We study the extension profile of a pair of jointly distributed finite-valued random variables $(X,Y)$, defined as the set of all triples of numbers \[ ( H(X|W), H(Y|W), I(X:Y|W) ) \] obtained by extending the pair with an auxiliary random variable $W$. This object captures structural properties of a joint distribution that are not determined by the entropies of $X$ and $Y$ and their mutual information. We associate with the extension profile its \emph{shape function} defined in terms of the Legendre--Fenchel transform of the profile's lower boundary. We establish upper and lower bounds on the shape function in terms of classical information-theoretic quantities. For random pairs $(X,Y)$ whose joint distribution is uniform on its support, we relate the shape function to combinatorial and spectral properties of the associated bipartite support graph. We show that for random pairs uniformly distributed on the edges of balanced expander graphs, the shape function is within $O(\log H(XY))$ of the pointwise infimum over all pairs, which proves the conjecture stated in [MR26]. These results connect the extension profile with non-Shannon-type information inequalities, Gács--Körner common information, and spectral graph theory. We apply these techniques to study an approximate version of a question posed by Zhen Zhang ([Zha03]) concerning the characterization of pairs with maximally non-extractable mutual information.

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