That's not how you do the mean field approximation in practice. In your Hamiltonian you have N particles or nodes, which you send to infinity in the thermodynam

That's not how you do the mean field approximation in practice. In your Hamiltonian you have N particles or nodes, which you send to infinity in the thermodynamic limit. In the mean field approximation you are just letting everybody interact with everybody and then scaling the interaction strength as 1/N so that energy scales as N. This is the core difference between the Edwards-Anderson and the Sherrington-Kirkpatrick models: instead of interacting with a finite set of neighbors you interact with everybody else (or an extensive number of neighbors).