Figure 1.

Expected values of $f_4$-statistics under specified admixture graph models. (A) The expected value of $f_4(A,B;C,D)$ is given by the intersection between the path from $A$ to $B$ with the path from $C$ to $D$. Under the model shown, $\mathrm{E}\left[f_4(A,B;C,D)\right]=0$. (B) The expected value of $f_4(A,D;B,C)$ is given by the intersection between the path from $A$ to $D$ with the path from $B$ to $C$. Under the model shown, $\mathrm{E}\left[f_4(A,D;B,C)\right]=y$. (C) With population $C$ admixed, the path from $B$ to $C$ can be decomposed into two components. Under the model shown, with a proportion of $\alpha$ $B$-related ancestry and $1-\alpha$ $D$-related ancestry, the former yields a path (lighter red) that has a weight of $\alpha$ but does not intersect the path from $A$ to $D$, while the latter yields a path (darker red) that has a weight of $1-\alpha$ and intersects the path from $A$ to $D$ over the branch with length $y$. In total, $\mathrm{E}\left[f_4(A,D;B,C)\right]=(1-\alpha )y$.