Rapid Mixing of Gibbs Sampling on Graphs that are Sparse on Average

0.19 MB·English

by  Elchanan Mossel

Rapid Mixing of Gibbs Sampling on Graphs that are Sparse on
Average
9 Elchanan Mossel∗ AllanSly †
0
0
January 29,2009
2

n
a
J

9 Abstract
2
Gibbssamplingalso knownasGlauberdynamicsisa populartechniqueforsamplinghighdimen-

]
R sional distributions defined on graphs. Of special interest is the behavior of Gibbs sampling on the
Erdo˝s-Re´nyirandomgraphG(n,d/n), whereeachedgeischosenindependentlywithprobabilityd/n
P
anddisfixed. WhiletheaveragedegreeinG(n,d/n)isd(1−o(1)),itcontainsmanynodesofdegree
.
h
oforderlogn/loglogn.
t
a The existence of nodes of almost logarithmic degrees implies that for many natural distributions
m
defined on G(n,p) such as uniformcoloring(with a constantnumberof colors) or the Ising modelat
[ anyfixedinversetemperatureβ,themixingtimeofGibbssamplingisatleastn1+Ω(1/loglogn). Recall

thattheIsingmodelwithinversetemperatureβ definedonagraphG = (V,E)isthedistributionover
3
{±}V givenbyP(σ) = 1 exp(β σ(v)σ(u)). Highdegreenodesposea technicalchallenge
v Z (v,u)∈E
3 in proving polynomial time mixing of the dynamics for many models including the Ising model and
P
0 coloring. Almost all known sufficient conditions in terms of β or number of colors needed for rapid
6 mixingofGibbssamplersarestatedintermsofthemaximumdegreeoftheunderlyinggraph.
3
Inthisworkweshowthatforeveryd < ∞andtheIsingmodeldefinedonG(n,d/n),thereexists
.
4 a β > 0, such that for all β < β with probability going to 1 as n → ∞, the mixing time of the
d d
0
dynamics on G(n,d/n) is polynomial in n. Our results are the first polynomial time mixing results
7
provenforanaturalmodelonG(n,d/n)ford>1wheretheparametersofthemodeldonotdependon
0
: n. TheyalsoprovidearareexamplewhereonecanproveapolynomialtimemixingofGibbssamplerin
v
asituationwheretheactualmixingtimeisslowerthannpolylog(n). Ourproofexploitsinnovelways
i
X the localtreelike structureof Erdo˝s-Re´nyirandomgraphs, comparisonand blockdynamicsarguments
r andarecentresultofWeitz.
a Ourresultsextendtomuchmoregeneralfamiliesofgraphswhicharesparseinsomeaveragesense
andtomuchmoregeneralinteractions. Inparticular,theyapplytoanygraphforwhicheveryvertexv
ofthegraphhasaneighborhoodN(v)ofradiusO(logn)inwhichtheinducedsub-graphisatreeunion
at most O(logn) edges and where for each simple path in N(v) the sum of the vertex degrees along
thepathisO(logn). Moreover,ourresultapplyalsointhecaseofarbitraryexternalfieldsandprovide
thefirstFPRASforsamplingtheIsingdistributioninthiscase. WefinallypresentanonMarkovChain
algorithmforsamplingthedistributionwhichiseffectiveforawiderrangeofparameters. Inparticular,
forG(n,d/n)itappliesforallexternalfieldsandβ <β ,wheredtanh(β )=1isthecriticalpointfor
d d
decayofcorrelationfortheIsingmodelonG(n,d/n).
Keywords: Erdo˝s-Re´nyiRandomGraphs,GibbsSamplers,GlauberDynamics,MixingTime,Isingmodel.
∗Email:[email protected],U.C.Berkeley.SupportedbyanAlfredSloanfellowshipinMathematics
andbyNSFgrantsDMS-0528488,DMS-0548249(CAREER)byDODONRgrantN0014-07-1-05-06.
†Email:[email protected],U.C.Berkeley.
1 Introduction
Efficient approximate sampling from Gibbs distributions is a central challenge of randomized algorithms.
