  We consider a quasi two-dimensional network connection growth model that
minimizes the wiring cost while maximizing the network connections, but at the
same time edge crossings are penalized or forbidden. This model is mapped to a
dilute anti-ferromagnetic Ising spin system with frustrations. We obtain
analytic results for the order-parameter or mean degree of the optimized
network using mean field theories. The cost landscape is analyzed in detail
showing complex structures due to frustration as the crossing penalty
increases. For the case of strictly no edge crossing is allowed, the mean-field
equations lead to a new algorithm that can effectively find the (near) optimal
solution even for this strongly frustrated system. All these results are also
verified by Monte Carlo simulations and numerical solution of the mean-field
equations. Possible applications and relation to the planar triangulation
problem is also discussed.
