TY - JOUR
T1 - Non-backtracking random walks mix faster
AU - Alon, Noga
AU - Benjamini, Itai
AU - Lubetzky, Eyal
AU - Sodin, Sasha
N1 - Funding Information:
E. Lubetzky’s research is partially supported by a Charles Clore Foundation Fellowship.
Funding Information:
N. Alon’s research is supported in part by the Israel Science Foundation, by a USA-Israeli BSF grant, and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University.
PY - 2007/8
Y1 - 2007/8
N2 - We compute the mixing rate of a non-backtracking random walk on a regular expander. Using some properties of Chebyshev polynomials of the second kind, we show that this rate may be up to twice as fast as the mixing rate of the simple random walk. The closer the expander is to a Ramanujan graph, the higher the ratio between the above two mixing rates is. As an application, we show that if G is a high-girth regular expander on n vertices, then a typical non-backtracking random walk of length n on G does not visit a vertex more than (1 + o(1))log n/log log n times, and this result is tight. In this sense, the multi-set of visited vertices is analogous to the result of throwing n balls to n bins uniformly, in contrast to the simple random walk on G, which almost surely visits some vertex Ω(log n) times.
AB - We compute the mixing rate of a non-backtracking random walk on a regular expander. Using some properties of Chebyshev polynomials of the second kind, we show that this rate may be up to twice as fast as the mixing rate of the simple random walk. The closer the expander is to a Ramanujan graph, the higher the ratio between the above two mixing rates is. As an application, we show that if G is a high-girth regular expander on n vertices, then a typical non-backtracking random walk of length n on G does not visit a vertex more than (1 + o(1))log n/log log n times, and this result is tight. In this sense, the multi-set of visited vertices is analogous to the result of throwing n balls to n bins uniformly, in contrast to the simple random walk on G, which almost surely visits some vertex Ω(log n) times.
KW - Balls and bins
KW - Expanders
KW - Girth
KW - Mixing rate
KW - Non-backtracking random walks
UR - http://www.scopus.com/inward/record.url?scp=34547947639&partnerID=8YFLogxK
U2 - 10.1142/S0219199707002551
DO - 10.1142/S0219199707002551
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AN - SCOPUS:34547947639
SN - 0219-1997
VL - 9
SP - 585
EP - 603
JO - Communications in Contemporary Mathematics
JF - Communications in Contemporary Mathematics
IS - 4
ER -