Examples include sampling from the uniform distribution over independent sets of a graph [27, 26, 6, 8],
sampling from the uniform distribution of perfect matchings in a graph [17], or sampling from the uni-
form distribution of colorings [12, 4, 5] of a graph. A natural family of approximate sampling techniques
is given by Gibbs samplers, also known as Glauber dynamics. These are reversible Markov chains that
have the desired distribution as their stationary distribution and where at each step the status of one vertex
isupdated. Itistypicallyeasytoestablishthatthechainswilleventuallyconvergetothedesireddistribution.
Studying the convergence rate of the dynamics is interesting from both the theoretical computer science
thestatistical physics perspectives. Approximate convergence intimepolynomial inthesizeofthesystem,
sometimes called rapid mixing, is essential in computer science applications. The convergence rate is also
of natural interest in physics where the dynamical properties of such distributions are extensively studied,
see e.g. [20]. Much recent work has been devoted to determining sufficient and necessary conditions for
rapid convergence of Gibbs samplers. A common feature to most of this work [27, 26, 6, 8, 12, 4, 18, 22]
isthattheconditions forconvergence arestatedintermsofthemaximaldegreeoftheunderlying graph. In
particular,theseresultsdonotallowfortheanalysisofthemixingrateofGibbssamplersontheErdo˝s-Re´nyi
random graph, which is sparse on average, but has rare denser sub-graphs. Recent work has been directed
at showing how to relax statements so that they do not involve maximal degrees [5, 13], but the results are
not strong enough to imply rapid mixing of Gibbs sampling for the Ising model on G(n,d/n) for d > 1
andanyβ > 0orforsampling uniform colorings fromG(n,d/n)ford > 1and1000dcolors. Thesecond
challenge ispresented asthemajoropenproblem of[5].
In this paper we give the first rapid convergence result of Gibbs samplers for the Ising model on Erdo˝s-
Re´nyi random graphs in terms of the average degree and β only. Our results hold for the Ising model
allowing different interactions and arbitrary external fields. We note that there is an FPRAS that samples
from the Ising model on any graph [16] as long as all the interactions are positive and the external field is
thesamefor allvertices. However, these results do notprovide aFPRASinthe case wheredifferent nodes
havedifferent externalfieldsaswedohere.
Ourresultsarefurtherextendedtomuchmoregeneralfamiliesofgraphsthatare“tree-like” and“sparseon
average”. These are graph where every vertex has a radius O(logn) neighborhood which is a tree with at
mostO(logn)edges added and wherefor each simple path in theneighborhood, the sum ofdegrees along
the path is O(logn). An important open problem [5] is to establish similar conditions for other models
definedongraphs, suchastheuniform distribution overcolorings.
Below we define the Ising model and Gibbs samplers and state our main result. Some related work and a
sketchoftheproofarealsogivenastheintroduction. Section2givesamoredetailedproofthoughwehave
nottriedtooptimizeanyoftheparametersinproofsbelow.
1.1 The IsingModel
The Ising model is perhaps the simplest model defined on graphs. This model defines a distribution on
labelings of the vertices of the graph by + and −. The Ising model has various natural generalizations
including the uniform distribution over colorings. The Ising model with varying parameters is of use in a
varietyofareasofmachinelearning, mostnotablyinvision, seee.g.[9].
Definition1.1 The (homogeneous) Ising model on a (weighted) graph G with inverse temperature β is a
2
distribution onconfigurations {±}V suchthat
1
P(σ) = exp(β σ(v)σ(u)) (1)
Z(β)
{v,u}∈E
X
whereZ(β)isanormalizing constant.
Moregenerally, wewillbeinterested in(inhomogeneous) Isingmodelsdefinedby:
1
P(σ) = exp( β σ(v)σ(u)+ h σ(v)), (2)
u,v v
Z(β)
{v,u}∈E v
X X
where h are arbitrary and where β ≥ 0 for all u and v. In the more general case we will write
v u,v
β = max β .
u,v u,v
1.2 Gibbs Sampling
The Gibbs sampler is a Markov chain on configurations where a configuration σ is updated by choosing a
vertex v uniformly atrandom and assigning itaspin according tothe Gibbs distribution conditional on the
spinsonG−{v}.
