config/libreoffice/4;
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Report/Bibliography.bib

@@ -1333,4 +1333,27 @@ of 27},
 	Pages = {80},
 	Abstract = {<h4>Background</h4>Numerous centrality measures have been introduced to identify "central" nodes in large networks. The availability of a wide range of measures for ranking influential nodes leaves the user to decide which measure may best suit the analysis of a given network. The choice of a suitable measure is furthermore complicated by the impact of the network topology on ranking influential nodes by centrality measures. To approach this problem systematically, we examined the centrality profile of nodes of yeast protein-protein interaction networks (PPINs) in order to detect which centrality measure is succeeding in predicting influential proteins. We studied how different topological network features are reflected in a large set of commonly used centrality measures.<h4>Results</h4>We used yeast PPINs to compare 27 common of centrality measures. The measures characterize and assort influential nodes of the networks. We applied principal component analysis (PCA) and hierarchical clustering and found that the most informative measures depend on the network's topology. Interestingly, some measures had a high level of contribution in comparison to others in all PPINs, namely Latora closeness, Decay, Lin, Freeman closeness, Diffusion, Residual closeness and Average distance centralities.<h4>Conclusions</h4>The choice of a suitable set of centrality measures is crucial for inferring important functional properties of a network. We concluded that undertaking data reduction using unsupervised machine learning methods helps to choose appropriate variables (centrality measures). Hence, we proposed identifying the contribution proportions of the centrality measures with PCA as a prerequisite step of network analysis before inferring functional consequences, e.g., essentiality of a node.},
 	URL = {https://europepmc.org/articles/PMC6069823},
+}
+
+@Article{Katz,
+  author={Leo Katz},
+  title={{A new status index derived from sociometric analysis}},
+  journal={Psychometrika},
+  year=1953,
+  volume={18},
+  number={1},
+  pages={39-43},
+  month={March},
+  keywords={},
+  doi={10.1007/BF02289026},
+  abstract={No abstract is available for this item.},
+  url={https://ideas.repec.org/a/spr/psycho/v18y1953i1p39-43.html}
+}
+
+@article{ModKatz,
+  title={Katz centrality of Markovian temporal networks: Analysis and optimization},
+  author={Masaki Ogura and Victor M. Preciado},
+  journal={2017 American Control Conference (ACC)},
+  year={2017},
+  pages={5001-5006}
 }

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

@@ -27,9 +27,12 @@
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 \@writefile{toc}{\contentsline {subsection}{\numberline {4.1}Introduction}{4}{}\protected@file@percent }
+\citation{Katz}
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-\@writefile{toc}{\contentsline {subsection}{\numberline {4.3}Betweenness}{6}{}\protected@file@percent }
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+\newlabel{eq:between}{{1}{5}}
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+\newlabel{eq:Katz}{{2}{6}}
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@@ -53,6 +56,7 @@
 \bibcite{noauthor_health_1996}{10}
 \bibcite{PCI}{11}
 \bibcite{PMID:30064421}{12}
+\bibcite{Katz}{13}
 \bibstyle{ieeetr}
 \@writefile{toc}{\contentsline {section}{Bibliography}{7}{}\protected@file@percent }
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Report/Schrick-Noah_CS-7863_Final-Report.bbl

@@ -58,4 +58,8 @@ M.~Ashtiani, A.~Salehzadeh-Yazdi, Z.~Razaghi-Moghadam, H.~Hennig,
   centrality measures for protein-protein interaction networks,'' {\em BMC
   systems biology}, vol.~12, p.~80, July 2018.
 
+\bibitem{Katz}
+L.~Katz, ``{A new status index derived from sociometric analysis},'' {\em
+  Psychometrika}, vol.~18, pp.~39--43, March 1953.
+
 \end{thebibliography}

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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}
 

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}{}%

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)

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)
 }