We systematically treat every-pair interactions (a) that exhibit power-law dependence on the Euclidean distance and (b) act in structures that can be characterized using fractal geometry. We analytically derive the mean interaction field of the cells and find that (i) in a long-range interaction regime, the mean interaction field increases following a power-law with the size of the system, (ii) in a short-range interaction regime, the field saturates, and (iii) in the intermediate range it follows a logarithmic behavior. For long-range interactions, the theoretical calculations align closely with the numerical simulations. For short-range interactions, we observe that discreteness significantly affects the results, which requires an expansion that substantially improves the accuracy of the analytical expression. We conclude with applications from the Urban Heat Island (UHI) and the urban scaling contexts.