Table 1.

Local, Quasi-local, and Global Similarity indices.

Local index	Matrix form	Global extension	Quasi-local extension
$CN(u,v)=\left|\mathrm{\Gamma },(,u,),\cap ,\mathrm{\Gamma },(,v,)\right|$	$A^2$	${\left(I,-,\beta ,A\right)}^{-1}-I$	$A^2+\beta A^3$
$RA(u,v)={\sum }_{w\in \{\mathrm{\Gamma }(u)\cap \mathrm{\Gamma }(v)\}}\frac{1}{\left|\mathrm{\Gamma },(,w,)\right|}$	$A{D}^{-1}A$	$A{\left(I,-,\beta ,D^{-1},A\right)}^{-1}-A$	$A{D}^{-1}A+\beta A{D}^{-1}A{D}^{-1}A$
$AA(u,v)={\sum }_{w\in \{\mathrm{\Gamma }(u)\cap \mathrm{\Gamma }(v)\}}\frac{1}{log\left|\mathrm{\Gamma },(,w,)\right|}$	$A{\left(log,D\right)}^{-1}A$	$A{\left(I,-,\beta ,{(logD)}^{-1},A\right)}^{-1}-A$	$A{\left(log,D\right)}^{-1}A+\beta A{\left(log,D\right)}^{-1}A{\left(log,D\right)}^{-1}A$
$SO(u,v)=\frac{2\left|\mathrm{\Gamma },(,u,),\cap ,\mathrm{\Gamma },(,v,)\right|}{|\mathrm{\Gamma }(u)|+|\mathrm{\Gamma }(v)|}$	$2{\left(D,{\left(A^2\right)}_{\mathit{ij}}^{-1},+,{\left(A^2\right)}_{\mathit{ij}}^{-1},D\right)}_{\mathit{ij}}^{-1}$	$2{\left(D,{\left({\left(I,-,\beta ,A\right)}^{-1},-,I\right)}_{\mathit{ij}}^{-1},+,{\left({\left(I,-,\beta ,A\right)}^{-1},-,I\right)}_{\mathit{ij}}^{-1},D\right)}_{\mathit{ij}}^{-1}$	$2{\left(D,{\left(A^2,+,\beta ,A^3\right)}_{\mathit{ij}}^{-1},+,{\left(A^2,+,\beta ,A^3\right)}_{\mathit{ij}}^{-1},D\right)}_{\mathit{ij}}^{-1}$
$SA(u,v)=\frac{\left|\mathrm{\Gamma },(,u,),\cap ,\mathrm{\Gamma },(,v,)\right|}{\sqrt{|\mathrm{\Gamma }(u)|\times |\mathrm{\Gamma }(v)|}}$	$D^{-\frac{1}{2}}{A}^2{D}^{-\frac{1}{2}}$	$D^{-\frac{1}{2}}\left({\left(I,-,\beta ,A\right)}^{-1},-,I\right)D^{-\frac{1}{2}}$	$D^{-\frac{1}{2}}{A}^2{D}^{-\frac{1}{2}}+\beta D^{-\frac{1}{2}}{A}^3{D}^{-\frac{1}{2}}$
$LHN(u,v)=\frac{\left|\mathrm{\Gamma },(,u,),\cap ,\mathrm{\Gamma },(,v,)\right|}{|\mathrm{\Gamma }(u)|\times |\mathrm{\Gamma }(v)|}$	$D^{-1}{A}^2{D}^{-1}$	$D^{-1}\left({\left(I,-,\beta ,A\right)}^{-1},-,I\right)D^{-1}$	$D^{-1}{A}^2{D}^{-1}+\beta D^{-1}{A}^3{D}^{-1}$
$HP(u,v)=\frac{\left|\mathrm{\Gamma },(,u,),\cap ,\mathrm{\Gamma },(,v,)\right|}{min(|\mathrm{\Gamma }(u)|,|\mathrm{\Gamma }(v)|)}$	${\left(min,\left(D,{\left(A^2\right)}_{\mathit{ij}}^{-1},,,{\left(A^2\right)}_{\mathit{ij}}^{-1},D\right)\right)}_{\mathit{ij}}^{-1}$	${\left(min,\left(D,{\left({\left(I,-,\beta ,A\right)}^{-1},-,I\right)}_{\mathit{ij}}^{-1},,,{\left({\left(I,-,\beta ,A\right)}^{-1},-,I\right)}_{\mathit{ij}}^{-1},D\right)\right)}_{\mathit{ij}}^{-1}$	${\left(min,\left(D,{\left(A^2,+,\beta ,A^3\right)}_{\mathit{ij}}^{-1},,,{\left(A^2,+,\beta ,A^3\right)}_{\mathit{ij}}^{-1},D\right)\right)}_{\mathit{ij}}^{-1}$
$HD(u,v)=\frac{\left|\mathrm{\Gamma },(,u,),\cap ,\mathrm{\Gamma },(,v,)\right|}{max(|\mathrm{\Gamma }(u)|,|\mathrm{\Gamma }(v)|)}$	${\left(max,\left(D,{\left(A^2\right)}_{\mathit{ij}}^{-1},,,{\left(A^2\right)}_{\mathit{ij}}^{-1},D\right)\right)}_{\mathit{ij}}^{-1}$	${\left(max,\left(D,{\left({\left(I,-,\beta ,A\right)}^{-1},-,I\right)}_{\mathit{ij}}^{-1},,,{\left({\left(I,-,\beta ,A\right)}^{-1},-,I\right)}_{\mathit{ij}}^{-1},D\right)\right)}_{\mathit{ij}}^{-1}$	${\left(max,\left(D,{\left(A^2,+,\beta ,A^3\right)}_{\mathit{ij}}^{-1},,,{\left(A^2,+,\beta ,A^3\right)}_{\mathit{ij}}^{-1},D\right)\right)}_{\mathit{ij}}^{-1}$

Here A represents the adjacency matrix of the network, I is the identity matrix with size equal to the size of the matrix A, and D represents the diagonal degree matrix whose ith diagonal element is the degree of the ith node of the graph. Furthermore, $A^{-1}$ represents the inverse of the matrix A, while $A_{\mathit{ij}}^{-1}$ represents the element-wise inverse operation.