Evolving network - simulation study. From regular lattice to scale free   network
  The Watts-Strogatz algorithm of transferring the square lattice to a small world network is modified by introducing preferential rewiring constrained by connectivity demand. The evolution of the network is two-step: sequential preferential rewiring of edges controlled by $p$ and updating the information about changes done. The evolving system self-organizes into stationary states. The topological transition in the graph structure is noticed with respect to $p$. Leafy phase - a graph formed by multiple connected vertices (graph skeleton) with plenty of leaves attached to each skeleton vertex emerges when $p$ is small enough to pretend asynchronous evolution. Tangling phase where edges of a graph circulate frequently among low degree vertices occurs when $p$ is large. There exist conditions at which the resulting stationary network ensemble provides networks which degree distribution exhibit power-law decay in large interval of degrees. 
Danuta Makowiec