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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ib/paper-4/9h/c/solution

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Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 4 / 9H / c / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-03

Let

D_{i}

be the degree of a vertex and put

π_{i} = \frac{D _{i}}{\sum _{k} D _{k}} .

(1)

i

and

j

are adjacent, then

π_{i} p_{ij} = 1/ \sum_{k} D_{k} = π_{j} p_{ji}

; if they are not adjacent, both sides vanish. Thus the walk is in detailed balance, and its invariant distribution is proportional to vertex degree.

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