Definition1.2 Givenagraph G = (V,E) andaninverse temperature β,theGibbssampler isthe discrete
time Markov chain on {±}V where given the current configuration σ the next configuration σ′ is obtained
bychoosingavertexv inV uniformlyatrandomand
• Lettingσ′(w) = σ(w)forallw 6= v.
• σ′(v)isassigned thespin+withprobability
exp(h + β σ(u))
v u:(v,u)∈E u,v
.
exp(h + β σ(u))+exp(−h − β σ(u))
v u:(v,u)∈E u,v P v u:(v,u)∈E u,v
P P
We will be interested in the time it takes the dynamics to get close to the distributions (1) and (2). The
mixingtime τ ofthechainisdefinedasthenumberofstepsneededinordertoguarantee thatthechain,
mix
starting from an arbitrary state, is within total variation distance 1/2e from the stationary distribution. We
willboundthemixingtimebytherelaxation timedefinedbelow.
It is well known that Gibbs sampling is a reversible Markov chain with stationary distribution P. Let
1 = λ > λ ≥ ... ≥ λ ≥ −1 denote the eigenvalues of the transition matrix of Gibbs sampling. The
1 2 m
spectral gap is denoted by max{1−λ ,1−|λ |} and the relaxation time τ is the inverse of the spectral
2 m
gap. Therelaxation timecanbegivenintermsoftheDirichletformoftheMarkovchainbytheequation
2 P(σ)(f(σ))2
τ = sup σ : P(σ)f(σ) 6= 0 (3)
Q(σ,τ)(f(σ)−f(τ))2
( σ6=τ P σ )
X
where f : {±}V → R is any fuPnction on configurations, Q(σ,τ) = P(σ)P(σ → τ) and P(σ → τ) is
transition probability from σ to τ. We use the result that for reversible Markov chains the relaxation time
satisfies
1
τ ≤ τ ≤ τ 1+ log(minP(σ))−1 (4)
mix
2 σ
(cid:18) (cid:19)
3
whereτ isthemixingtime(seee.g. [2])andsobyboundingtherelaxationtimewecanboundthemixing
mix
timeuptoapolynomialfactor.
While our results are given for the discrete time Gibbs Sampler described above, it will at times be conve-
nient to consider the continuos time version of the model. Here sites are updated at rate 1 by independent
Poissonclocks. Thetwochainsarecloselyrelated,therelaxationtimeofthecontinuoustimeMarkovchain
isntimestherelaxation timeofthediscrete chain(seee.g. [2]).
Forourproofsitwillbeusefultousethenotionofblockdynamics. TheGibbssamplercanbegeneralizedto
updateblocks ofvertices ratherthanindividual vertices. ForblocksV ,V ,...,V ⊂ V withV = ∪ V the
1 2 k i i
blockdynamicsoftheGibbssamplerupdatesaconfigurationσbychoosingablockV uniformlyatrandom
i
andassigningthespinsinV accordingtotheGibbsdistributionconditionalonthespinsonG−{V }. There
i i
isalsoacontinuousanaloginwhichtheblockseachupdateatrate1. Incontinuoustime,therelaxationtime
ofthe Gibbs sampler can begiven in termsof the relaxation timeof the block dynamics and therelaxation
timesoftheGibbssamplerontheblocks.
Proposition 1.3 Incontinuous timeifτ istherelaxation timeoftheblock dynamics and τ isthemax-
block i
imum the relaxation time on V given any boundary condition from G − {V } then by Proposition 3.4 of
i i
[20]
τ ≤ τ (maxτ )max{#j :v ∈ V }. (5)
block i j
i v∈V
1.2.1 MonotoneCoupling
For two configurations X,Y ∈ {−,+}V we let X < Y denote that X is greater than or equal to Y
pointwise. When all the interactions β are positive, it is well known that the Ising model is a monotone
ij
systemunderthispartialordering, thatisifX < Y then,
P σ = +|σ = X ≥ P σ = +|σ = Y .
