Figure 1
The configuration model represented (upper left) as a conventional edge rewiring process and (right) as an urn problem. In the former case, once the out-stub \((A,\cdot )\) has been sampled for rewiring, then one in-stub is sampled uniformly at random from those available, to draw a new multi-edge. If we represent each possible combination of an out-stub and an in-stub as a ball, we arrive at the urn problem without replacement. For the shown vertices, the odds of observing a multi-edge \((A,B)\) are three times higher than of observing a multi-edge between \((A,D)\) and \(1.5\) times higher than of observing a multi-edge between \((A,C)\) in both model representations.
