## root / latex / report_feb_16.tex @ 4ca27bae

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1 | 46d9d2ec | Quynh PX Nguyen | \documentclass[12pt]{article} |
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2 | % This first part of the file is called the PREAMBLE. It includes |
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3 | % customizations and command definitions. The preamble is everything |
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4 | % between \documentclass and \begin{document}. |
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6 | \usepackage[margin=1in]{geometry} % set the margins to 1in on all sides |
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7 | \usepackage{graphicx} % to include figures |
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8 | \graphicspath{ {report_images/} } % '/' is needed |
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9 | \usepackage{epstopdf} % for the .ps image |
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10 | \epstopdfsetup{update} |
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11 | \DeclareGraphicsExtensions{.eps} |
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12 | \epstopdfDeclareGraphicsRule{.eps}{pdf}{.pdf}{ps2pdf -dEPSCrop -dNOSAFER #1 \OutputFile} |
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13 | \usepackage[noabbrev,capitalise]{cleveref} |
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63 | \newcommand{\comb}[2]{\binom{#1^2 + #2^2}{#1+#2}} |
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65 | \usepackage[T1]{fontenc} |
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70 | } |
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72 | \usepackage{booktabs} % To thicken table lines |
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74 | \usepackage{algorithm} |
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75 | \usepackage[noend]{algpseudocode} % For pseudocode |
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76 | |||

77 | \begin{document} |
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78 | |||

79 | \title{Thesis - Report Feb 10 - Heuristic Betweenness Centrality for Weighted Graph} |
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80 | |||

81 | \author{Quynh Nguyen - DISI - University of Trento} |
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82 | |||

83 | \maketitle |
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84 | \clearpage |
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85 | |||

86 | \section{Introduction} |
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87 | \subsection{Notation} |
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88 | \textbf{BBC}: the score obtained by Brandes Betweenness Centrality |
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89 | |||

90 | \textbf{BBCT}: the score obtained by Brandes BC with the inclusion of targets |
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91 | |||

92 | \textbf{HBCU}: the heuristic way to obtain BC score, for unweighted graph |
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93 | |||

94 | \textbf{HBCW}: the heuristic way to obtain BC score, for weighted graph |
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95 | |||

96 | \subsection{Dataset} |
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97 | There are 5 datasets in total: |
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98 | |||

99 | \begin{itemize} |
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100 | \item Unweighted graphs |
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101 | \begin{itemize} |
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102 | \item \emph{simple}: this is the graph getting from paper \cite{PuzisZEDB12} |
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103 | \item \emph{ninux_unweighted_connected}: all the vertices and edges are the same as \emph{ninux_30_1}, but the costs for all edges are 1. It's |
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104 | \end{itemize} |
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105 | \item Weighted graphs |
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106 | \begin{itemize} |
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107 | \item \emph{ninux_30_1} |
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108 | \item \emph{olsr-netjson} |
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109 | \item \emph{jsoninfo_topo} |
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110 | \end{itemize} |
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111 | \end{itemize} |
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112 | |||

113 | \subsection{Comparison between BBC, BBCT, HBCU} |
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114 | We have BBC $\sim$ HBCU, there are constant difference between the scores obtained by these 2 methods. |
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115 | |||

116 | And we have BBCT = HBCU. See \ref{fig:feb16_jsoninfo_topo_unweighted} |
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117 | |||

118 | \begin{figure}[h] |
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119 | \caption{BBCT \& HBCU for jsoninfo_topo. Their results are the same} |
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120 | \label{fig:feb16_jsoninfo_topo_unweighted} |
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121 | \includegraphics[]{feb16_jsoninfo_topo_unweighted.eps} |
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122 | \end{figure} |
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123 | |||

124 | \subsection{Purpose} |
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125 | This report will |
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126 | \begin{itemize} |
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127 | \item Show how HBCW is obtained |
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128 | \item Comparing HBCU and HBCW |
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129 | \end{itemize} |
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130 | |||

131 | \section{HBC for weighted graph} |
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132 | \subsection{Method} |
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133 | When calculating the BC for each bi-connected component, \emph{weight_map} is included. As a result, HBCW will take into account the weight of each edge when calculating the number of shortest paths $\sigma$. And the $\sigma$ will affect the final BC score. |
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134 | |||

135 | \subsection{Result} |
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136 | For \textbf{unweighted graph}, BBCT = HBCW. |
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137 | |||

138 | For \textbf{weighted graph}, BBCT = HBCW with these dataset: |
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139 | \begin{itemize} |
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140 | \item \emph{ninux_30_1}. See \ref{fig:feb16_ninux_30_1_weighted} |
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141 | \item \emph{olsr-netjson} |
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142 | \end{itemize} |
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143 | |||

144 | And BBCT != HBCW in the dataset \emph{jsoninfo_topo}. See \ref{fig:feb16_jsoninfo_topo_weighted}. Compare to the result obtained by HBCU in \ref{fig:feb16_jsoninfo_topo_unweighted} |
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145 | |||

146 | \begin{figure}[h] |
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147 | \caption{BBCT \& HBCW for ninux_30_1} |
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148 | \label{fig:feb16_ninux_30_1_weighted} |
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149 | \includegraphics[]{feb16_ninux_30_1_weighted.eps} |
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150 | \end{figure} |
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151 | \begin{figure}[h] |
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152 | \caption{BBCT \& HBCW for jsoninfo_topo. Their results are different} |
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153 | \label{fig:feb16_jsoninfo_topo_weighted} |
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154 | \includegraphics[]{feb16_jsoninfo_topo_weighted.eps} |
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155 | \end{figure} |
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156 | |||

157 | \clearpage |
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158 | \bibliography{references}{} |
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159 | \bibliographystyle{plain} |
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160 | |||

161 | \end{document} |