Table 4 Evaluation of EigenVector Centrality (EVC) for Eq. (2).

\(\begin{aligned} EV1 & = \left[ {\begin{array}{*{20}l} 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill \\ 1 \hfill & 0 \hfill & 1 \hfill & 1 \hfill & 1 \hfill & 0 \hfill \\ 0 \hfill & 1 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 1 \hfill \\ 0 \hfill & 1 \hfill & 1 \hfill & 0 \hfill & 1 \hfill & 1 \hfill \\ 1 \hfill & 1 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 1 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & 1 \hfill & 1 \hfill & 0 \hfill \\ \end{array} } \right]\left[ {\begin{array}{*{20}l} 1 \hfill \\ 1 \hfill \\ 1 \hfill \\ 1 \hfill \\ 1 \hfill \\ 1 \hfill \\ \end{array} } \right] \\ & = \left[ {\begin{array}{*{20}l} 2 \hfill \\ 4 \hfill \\ 3 \hfill \\ 4 \hfill \\ 4 \hfill \\ 3 \hfill \\ \end{array} } \right] \\ \end{aligned}\)

\(\begin{aligned} Normalized value &=n1 \\ &=\sqrt{{2}^{2}+{4}^{2}+{3}^{2}+{4}^{2}+{4}^{2}+{3}^{2}}\\ & =8.3666 \end{aligned}\)

\(\begin{aligned} EVC1 & = \frac{{EV1}}{{n1}} = \left[ {\begin{array}{*{20}l} {0.2390} \hfill \\ {0.4781} \hfill \\ {0.3586} \hfill \\ {0.4781} \hfill \\ {0.4781} \hfill \\ {0.3586} \hfill \\ \end{array} } \right] \\ & \to Iteration\# 1 \end{aligned}\)

\(\begin{aligned} EV3 & = \left[ {\begin{array}{*{20}l} 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill \\ 1 \hfill & 0 \hfill & 1 \hfill & 1 \hfill & 1 \hfill & 0 \hfill \\ 0 \hfill & 1 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 1 \hfill \\ 0 \hfill & 1 \hfill & 1 \hfill & 0 \hfill & 1 \hfill & 1 \hfill \\ 1 \hfill & 1 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 1 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & 1 \hfill & 1 \hfill & 0 \hfill \\ \end{array} } \right]\left[ {\begin{array}{*{20}l} {0.2760} \hfill \\ {0.4485} \hfill \\ {0.3795} \hfill \\ {0.4830} \hfill \\ {0.4485} \hfill \\ {0.3795} \hfill \\ \end{array} } \right] \\ & = \left[ {\begin{array}{*{20}l} {0.8971} \hfill \\ {1.5872} \hfill \\ {1.3111} \hfill \\ {1.6562} \hfill \\ {1.5872} \hfill \\ {1.3111} \hfill \\ \end{array} } \right] \\ \end{aligned}\)

\(\begin{aligned} n3 & = \sqrt {\begin{array}{*{20}l} {0.8971^{2} + 1.5872^{2} + 1.3111^{2} } \\ { + 1.6562^{2} + 1.5872^{2} + 1.3111^{2} } \\ \end{array} } \\ & = 3.4675 \\ \end{aligned}\)

\(\begin{aligned} EVC3 & = \frac{{EV3}}{{n3}} = \left[ {\begin{array}{*{20}l} {0.2587} \hfill \\ {0.4577} \hfill \\ {0.3781} \hfill \\ {0.4776} \hfill \\ {0.4577} \hfill \\ {0.3781} \hfill \\ \end{array} } \right] \\ & \to Iteration\# 3 \end{aligned}\)

\(\begin{aligned} EV2 & = \left[ {\begin{array}{*{20}l} 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill \\ 1 \hfill & 0 \hfill & 1 \hfill & 1 \hfill & 1 \hfill & 0 \hfill \\ 0 \hfill & 1 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 1 \hfill \\ 0 \hfill & 1 \hfill & 1 \hfill & 0 \hfill & 1 \hfill & 1 \hfill \\ 1 \hfill & 1 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 1 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & 1 \hfill & 1 \hfill & 0 \hfill \\ \end{array} } \right]\left[ {\begin{array}{*{20}l} {0.2390} \hfill \\ {0.4781} \hfill \\ {0.3586} \hfill \\ {0.4781} \hfill \\ {0.4781} \hfill \\ {0.3586} \hfill \\ \end{array} } \right] \\ & = \left[ {\begin{array}{*{20}l} {0.9562} \hfill \\ {1.5538} \hfill \\ {1.3148} \hfill \\ {1.6733} \hfill \\ {1.5538} \hfill \\ {1.3148} \hfill \\ \end{array} } \right] \\ \end{aligned}\)

