Graph Global Overlap Measure: Katz Index

The Katz index is

$$ \mathbf S_{\text{Katz}}[u,v] = \sum_{i=1}^\infty \beta^i \mathbf A^i[u, v], $$

where $\mathbf A^i[u, v]$ is the matrix $\mathbf A$ to the $i$th power. Some for $\beta^i$. The Katz index describes the similarity between of node $u$ and node $v$.

The index is proved to be the following

$$ \mathbf S_{\text{Katz}} = (\mathbf I - \beta \mathbf A)^{-1} - \mathbf I. $$

Parameter $\beta$

The parameter $\beta$ is the punishment for path length. If $\beta < 1$, each increase in power will make the whole term $\beta^i A^i$ smaller. If a path between $u$ and $v$ is longer, i.e., $i$ is larger, then the weight $\beta^i$ is smaller.