  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.
