LHN similarity is defined as:

$\mathbf{S}_{LNH} [u,v]=\mathbf{I}[u,v]+\frac{2m}{d_ud_v}\sum_{i=1}^{\infty}\beta^i \lambda_1^{1-i} \mathbf{A}^i [u,v]$

where $d_u$ and $d_v$ are degree of nodes $u$ and $v$, and $\lambda_1$ is the largest eigenvalue, and $m$ is total number of edges in the graph. And it can be proved that:

$\mathbf{S}_{LNH}=2\alpha m\lambda_1\mathbf{D}^{-1}(\mathbf{I}-\frac{\beta}{\lambda_1} \mathbf{A})^{-1}\mathbf{D}$

The idea of LHN is that, becasue Katz similarity gives much more higher scores for high degree nodes, it wants to solve this issue by normalizing actual number of observed path using expected number of path, whcich is $\frac{\mathbf{A}^i}{\mathbb{E}[\mathbf{A}^i]}$. And $\mathbb{E}[\mathbf{A}^i]$ can be estimated through $d_u, d_v$ and $\lambda_1$.