Another way this could turn out: If incoming degree is anti-correlated with outgoing degree, the effect of power-seeking may be washed out by it being hard, so we should expect worse than optimal policies with maybe more, maybe less powerseekyness as the optimal policy. Depending on the particulars of the environment. The next question is what particulars? Perhaps the extent of decorrelation, maybe varying the ratio of the two exponents is a better idea. Perhaps size becomes a factor. In sufficiently large environments, maybe figuring out how to access one of many power nodes becomes easier on average than figuring out how to access the single goal node. The number & relatedness of rewarding nodes also seems relevant. If there are very few, then we expect finding a power node becomes easier than finding a reward node. If there are very many, and/or they each lead into each other, then your chances of finding a reward node increase, and given you find a reward node, your chances of finding more increase, so power is not so necessary.
Another way this could turn out: If incoming degree is anti-correlated with outgoing degree, the effect of power-seeking may be washed out by it being hard, so we should expect worse than optimal policies with maybe more, maybe less powerseekyness as the optimal policy. Depending on the particulars of the environment. The next question is what particulars? Perhaps the extent of decorrelation, maybe varying the ratio of the two exponents is a better idea. Perhaps size becomes a factor. In sufficiently large environments, maybe figuring out how to access one of many power nodes becomes easier on average than figuring out how to access the single goal node. The number & relatedness of rewarding nodes also seems relevant. If there are very few, then we expect finding a power node becomes easier than finding a reward node. If there are very many, and/or they each lead into each other, then your chances of finding a reward node increase, and given you find a reward node, your chances of finding more increase, so power is not so necessary.