v V\{v} V\{v} v V\{v} V\{v}
As it is a monotne sys(cid:0)tem there exists a coupling(cid:1)of Ma(cid:0)rkov chains {Xtx}x∈{−,+}V(cid:1) such that marginally
each has the lawofthe Gibbs Samplerwith starting configurations Xx = X and further that ifx < y then
0
for all t, Xx < Xy. This is referred to as the monotone coupling and can be constructed as follows: let
t t
v ,...bearandomsequence ofverticesupdated bytheGibbsSamplerandassociate withthemiidrandom
1
variablesU ,...distributed asU[0,1]whichdeterminehowthesiteisupdated. Attheithupdatethesitev
1 i
isupdated to+if
exp(h + β σ(u))
v u:(v,u)∈E u,v
U ≤
i
exp(h + β σ(u))+exp(−h − β σ(u))
v u:(v,u)∈E u,v P v u:(v,u)∈E u,v
and to − otherwise. It is well kPnown that such transitions preserve tPhe partial ordering which guarantees
that if x < y then Xx < Xy by the monotonicity ofthe system. Inparticular this implies that itisenough
t t
tobounded thetimetakentocouplefromtheall+andall−starting configurations.
1.3 Erdo˝s-Re´nyi Random Graphs and OtherModels ofgraphs
The Erdo˝s-Re´nyi random graph G(n,p), is the graph with n vertices V and random edges E where each
potential edge (u,v) ∈ V ×V ischosen independently withprobability p. Wetake p = d/n where d ≥ 1
is fixed. In the case d < 1, it is well known that with high probability all components of G(n,p) are of
logarithmicsizewhichimpliesimmediatelythatthedynamicsmixinpolynomial timeforallβ.
4
Foravertex v in G(n,d/n) let V(v,l) = {u ∈ G : d(u,v) ≤ l}, the set of vertices within distance l of v,
let S(v,l) = {u ∈ G : d(u,v) = l}, let E(v,l) = {(u,w) ∈ G : u,w ∈ V(v,l)} and let B(v,l) be the
graph(V(v,l),E(v,l)).
Ourresultsonlyrequiresomesimplefeaturesoftheneighborhoods ofallvertices inthegraph.
Definition1.4 LetG= (V,E)beagraphandv avertexinG. Lett(G)denotethetreeaccessofG,i.e.,
t(G) = |E|−|V|+1.
Wecallapathv ,v ,...selfavoiding ifforalli 6= j itholdsthatv 6= v . Weletthemaximalpathdensity
1 2 i j
mbedefinedby
m(G,v,l) = max d
u
Γ
u∈Γ
X
wherethe maximum is taken over allself-avoiding paths Γ starting at v withlength at mostl and d isthe
u
degreeofnodeu. Wewritet(v,l)fort(B(v,l))andm(v,l)form(B(v,l),v,l).
1.4 OurResults
Throughout wewillbeusing thetermwithhigh probability tomeanwithprobability 1−o(1)asngoesto
∞.
Theorem1.5 LetGbearandom graphdistributed asG(n,d/n). When
1
tanh(β) < ,
e2d
there exists constant a C = C(d) such that the mixing time of the Glauber dynamics is O(nC) with high
probability (probability 1− o(1)) over the graph as n goes to ∞. The result holds for the homogeneous
model(1)andfortheinhomogeneous model(2)provided|h |≤ 100βnforallv.
v
NoteinthetheoremabovetheO(·)bounddependsonβ. Itmaybeviewedasaspecialcaseofthefollowing
moregeneralresult.
Theorem1.6 Let G = (V,E) be any graph on n vertices satisfying the following properties. There exist
a> 0,0 < b< ∞and0< c < ∞suchthatforallv ∈ V itholdsthat
t(v,alogn)≤ blogn, m(v,alogn)≤ clogn.
Thenif
a
tanh(β) < ,
e1/a(c−a)
there exists constant aC = C(a,b,c,β) suchthat themixing timeoftheGlauber dynamics isO(nC). The
result holds for the homogeneous model (1) and for the inhomogeneous model (2) provided |h | ≤ 100βn
v
forallv.
