Optimized two-dimensional Networks with edge crossing cost: frustrated anti-ferromagnetic spin system

TL;DR

This paper models and analyzes a quasi two-dimensional network optimization problem with edge crossing penalties, mapping it to a frustrated anti-ferromagnetic spin system, and develops algorithms for near-optimal solutions.

Contribution

It introduces a novel mapping of the network problem to a frustrated anti-ferromagnetic Ising model and develops a new algorithm for strictly crossing-free network optimization.

Findings

Complex cost landscape due to frustration analyzed

New algorithm effectively finds near-optimal solutions in strongly frustrated systems

Results verified by Monte Carlo simulations and mean-field analysis

Abstract

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.