Table 5 Evaluation of Katz Centrality for Fig. 2b.
From: Dangling centrality highlights critical nodes by evaluating network stability through link removal
\(\begin{aligned} C_{{katz}} & = 1*\left( {\left[ {\begin{array}{*{20}l} 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill \\ \end{array} } \right]} \right. \\ & \quad - 0.2{\text{*}}\left. {\left[ {\begin{array}{*{20}l} 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill \\ 1 \hfill & 0 \hfill & 1 \hfill & 1 \hfill & 1 \hfill & 0 \hfill \\ 0 \hfill & 1 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 1 \hfill \\ 0 \hfill & 1 \hfill & 1 \hfill & 0 \hfill & 1 \hfill & 1 \hfill \\ 1 \hfill & 1 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 1 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & 1 \hfill & 1 \hfill & 0 \hfill \\ \end{array} } \right]} \right)^{{ - 1}} *\left[ {\begin{array}{*{20}l} 1 \hfill \\ 1 \hfill \\ 1 \hfill \\ 1 \hfill \\ 1 \hfill \\ 1 \hfill \\ \end{array} } \right] \\ \end{aligned}\) \(\begin{aligned} C_{{katz}} & = \left( {\left[ {\begin{array}{*{20}l} 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill \\ \end{array} } \right]} \right. \\ & \quad - \left. {\left[ {\begin{array}{*{20}l} {0.0} \hfill & {0.2} \hfill & {0.0} \hfill & {0.0} \hfill & {0.2} \hfill & {0.0} \hfill \\ {0.2} \hfill & {0.0} \hfill & {0.2} \hfill & {0.2} \hfill & {0.2} \hfill & {0.0} \hfill \\ {0.0} \hfill & {0.2} \hfill & {0.0} \hfill & {0.2} \hfill & {0.0} \hfill & {0.2} \hfill \\ {0.0} \hfill & {0.2} \hfill & {0.2} \hfill & {0.0} \hfill & {0.2} \hfill & {0.2} \hfill \\ {0.2} \hfill & {0.2} \hfill & {0.0} \hfill & {0.2} \hfill & {0.0} \hfill & {0.2} \hfill \\ {0.0} \hfill & {0.0} \hfill & {0.2} \hfill & {0.2} \hfill & {0.2} \hfill & 0 \hfill \\ \end{array} } \right]} \right)^{{ - 1}} *\left[ {\begin{array}{*{20}l} 1 \hfill \\ 1 \hfill \\ 1 \hfill \\ 1 \hfill \\ 1 \hfill \\ 1 \hfill \\ \end{array} } \right] \\ \end{aligned}\) \(\begin{aligned} C_{{katz}} & = \left( {\left[ {\begin{array}{*{20}l} {1.00} \hfill & { - 0.2} \hfill & { - 0.2} \hfill & { - 0.2} \hfill & {0.00} \hfill \\ { - 0.2} \hfill & {1.00} \hfill & { - 0.2} \hfill & {0.00} \hfill & {0.00} \hfill \\ { - 0.2} \hfill & { - 0.2} \hfill & {1.00} \hfill & {0.00} \hfill & {0.00} \hfill \\ { - 0.2} \hfill & {0.00} \hfill & {0.00} \hfill & {1.00} \hfill & { - 0.2} \hfill \\ {0.00} \hfill & {0.00} \hfill & {0.00} \hfill & { - 0.2} \hfill & {1.00} \hfill \\ \end{array} } \right]} \right)^{{ - 1}} *\left[ {\begin{array}{*{20}l} 1 \hfill \\ 1 \hfill \\ 1 \hfill \\ 1 \hfill \\ 1 \hfill \\ 1 \hfill \\ \end{array} } \right] \\ & \quad = \left[ {\begin{array}{*{20}l} {2.4059} \hfill \\ {3.5146} \hfill \\ {3.0335} \hfill \\ {3.6192} \hfill \\ {3.5146} \hfill \\ {3.0335} \hfill \\ \end{array} } \right] \\ \end{aligned}\) |
