Table 1 Local, Quasi-local, and Global Similarity indices.
From: Path-based extensions of local link prediction methods for complex networks
Local index | Matrix form | Global extension | Quasi-local extension |
|---|---|---|---|
\(CN(u,v) = \left| \Gamma (u)\cap \Gamma (v)\right| \) | \(A^2\) | \(\left( I-\beta A\right) ^{-1}-I\) | \(A^2 + \beta A^3\) |
\(RA(u,v) = \sum _{w \in \{ \Gamma (u)\cap \Gamma (v) \}}\frac{1}{ \left| \Gamma (w)\right| }\) | \(AD^{-1}A\) | \(A\left( I-\beta D^{-1}A\right) ^{-1}-A\) | \(AD^{-1}A+\beta AD^{-1}AD^{-1}A\) |
\(AA(u,v) = \sum _{w \in \{ \Gamma (u)\cap \Gamma (v) \}}\frac{1}{\log \left| \Gamma (w)\right| }\) | \(A\left( \log D\right) ^{-1}A\) | \(A\left( I-\beta (\log D)^{-1}A\right) ^{-1}-A\) | \(A\left( \log D\right) ^{-1}A+\beta A\left( \log D\right) ^{-1}A\left( \log D\right) ^{-1}A\) |
\(SO(u,v) = \frac{2\left| \Gamma (u)\cap \Gamma (v)\right| }{|\Gamma (u)|+|\Gamma (v)|}\) | \(2\left( D\left( A^2\right) _{ij}^{-1}+\left( A^2\right) _{ij}^{-1}D\right) _{ij}^{-1}\) | \(2\left( D\left( \left( I-\beta A\right) ^{-1}-I\right) _{ij}^{-1}+\left( \left( I-\beta A\right) ^{-1}-I\right) _{ij}^{-1}D\right) _{ij}^{-1}\) | \(2\left( D\left( A^2+\beta A^3\right) _{ij}^{-1}+\left( A^2+\beta A^3\right) _{ij}^{-1}D\right) _{ij}^{-1}\) |
\(SA(u,v) = \frac{\left| \Gamma (u)\cap \Gamma (v)\right| }{\sqrt{|\Gamma (u)|\times |\Gamma (v)|}}\) | \( D^{-\frac{1}{2}}A^2D^{-\frac{1}{2}} \) | \(D^{-\frac{1}{2}}\left( \left( I-\beta A\right) ^{-1}-I\right) D^{-\frac{1}{2}}\) | \(D^{-\frac{1}{2}}A^2D^{-\frac{1}{2}}+\beta D^{-\frac{1}{2}}A^3D^{-\frac{1}{2}}\) |
\(LHN(u,v) = \frac{\left| \Gamma (u)\cap \Gamma (v)\right| }{|\Gamma (u)|\times |\Gamma (v)|}\) | \( D^{-1}A^2D^{-1} \) | \(D^{-1}\left( \left( I-\beta A\right) ^{-1}-I\right) D^{-1}\) | \( D^{-1}A^2D^{-1}+\beta D^{-1}A^3D^{-1}\) |
\(HP(u,v) = \frac{\left| \Gamma (u)\cap \Gamma (v)\right| }{\min (|\Gamma (u)|, |\Gamma (v)|)} \) | \(\left( \min \left( D\left( A^2\right) _{ij}^{-1},\left( A^2\right) _{ij}^{-1}D\right) \right) _{ij}^{-1}\) | \(\left( \min \left( D\left( \left( I-\beta A\right) ^{-1}-I\right) _{ij}^{-1},\left( \left( I-\beta A\right) ^{-1}-I\right) _{ij}^{-1}D\right) \right) _{ij}^{-1}\) | \(\left( \min \left( D\left( A^2+\beta A^3\right) _{ij}^{-1},\left( A^2+\beta A^3\right) _{ij}^{-1}D\right) \right) _{ij}^{-1}\) |
\(HD(u,v) = \frac{\left| \Gamma (u)\cap \Gamma (v)\right| }{\max (|\Gamma (u)|, |\Gamma (v)|)} \) | \(\left( \max \left( D\left( A^2\right) _{ij}^{-1},\left( A^2\right) _{ij}^{-1}D\right) \right) _{ij}^{-1}\) | \(\left( \max \left( D\left( \left( I-\beta A\right) ^{-1}-I\right) _{ij}^{-1},\left( \left( I-\beta A\right) ^{-1}-I\right) _{ij}^{-1}D\right) \right) _{ij}^{-1}\) | \(\left( \max \left( D\left( A^2+\beta A^3\right) _{ij}^{-1},\left( A^2+\beta A^3\right) _{ij}^{-1}D\right) \right) _{ij}^{-1}\) |