Remark1.7 The condition that |h | ≤ 100βn for all v will be needed in the proof of the result in the
v
general case (2). However, we note that given Theorem 1.6 as a black box, it is easy to extend the result
and provide an efficient sampling algorithm in the general case without any bounds on the h . In the case
v
wheresomeoftheverticesvsatisfy|h |≥ 10βn,itiseasytoseethatthetargetdistribution satisfiesexcept
v
5
with exponentially small probability that σ = + for all v with h > 10βn and σ = − for all v with
v v v
h < −10βn. Thuswemayset σ = +whenh > 10βn and σ = −when h < 10βn and consider the
v v v v v
dynamics wherethesevalues arefixed. Doingsowilleffectively restrict thedynamics tothegraphspanned
bytheremaining vertices and willmodify thevalues ofh fortheremaining vertices; however, itiseasyto
v
see that all remaining vertices will have |h | ≤ 100βn. It is also easy to verify that if the original graph
v
satisfied the hypothesis of Theorem 1.6 then so does the restricted one. Therefore we obtain an efficient
samplingprocedure forthedesireddistribution.
1.5 Related Workand OpenProblems
Muchworkhasbeenfocusedontheproblemofunderstanding themixingtimeoftheIsingmodelinvarious
contexts. Inaseriesofresults[14,1,28]culminatingin[25]itwasshownthattheGibbssampleroninteger
latticemixesrapidlywhenthemodelhasthestrongspatialmixingproperty. InZ2strongspatialmixing,and
therefore rapidmixing, holdsintheentireuniqueness regime(seee.g. [21]). Ontheregulartreethemixing
time is always polynomial but is only O(nlogn) up to the threshold for extremity [3]. For completely
general graphs thebestknownresultsaregivenbytheDobrushin condition whichestablishes rapidmixing
whendtanh(β) < 1wheredisthemaximumdegree.
Most results for mixing rates of Gibbs samplers are stated in terms of the maximal degree. For example
manyresultshavefocusedonsamplinguniformcolorings, theresultareoftheform: foreverygraphwhere
all degrees are at most d if the number of colors q satisfies q ≥ q(d) then Gibbs sampling is rapidly
mixing [27, 26, 6, 8, 12, 4, 18, 22]. For example, Jerrum [15] showed that one can take q(d) = 2d.
Thenoveltyoftheresult presented hereisthatitallowsforthestudyofgraphs wheretheaverage degreeis
smallwhilesomedegreesmaybelarge.
Previous attempts at studying this problem, withbounded average degree but some large degrees, for sam-
pling uniform colorings yielded weaker results. In [5] it is shown that Gibbs sampling rapidly mixes
on G(n,d/n) if q = Ω ((logn)α) where α < 1 and that a variant of the algorithm rapidly mixes if
d
q ≥ Ω (loglogn/logloglogn). Indeed the main open problem of [5]is to determine if one can take q to
d
be a function of d only. Our results here provide a positive answer to the analogous question for the Ising
model. We further note that other results where the conditions on degree are relaxed [13] do not apply in
oursetting.
Thefollowingpropositions, whichareeasyandwellknown,establish thatford> 1andlargeβ themixing
timeisexponential innandthatforalld > 0andβ > 0themixingtimeismorethannpolylog(n).
Proposition 1.8 If d > 0 and β > 0 then with high probability the mixing time of the dynamics on
G(n,d/n)isatleastn1+Ω(1/loglogn).
Proof: TheprooffollowsfromthefactthatG(n,d/n)containsanisolatedstarwiths = Ω(logn/loglogn)
vertices with high probability and that the mixing time ofthe star is sexp(Ω(s)). Since thestar isupdated
withfrequency s/n,itfollowsthatthemixingtimeisatleast
(n/s)sexp(Ω(s)) = nexp(Ω(s)) = n1+Ω(1/loglogn).
