Continuous-time vertex-reinforced random walks on complete-like graphs

Shuo Qin; Pierre Tarres

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arXiv · 2311.13465

Continuous-time vertex-reinforced random walks on complete-like graphs

Shuo Qin·Pierre Tarres

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Abstract

We introduce the continuous-time vertex-reinforced random walk (cVRRW) as a continuous-time version of the vertex-reinforced random walk (VRRW), which might open a new perspective on the study of the VRRW. It has been proved by Limic and Volkov that for the VRRW on a complete-like graph $K_d \cup \partial K_d$, the asymptotic frequency of visits is uniform over the non-leaf vertices. We give short proofs of those results by establishing a stochastic approximation result for the cVRRW on complete-like graphs. We also prove that almost surely, the number of visits to each leaf up to time n divided by $n^{\frac{1}{d-1}}$ converges to a non-zero limit. We solve a conjecture by Limic and Volkov on the rate of convergence in the case of the complete graph.

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Shuo Qin, Pierre Tarres. 2023-11-22. Continuous-time vertex-reinforced random walks on complete-like graphs. https://arxiv.org/abs/2311.13465

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