cover times, blanket times, and majorizing measures Jian Ding U. C. Berkeley James R. Lee University...

cover times, blanket times, and majorizing measures

Jian DingU. C. Berkeley

James R. LeeUniversity of Washington

Yuval PeresMicrosoft Research

random walks on graphs

By putting conductances {cuv} on the edges of the graph,we can get any reversible Markov chain.

hitting and covering

Hitting time: H(u,v) = expected # of steps to hit v starting at u

Commute time: κ(u,v) = H(u,v) + H(v,u)(metric)

Cover time: tcov(G) = expected time to hit all vertices of G, starting from the worst vertex





pathcomplete graph

expander2-dimensional grid3-dimensional grid

complete d-ary tree

n2

n log nn log nn (log n)2

n log nn (log n)2/log d

orders of magnitude of some cover times

[coupon collecting][Broder-Karlin 88][Aldous 89, Zuckerman 90][Aldous 89, Zuckerman 90][Zuckerman 90]





regular treesrandom graphs

discrete torus, lattices

[Aldous 91][Cooper-Frieze 08][Dembo-Peres-Rosen-Zeitouni 04]

asymptotically optimal bounds





(1-o(1)) n ln n · tcov(G) · min(4n3/27, 2mn)

general bounds (n = vertices, m = edges)

[Feige’95, Alelinuas-Karp-Lipton-Lovasz-Rackoff’79][Feige’95, Matthews’88]

(conjecture: the completegraph is extremal)

electrical resistance




+u

v

Reff (u,v) = inverse of electrical current flowing from u to v

[Chandra-Raghavan-Ruzzo-Smolensky-Tiwari’89]:

If G has m edges, then for every pair u,v

κ(u,v) = 2m Reff (u,v)

(endows κ with specialgeometric properties)

computation

Hitting time: easy to compute in deterministic poly time by solving system of linear equations

H(u,u) = 0 H(u,v) = 1 + Ew»u H(w,v)

Cover time: easy to compute in deterministic exponential time

Approximations (deterministic, poly-time):

[Matthews’88, CRRST’89]

Augmented Matthews bound yields an O(log log n)2 approximation[Kahn-Kim-Lovasz-Vu’99]

For trees, there is an 1+² approximation for every ² > 0[Feige-Zeitouni’09]

maxu,v κ(u,v) yields an O(log n) approximation

Open question: Does there exist an O(1)-approximation for general graphs?[Aldous-Fill’94]

blanket times

Blanket times [Winkler-Zuckerman’96]:

¯-Blanket time is the expected first time T at which all the local times,

are within a factor of ¯.

blanket times, comparisons

Conjecture [Winkler-Zuckerman’96]: For every graph G and 0 < ¯ < 1, tblanket(G,¯) ³ tcov(G).

Proved for many special cases. True up to (log log n)2 by [KKLV’99]

Comparison of cover times:If G and G’ are two graphs on the same set of nodes andκG(u,v) · κG’(u,v) for all u,v 2 V, does it follow that

?

³ ³

main theorem

Talagrand introduced a functional on any metric space (X, d).

THEOREM: For any graph G,

where ³ denotes equivalence up to a universal constant.

°2(X ;d)

tcov(G) ³ [°2(G;p

· )]2

Some consequences:

- There is a deterministic O(1)-approximation to for any metric space, hence the same holds for tcov(G).

- Postively resolves the Winkler-Zuckerman blanket time conjectures.

- Bi-lipschitz stability. For instance, tcov(G) ³ tcov(G’) where G’ is a spectral sparsifier of G.

°2(X ;d)

main theorem

Talagrand introduced a functional on any metric space (X, d).

THEOREM: For any graph G, for any 0 < ¯ < 1,

where A . B denotes A · O(B).

°2(X ;d)

h°2(G;

p· )

i 2. tcov(G) · tblanket(G;¯ ) . ¯

h°2(G;

p· )

i 2

Some consequences:

- There is a deterministic O(1)-approximation to for any metric space, hence the same holds for tcov(G).

- Postively resolves the Winkler-Zuckerman blanket time conjectures.

- Bi-lipschitz stability. For instance, tcov(G) ³ tcov(G’) where G’ is a spectral sparsifier of G.

