Scale-free Network on Euclidean Space Optimized by Rewiring of Links

A Barab\'asi-Albert scale-free network is constructed whose nodes are the Poisson distributed random points within a unit square and links are the straight line connections among the nodes. The cost function, which is the total wiring length associated with a such a network defined on a two dimensional plane is optimized. The optimization process consists of random selection of a pair of links and rewiring them to reduce the total length of the pair but with the constraint that the degree as well as the out-degree and in-degree of each node are precisely maintained. The resulting optimized network has a small diameter as well as high clustering and the link length distribution has a stretched exponential tail.