  In the realm of Boltzmann-Gibbs statistical mechanics there are three well
known isomorphic connections with random geometry, namely (i) the
Kasteleyn-Fortuin theorem which connects the $\lambda \to 1$ limit of the
$\lambda$-state Potts ferromagnet with bond percolation, (ii) the isomorphism
which connects the $\lambda \to 0$ limit of the $\lambda$-state Potts
ferromagnet with random resistor networks, and (iii) the de Gennes isomorphism
which connects the $n \to 0$ limit of the $n$-vector ferromagnet with
self-avoiding random walk in linear polymers. We provide here strong numerical
evidence that a similar isomorphism appears to emerge connecting the energy
$q$-exponential distribution $\propto e_q^{-\beta_q \varepsilon}$ (with $q=4/3$
and $\beta_q \omega_0 =10/3$) optimizing, under simple constraints, the
nonadditive entropy $S_q$ with a specific geographic growth random model based
on preferential attachment through exponentially-distributed weighted links,
$\omega_0$ being the characteristic weight.
