Scattering matrix of matched a device with matched ports
The S matrix is only defined properly when all its ports are terminated by a matched load. The definition in Pozar is $$S_{ij}=\frac{V_{i}^{-}}{V_{j}^{+}}\Biggr|_{V_{k}^{+}=0\,\text{for}\,k\neq j}$$ \$V^+\$ for incoming waves and \$V^-\$ for outgoing (reflected) waves

The power of the S matrix is that it defines how your n-port behaves together with your other system components at the same characteristic impedance. This makes us able to build system blocks, then connect them together knowing how they will behave. 

If the n-port is not terminated properly, since it is a linear definition, the output on a port will be a superposition of the signal and reflections from the connected network(s) 

To summarize, the actual S parameters are linear, thus they are independent of the voltage across them (no saturation).
But the assumption that the scattering parameters of a device will be the same no matter to what network it is connected is not correct, to quote the book. 
"An important point to understand about scattering parameters is that the reflection coefficient looking into port n is not equal to \$S_{nn}\$ unless all other ports are matched"