= Solution
Use <degree-preserving edge rewiring>, often called a double-edge swap. Select two <edges> $(u,v)$ and $(x,y)$ with four distinct endpoints, then replace them by $(u,y)$ and $(x,v)$, choosing the reconnection symmetrically. Reject a proposal that would create a <self-loop> or an already present edge if the target is a <simple graph>. Each endpoint loses one edge and gains one, so every <degree of a vertex>, as well as the numbers of <vertices> and <edges>, is preserved exactly.
Repeat many such random proposals to generate reference networks. With symmetric proposals, retaining rejected steps as self-transitions, the uniform distribution on the reachable degree-preserving realizations is stationary. Adequate mixing must be checked; a few successful swaps do not establish an independent random sample. Degree preservation alone does not preserve connectedness, so impose and check that extra restriction if the <graph distance> comparison requires connected networks.
Back to article page