\($$ \begin{aligned} n2 & = \sqrt {\begin{array}{*{20}l} {0.9562^{2} + 1.5538^{2} + 1.3148^{2} } \\ { + 1.6733^{2} + 1.5538^{2} + 1.3148^{2} } \\ \end{array} } \\ & = 3.4641 \\ \end{aligned} $$\)

\(\begin{aligned} EVC2 &= \frac{{EV2}}{{n2}} = \left[ {\begin{array}{*{20}l} {0.2760} \hfill \\ {0.4485} \hfill \\ {0.3795} \hfill \\ {0.4830} \hfill \\ {0.4485} \hfill \\ {0.3795} \hfill \\ \end{array} } \right] \\ &\to Iteration\# 2 \end{aligned}\)

\(\begin{aligned} EV4 & = \left[ {\begin{array}{*{20}l} 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill \\ 1 \hfill & 0 \hfill & 1 \hfill & 1 \hfill & 1 \hfill & 0 \hfill \\ 0 \hfill & 1 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 1 \hfill \\ 0 \hfill & 1 \hfill & 1 \hfill & 0 \hfill & 1 \hfill & 1 \hfill \\ 1 \hfill & 1 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 1 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & 1 \hfill & 1 \hfill & 0 \hfill \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {0.2587} \\ {0.4577} \\ {0.3781} \\ {0.4776} \\ {0.4577} \\ {0.3781} \\ \end{array} } \right] \\ & = \left[ {\begin{array}{*{20}c} {0.9154} \\ {1.5722} \\ {1.3134} \\ {1.6717} \\ {1.5722} \\ {1.3134} \\ \end{array} } \right] \\ \end{aligned}\)

\(\begin{aligned} n4 & = \sqrt {\begin{array}{*{20}l} {0.9154^{2} + 1.5722^{2} + 1.3134^{2} } \\ { + 1.6717^{2} + 1.5722^{2} + 1.3134^{2} } \\ \end{array} } \\ & = 3.4679 \\ \end{aligned} $$\)

\(\begin{aligned} EVC4 &= \frac{{EV4}}{{n4}} = \left[ {\begin{array}{*{20}l} {0.2640} \hfill \\ {0.4533} \hfill \\ {0.3787} \hfill \\ {0.4820} \hfill \\ {0.4533} \hfill \\ {0.3787} \hfill \\ \end{array} } \right] \\ & \to Iteration\# 4 \end{aligned}\)

\(\begin{aligned} EV5 & = \left[ {\begin{array}{*{20}l} 0 \hfill & 1 \hfill & 0 \hfill & 0 \hfill & 1 \hfill & 0 \hfill \\ 1 \hfill & 0 \hfill & 1 \hfill & 1 \hfill & 1 \hfill & 0 \hfill \\ 0 \hfill & 1 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 1 \hfill \\ 0 \hfill & 1 \hfill & 1 \hfill & 0 \hfill & 1 \hfill & 1 \hfill \\ 1 \hfill & 1 \hfill & 0 \hfill & 1 \hfill & 0 \hfill & 1 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & 1 \hfill & 1 \hfill & 0 \hfill \\ \end{array} } \right]\left[ {\begin{array}{*{20}l} {0.2640} \hfill \\ {0.4533} \hfill \\ {0.3787} \hfill \\ {0.4820} \hfill \\ {0.4533} \hfill \\ {0.3787} \hfill \\ \end{array} } \right] \\ & = \left[ {\begin{array}{*{20}l} {0.9067} \hfill \\ {1.5781} \hfill \\ {1.3141} \hfill \\ {1.6642} \hfill \\ {1.5781} \hfill \\ {1.3141} \hfill \\ \end{array} } \right] \\ \end{aligned}\)

\(\begin{aligned} n5 =\sqrt{{0.9067}^{2}+{1.5781}^{2}+{1.3141}^{2}+{1.6642}^{2}+{1.5781}^{2}+{1.3141}^{2}}\\ =3.4679 \to \lambda (Principal Eigenvalue) \end{aligned}\)

\(\begin{aligned} EVC5 = \frac{{EV5}}{{n5}} = \left[ {\begin{array}{*{20}l} {0.2615} \hfill \\ {0.4551} \hfill \\ {0.3789} \hfill \\ {0.4799} \hfill \\ {0.4551} \hfill \\ {0.3789} \hfill \\ \end{array} } \right] \\ \to Iteration\# 5 \end {aligned}\)