Growth of bilinear maps

For a bilinear map $*:\mathbb R^d\times \mathbb R^d\to R^d$ of nonnegative coefficients and vector $s\in positive entries, among an exponentially number ways combining $n$ instances $s$ using $n-1$ applications $*$ for given $n$, we are interested in the largest entry over all resulting vectors. An asymptotic behavior is that $n$-th root this converges to growth rate $\lambda$ when tends infinity. In paper, prove existence limit by special structure called linear pattern. We also pose question on possibility relation between whether algebraic.