VRRW on complete-like graphs: almost sure behavior
The authors study the vertex-reinforced random walk (VRRW) on complete-like graphs, i.e., complete graphs ornamented with finitely many leaves at each vertex. They combine martingale and large deviation techniques to prove that almost surely such VRRW spends positive and equal proportions of time on each of its nonleaf vertices. Previously [\textit{S. Volkov}, Ann. Probab. 29, No.~1, 66--91 (2001; Zbl 1031.60089)], it was shown that this behavior occurs only up to an event of positive probability. The convergence rate of the empirical occupation measure is studied as well.
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