Table 1 Formula for link prediction methods used in this paper. Here \(\alpha\), \(\beta\), c and t are the tunable parameters. \(\Gamma _i\) represents the set of neighbours of node i, while \(|\Gamma _i|\) represents the degree of the node i. \([A]_{ij}\) represents the (i, j)-th entry of the matrix A, \(A^T\) represents the transpose of the matrix A, and \(d_{ij}\) is the length of the shortest distance between node i and node j. P is the transition matrix whose (i, j)-th entry represents the probability that a random walker staying at node i will walk to j in the next step. It is computed as \(P=D^{-1}A\). \(\mathbf {e_i}\) is an \(n\times 1\) vector with the i-th element equal to 1 and remaining are 0.
From: Link prediction in complex network using information flow
Method | Formula |
|---|---|
Common neighbour | \(CN(i,j) = |\Gamma _i \cap \Gamma _j |= [A^2]_{ij}\) |
Adamic adar | \(AA(i,j) = \sum _{k \in \{ \Gamma _i\cap \Gamma _j \}}\frac{1}{\log |\Gamma _k|}\) |
Resource allocation | \(RA(i,j) = \sum _{k \in \{ \Gamma _i \cap \Gamma _j \}}\frac{1}{|\Gamma _k|}\) |
Preferential attachement | \(PA(i,j) = |\Gamma _i|\times |\Gamma _j|\) |
Matrix forest index | \(MFI(i,j) = \left( I+L\right) ^{-1}\) |
Linear optimisation | \(LO(i,j) = A\left( \alpha \left( \alpha A^TA+I\right) ^{-1}A^TA\right)\) |
CN and distance | \(CND(i,j) = \frac{\left|\Gamma _i\cap \Gamma _j\right|}{2}+\frac{1-A(i,j)}{d_{i,j}}\) |
Superposed random walk | \(SRW(i,j) = \sum _{l=1}^{t} \left( \frac{|\Gamma _i|\pi _{ij}(t)}{2|E |} +\frac{|\Gamma _j|\pi _{ji}(t)}{2|E|} \right)\) where \(\mathbf {\pi _i}(t)=P^T\mathbf {\pi _i}(t-1)\) |
Random walk with restart | \(RWR = q_{xy}+q_{yx}\) where \({q_x}=(1-c)\left( I-cP^T\right) ^{-1}{e_x}\) |
Katz index | \(Katz(i,j) = \left( I-\beta A\right) ^{-1}-I\) |
Local path | \(LP(i,j)=[A^2+\beta A^3]_{ij}\) |