°2(X ;d)

a fast randomized algorithm

THEOREM: If g is an n-dimensional Gaussian, then

D = diagonal degree matrixA = adjacency matrix of G

THEOREM: For m-edge graphs, there is an O(m polylog(m))-time randomized algorithm to compute an O(1)-approximation to the cover time.Uses [Spielman-Teng] and [Spielman-

Srivistava]

main theorem

THEOREM: For any graph G and δ 2 (0,1),


tcov(G) ³ [°2(G;p

· )]2 ³ ± tblanket(G;±)

°2(S;d)

Gaussian processes

Consider a Gaussian process {Xu : u 2 S} with E (Xu)=0 8 u 2 S

(i.e. every linear combination ®1X1 + + ®kXk is normal)

Such a process comes with a natural metric

d(s; t) =q

E jX s ¡ X tj2

transforming (S,d) into a metric space.

Equivalently, for S finite, consider S µ Rn, and the process Xu = hg, ui for u 2 Swhere g=(g1, …, gn) is an i.i.d. N(0,1) vector.

PROBLEM: What is E max { Xu : u 2 S } ?

Gaussian processes

PROBLEM: What is E max { Xu : u 2 S } ?

®

If random variables are “independent,”expect the union bound to be tight.

Gaussian concentration:

Expect max for k points is about

Gaussian processes


Sudakov minoration:

Gaussian processes

covering trees

Recursively partition into pieces of diameter j=0, 1, 2, …

4¡ jValue of this path iswhere dj is the sequence of degreesdown the path

Pj 4¡ j

plogdj

valc(T) = maxP

j 4¡ jp

logdj max over all root-leaf paths

°2(S;d) = minvalc(T) min over all covering trees

(S;d)

covering trees

valc(T) = maxP

j 4¡ jp

logdj max over all root-leaf paths

°2(S;d) = minvalc(T) min over all covering trees

(S;d)

packing trees

valp(T) = minP

j 4¡ jp

logdj min over all root-leaf paths

(S;d)

Main technical theorem:

maxT valp(T) ³ minT 0 valc(T0) = °2(S;d)

majorizing measure theorem

Majorizing measures theorem (Talagrand):

(Recalling that d(s; t) =q

E jX s ¡ X tj2.)

main theorem [Ding, L, Peres]

THEOREM: For any graph G and δ 2 (0,1),


tcov(G) ³ [°2(G;p

· )]2 ³ ± tblanket(G;±)

°2(S;d)

hints of a connection


Sudakov minoration:


Sudakov minoration:

Matthew’s bound (1988):



KKLV concentration:

Here, Nt(w) denotes the number of visits to w when the random walk startedat u has returned to u for the (t deg(u))th time.


- Trees + KKLV concentration suffice for upper bound

- [Barlow-Ding-Nachmias-Peres] prove the “Dudley version”

an isomorphism theoremT he local time of v at time t is de¯ned by

L vt =

# visits to vdeg(v)

:

an isomorphism theorem

a problem on Gaussian processes

Gaussian process:

a problem on Gaussian processes

Gaussian process:

We need strong estimates on the size of this window as ε 0.(want to get a point there with probability at least 0.1)Problem: Majorizing measures handles first moments, but we need second moment bounds.

percolation on trees and the DGFF

First and second moments agree for

percolation on balanced trees

Problem: General Gaussian processes behaves nothing like percolation!

Resolution: Processes coming from the Isomorphism Theorem all arise from a “discrete Gaussian free field.”

percolation on trees and the DGFF

First and second moments agree for

percolation on balanced trees

For DGFFs, using electrical network theory, we show that it is possibleto select a subtree of the MM tree and a delicate filtration of theprobability space so that the Gaussian process can be coupled to apercolation process.

open questions

Holds for - complete graph - complete d-ary tree - discrete torus

QUESTION: Is there a deterministic, polynomial-time (1+²)-approximation to the cover time for every ² > 0 ?

QUESTION: Is the standard deviation of the time-to-cover bounded by the maximum hitting time?

cover times, blanket times, and majorizing measures Jian Ding U. C. Berkeley James R. Lee University...

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