Cover time

In mathematics, the cover time of a finite Markov chain is the number of steps taken by the chain, from a given starting state, until the first step at which all states have been reached. It is a random variable that depends on the Markov chain and the choice of the starting state. The cover time of a connected undirected graph is the cover time of the Markov chain that takes a random walk on the graph, at each step moving from one vertex to a uniformly-random neighbor of that vertex.

Applications

Cover times of graphs have been extensively studied in theoretical computer science for applications involving the complexity of st-connectivity, algebraic graph theory and the study of expander graphs, and modeling Token Ring computer networking technology.

In different classes of graphs

A classical problem in probability theory, the coupon collector's problem, can be interpreted as the result that the expected cover time of a complete graph $K_{n}$ is $n\ln n(1+o(1))$. For every other $n$-vertex graph, the expected cover time is at least as large as this formula. Any $n$-vertex regular expander graph also has expected cover time $\Theta (n\log n)$ from any starting vertex, and more generally the cover time of any regular graph is $$O\left({\frac {n\log n}{1-\lambda _{2}}}\right),$$ where $\lambda _{2}$ is the second-largest eigenvalue of the graph, normalized so that the largest eigenvalue is one. For arbitrary $n$-vertex graphs, from any starting vertex, the cover time is at most $$\left({\frac {4}{27}}+o(1)\right)n^{3},$$ and there exist graphs whose expected cover time is this large. In planar graphs, the expected cover time is $\Omega (n\log ^{2}n)$ and $O(n^{2})$.

See also

• Hitting time, the number of steps until a set of states is first reached