Last-Coverage Probabilities of Random Walks on Cycle Graphs for Symmetric, Biased, and Time-Dependent Walks
This paper studies the last-coverage vertex of a nearest-neighbour random walk on a cycle: the vertex most likely to be the final one discovered. We show that the visited vertices always form a single contiguous arc of the ring, a property we call the Connected Arc Property. This lets the walk be replaced by a two-state time-inhomogeneous Markov chain, which we call the interval-expansion representation, in which each expansion step has a probability given by the classical gambler's ruin formula. Within this framework we prove that the last-coverage probability of node k in an n-node cycle is (1-r)r^(n-1-k)/(1-r^(n-1)), where r = (1-p)/p and p is the clockwise transition probability. As p approaches 1/2, we recover the uniform distribution, giving an independent derivation of a classical result. For a walk with a deterministic time-varying bias, we give an exact series representation for the last-coverage probabilities, together with a finite truncation that serves as a computable, error-bounded approximation. All results are validated by Monte Carlo simulation, and we run experiments on sinusoidal time-varying bias. Finally, we compare generalized Petersen graphs with Erdos-Renyi graphs of matching size, finding that structural symmetry is associated with more uniform last-coverage distributions.