On the Cover Time for Random Walks on Random Graphs

  • Authors:

  • Johan Jonasson

  • Affiliations:

  • Chalmers University of Technology, S-412 96 Göteborg, Sweden, (e-mail: expect@math.chalmers.se)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 1998

Quantified Score

Hi-index 0.00

Visualization

Abstract

The cover time, C, for a simple random walk on a realization, GN, of 𝒢(N, p), the random graph on N vertices where each two vertices have an edge between them with probability p independently, is studied. The parameter p is allowed to decrease with N and p is written on the form f(N)/N, where it is assumed that f(N)≥c log N for some c1 to asymptotically ensure connectedness of the graph. It is shown that if f(N) is of higher order than log N, then, with probability 1−o(1), (1−ε)N log N≤E[C∣GN] ≤(1+ε)N log N for any fixed ε0, whereas if f(N)=O(log N), there exists a constant a0 such that, with probability 1−o(1), E[C∣GN] ≥(1+a)N log N. It is furthermore shown that if f(N) is of higher order than (log N)3 then Var(C∣GN)= o((N log N)2) with probability 1−o(1), so that with probability 1−o(1), the stronger statement that (1−ε)N log N≤C≤(1+ε)N log N holds.