= Degree-biased vertex distribution
For a finite <undirected graph> with $m>0$ <edges>, the degree-biased <probability distribution> is
$$
\pi(v)=\frac{\deg(v)}{2m}.
$$
Choose 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 $\deg(v)/m$ on that part and zero on the other. This conditional sampling convention differs from the <probability distribution> on the whole <vertex set>.
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