The independence numbers of powers of graphs have been long studied, under
several definitions of graph products, and in particular, under the strong
graph product. We show that the series of independence numbers in strong powers
of a fixed graph can exhibit a complex structure, implying that the Shannon
Capacity of a graph cannot be approximated (up to a sub-polynomial factor of
the number of vertices) by any arbitrarily large, yet fixed, prefix of the
series. This is true even if this prefix shows a significant increase of the
independence number at a given power, after which it stabilizes for a while