## root / latex / report_feb_10.tex @ 4ca27bae

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\documentclass[12pt]{article} |
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\usepackage{booktabs} % To thicken table lines |

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\usepackage{algorithm} |

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\usepackage[noend]{algpseudocode} % For pseudocode |

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\begin{document} |

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\title{Thesis - Report Feb 10 - Heuristic Betweenness Centrality vs Brandes Betweenness Centrality} |

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\author{Quynh Nguyen - DISI - University of Trento} |

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\maketitle |

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\clearpage |

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\section{The discrepency between Heuristic Betweenness Centrality (HBC) and Brandes Betweenness Centrality (BBC)} |

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\textbf{Summary:} the gap between the default version (of both networkx and BGL) to calculate BC is due to the inclusion of the source. |

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Algorithm 2 in \cite{Brandes2008136} show the variant of calculating betweenness centrality with endpoints included. The paper also mentions that we can "restrict the inclusion of endpoints to either sources or targets". |

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\begin{algorithm} |

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\caption{Including endpoints (both sources + targets)}\label{accumulation_endpoints} |

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\begin{algorithmic}[1] |

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\Procedure{accumulation}{} |

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\State $c_B[s] \leftarrow c_B[s] + (|S| - 1)$ |

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\For{$v \in V$} |

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$\delta[v] \leftarrow 0$ |

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\EndFor |

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\While{$S$ not empty} |

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\State pop $w \leftarrow S$ |

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\For{$v \in Pred[w]$} |

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$\delta[v] \leftarrow \delta[v] + \frac{\sigma[v]}{\sigma[w]} (1 + \delta[w])$ |

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\EndFor |

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\If{$w \neq s$} |

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$c_B[w] \leftarrow c_B[w] + \delta[w] + 1$ \Comment{$w$ is target of $s$ once} |

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\EndIf |

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\EndWhile |

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\EndProcedure |

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\end{algorithmic} |

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\end{algorithm} |

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\begin{algorithm} |

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\caption{Including targets}\label{accumulation_targets} |

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\begin{algorithmic}[1] |

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\Procedure{accumulation}{} |

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\For{$v \in V$} |

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$\delta[v] \leftarrow 0$ |

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\EndFor |

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\While{$S$ not empty} |

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\State pop $w \leftarrow S$ |

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\For{$v \in Pred[w]$} |

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$\delta[v] \leftarrow \delta[v] + \frac{\sigma[v]}{\sigma[w]} (1 + \delta[w])$ |

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\EndFor |

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\If{$w \neq s$} |

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$c_B[w] \leftarrow c_B[w] + \delta[w] + 1$ \Comment{$w$ is target of $s$ once} |

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\EndIf |

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\EndWhile |

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\EndProcedure |

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\end{algorithmic} |

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\end{algorithm} |

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\begin{algorithm} |

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\caption{Heuristic Betweenness Centrality}\label{accumulation_heuristic} |

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\begin{algorithmic}[1] |

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\Procedure{accumulation}{} |

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\For{$v \in V$} |

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$\delta[v] \leftarrow 0$ |

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\EndFor |

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\While{$S$ not empty} |

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\State pop $w \leftarrow S$ |

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\State $h \leftarrow $ get_the_communication_intensity_between $(w, s)$ |

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\For{$v \in Pred[w]$} |

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$\delta[v] \leftarrow \delta[v] + \frac{\sigma[v]}{\sigma[w]} (h + \delta[w])$ |

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\EndFor |

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\If{$w \neq s$} |

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$c_B[w] \leftarrow c_B[w] + \delta[w] + h$ |

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\EndIf |

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\EndWhile |

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\EndProcedure |

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\end{algorithmic} |

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\end{algorithm} |

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\begin{figure}[h] |

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\caption{Comparison between 3 different accumulation functions \ref{accumulation_endpoints}, \ref{accumulation_targets}, \ref{accumulation_heuristic}} |

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\label{fig:comparison_including_either_source_target} |

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\includegraphics[]{comparison_including_either_source_target.eps} |

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\end{figure} |

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\section{The BC inter} |

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Lemma 4.2 in \cite{PuzisZEDB12} shows that the BC for articulation points are the sum of BC in each bi-connected components, subtracted by the \texttt{BC inter}. However, when I subtract the \texttt{BC inter} from the \texttt{BC sum}, the results are not meaningful as shown in figure \ref{fig:substracted_by_bc_inter} and \ref{fig:not_substracted_by_bc_inter} |

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\begin{figure}[h] |

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\caption{Heuristic BC is NOT subtracted by BC inter} |

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\label{fig:not_substracted_by_bc_inter} |

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\includegraphics[]{feb10_bc_inter.eps} |

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\end{figure} |

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\begin{figure}[h] |

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\caption{Heuristic BC is subtracted by BC inter} |

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\label{fig:substracted_by_bc_inter} |

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\includegraphics[]{feb10_subtracted_by_bc_inter.eps} |

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\end{figure} |

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\clearpage |

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\bibliography{references}{} |

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\bibliographystyle{plain} |

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\end{document} |