Simrank is a recursive similarity measure between two nodes in a graph. Below is the Simrank definition for nodes x and y.
$$ S(x,y) = \begin{cases} 1, & \text{if } x=y \newline 0, & \text{if } |I(x)|=0 \text{ or } |I(y)|=0 \newline \frac{C}{|I(x)| |I(y)|} \sum_{a\in I(x)} \sum_{b\in I(y)} S(a,b), & \text{otherwise} \end{cases} $$
Simrank distance is defined as $d(x, y) = 1 - S(x, y)$.
Prove that: $d(x, y) + d(y, z) \ge d(x, z)$
I was able to work upto: $S(x, y) + S(y, z) - S(x, z) \le 1$.
I have been stuck after that equation. Any help is appreciated.