Similarly we can look at the expected _Staked Dilution_ (i.e. _Adjusted Staking Yield_) and _Un-staked Dilution_ as previously defined. Again, _dilution_ in this context is defined as the change in fractional representation (i.e. ownership) of a set of tokens within a larger set. In this sense, dilution can be a positive value: an increase in fractional ownership (staked dilution / _Adjusted Staking Yield_), or a negative value: a decrease in fractional ownership (un-staked dilution).
We are interested in the relative change in ownership of staked vs un-staked tokens as the overall token pool increases with inflation issuance. As discussed, this issuance is distributed only to staked token holders, increasing the staked token fractional representation of the _Total Current Supply_.
Due to this relative change in representation, the proportion of stake of any token holder will also change as a function of the _Inflation Schedule_ and the proportion of all tokens that are staked.
Of initial interest, however, is the _dilution of **un-staked** tokens_, or $D_{us}$. In the case of un-staked tokens, token dilution is only a function of the _Inflation Schedule_ because the amount of un-staked tokens doesn't change over time.
This can be seen by explicitly calculating un-staked dilution as $D_{us}$. The un-staked proportion of the token pool at time $t$ is $P_{us}(t_{N})$ and $I_{t}$ is the incremental inflation rate applied between any two consecutive time points. $SOL_{us}(t)$ and $SOL_{total}(t)$ is the amount of un-staked and total SOL on the network, respectively, at time $t$. Therefore $P_{us}(t) = SOL_{us}(t)/SOL_{total}(t)$.
So as guessed, this dilution is independent of the total proportion of staked tokens and only depends on inflation rate. This can be seen with our example _Inflation Schedule_ here:
We can do a similar calculation to determine the _dilution_ of staked token holders, or as we've defined here as the **_Adjusted Staked Yield_**, keeping in mind that dilution in this context is an _increase_ in proportional ownership over time. We'll use the terminology _Adjusted Staked Yield_ to avoid confusion going forward.
To see the functional form, we calculate, $Y_{adj}$, or the _Adjusted Staked Yield_ (to be compared to _D\_{us}_ the dilution of un-staked tokens above), where $P_{s}(t)$ is the staked proportion of token pool at time $t$ and $I_{t}$ is the incremental inflation rate applied between any two consecutive time points. The definition of $Y_{adj}$ is therefore:
As seen in the plot above, the proportion of staked tokens increases with inflation issuance. Letting $SOL_s(t)$ and $SOL_{\text{total}}(t)$ represent the amount of staked and total SOL at time $t$ respectively:
Where $SOL_{\text{total}}(t_1)\times I(t_1)$ is the additional inflation issuance added to the staked token pool. Now we can write $Y_{adj}$ in common terms $t_1 = t$:
So we see that the _Adjusted Staked Yield_ a function of the inflation rate and the percent of staked tokens on the network. We can see this plotted for various staking fractions here:
It is also clear that in all cases, dilution of un-staked tokens $>$ adjusted staked yield (i.e. dilution of staked tokens). Explicitly we can look at the _relative dilution of un-staked tokens to staked tokens:_ $D_{us}/Y_{adj}$. Here the relationship to inflation drops out and the relative dilution, i.e. the impact of staking tokens vs not staking tokens, is purely a function of the % of the total token supply staked. From above
Where we can see a primary dependence of the relative dilution of un-staked tokens to staked tokens is on the function of the proportion of total tokens staked. As shown above, the proportion of total tokens staked changes over time (i.e. $P_s = P_s(t)$ due to the re-staking of inflation issuance thus we see relative dilution grow over time as:
As might be intuitive, as the total fraction of staked tokens increases the relative dilution of un-staked tokens grows dramatically. E.g. with $80\%$ of the network tokens staked, an un-staked token holder will experience ~$400\%$ more dilution than a staked holder.
Again, this represents the change in fractional change in ownership of staked tokens and illustrates the built-in incentive for token holder to stake their tokens to earn _Staked Yield_ and avoid _Un-staked Dilution_.