Degree-biased vertex distribution

For a finite undirected graph with $m>0$ edges, the degree-biased probability distribution isChoose an edge uniformly, then choose one of its two endpoint incidences uniformly. A vertex is selected once for each incident edge end, so the handshaking lemma normalizes the probabilities. For the three-vertex path graph, the probabilities are $(1/4,1/2,1/4)$.

In a bipartite graph, choosing only the endpoint in one specified part instead gives $\mathrm{deg}(v)/m$ on that part and zero on the other. This conditional sampling convention differs from the probability distribution on the whole vertex set.