We numerically investigate a geographical d-dimensional (d=1,2,3,4$$d=1,2,3,4$$) Bianconi–Barabási-like model, characterized by preferential attachment growth mechanisms influenced by Euclidean distances and weighted edges.

The weights of the edges follow a predetermined random probability distribution. This model is implemented through a straightforward energy-driven dynamics and exhibits the distribution of ’energy’ per site in its quasi-stationary state.

Across all networks generated by this model, we observe q-exponential energy distributions over the entire parameter space, which exhibits that this model belongs to the realm of nonadditive q-entropies. Additionally, the time evolution of the site energies, characterized by the dynamic βε-$$beta _{varepsilon }-$$exponent, is analyzed.

R. Oliveira, L.R. da Silva, C. Tsallis, Energy distribution in long-range-interacting weighted geographic networks, The European Physical Journal Plus 139(11) (2024) 975.