(cid:4)
Proposition 1.9 If d > 1 then there exists β′ such that if β > β′ then the with probability going to 1, the
d d
mixingtimeofthedynamicsonG(n,d/n)isexp(Ω(n)).
6
Proof: Theclaim followsfromexpansion properties ofG(n,d/n). Itiswellknownthatifd > 1thenwith
high probability G(n,d/n) contains a core C of size at least α n such that that every S ⊂ C of size at
d
least α /4n has atleast γ n edges between C and S \C. Let Abe the set of configurations σ such that σ
d d
restrictedtoC hasatleastα /4+’sandatleastα /4−’s. ThenP(A) ≤ 2nexp(β|E|−2βγ n)/Z. Onthe
d d d
otherhandif+denotestheall+statethenP(+) = P(−) = exp(β|E|)/Z. Thusbystandardconductance
arguments, themixingtimeisexponential innwhen2exp(−2βγ ) < 1. (cid:4)
d
It is natural to conjecture that properties of the Ising model on the branching process with Poisson(d)
offspring distribution determines the mixing timeof the dynamics on G(n,d/n). In particular, itis natural
to conjecture that the critical point for uniqueness of Gibbs measures plays a fundamental role [10, 24]
as results of similar flavor were recently obtained for the hard-core model on random bi-partite d regular
graphs[23].
Conjecture1.10 Ifdtanh(β) > 1thenwithhigh probability over G(n,d/n)themixingtimeoftheGibbs
sampler is exp(Ω(n)). If d > 1 and dtanh(β) < 1 then with high probability over G(n,d/n) the mixing
timeoftheGibbssamplerispolynomial inn.
After proposing the conjecture we have recently learned that Antoine Gerschenfeld and Andrea Montanari
have found an elegant proof for estimating the partition function (that is the normalizing constant Z(β))
for the Ising model on random d-regular graphs [11]. Their result together with a standard conductance
argument showsexponentially slowmixingabovetheuniqueness threshold whichinthecontext ofrandom
regulargraphsis(d+1)tanh(β) = 1.
1.6 ProofTechnique
Ourprooffollowsthefollowingmainsteps.
• Analysis ofthe mixing timeforGibbs sampling ontrees ofvarying degrees. Wefindabound onthe
mixing time on trees in terms of the maximal sum of degrees along any simple path from the root.
This implies that for all β if we consider a tree where each node has number of descendants that
has Poisson distribution with parameter d − 1 then with high probability the mixing time of Gibbs
sampling on the tree is polynomial in its size. The motivation for this step is that we are looking at
tree-like graphs Notehowever, that theresults established here hold for allβ, whilerapid mixing for
G(n,d/n)doesnotholdforallβ. Ouranalysishereholdsforallboundaryconditionsandallexternal
fieldsonthetree.
• We next use standard comparison arguments to extend the result above to case where the graph is
a tree with a few edges added. Note that with high probability for all v ∈ G(n,d/n) the induced
subgraphB(v, 1log n)onallverticesofdistanceatmost 1log nfromvisatreewithatmostafew
2 d 2 d
edgesadded. (Notethisstillholdsforallβ).
• We next consider the effect of the boundary on the root of the tree. We show that for tree of alogn
levels, the total variation distance of the conditional distribution at the root given all + boundary
conditionsandall−boundaryconditionsisn−1−Ω(1)withprobability1−n−1−Ω(1) providedβ < β
d
issufficientlysmall(thisistheonlystepwherethefactthatβ issmallisused).
• Usingtheconstruction ofWeitz[27]andaLemmafrom[18,3]weshowthatthespatialdecayestab-
lishedinthepreviousstepalsoholdswithprobability 1−o(1)forallneighborhoods B(v,alogn)in
thegraph.
7
• Theremainingstepsusethefactthatastrongenoughdecayofcorrelationinsideblockseachofwhich
is rapidly mixing implies that the dynamics on the full graph is rapidly mixing. This idea is taken
from[7].
• Inordertoshowrapidmixingitsufficestoexhibitacouplingofthedynamicsstartingatall+andall
−thatcoupleswithprobability atleast1/2inpolynomialtime. Weshowthatthemonotonecoupling
(wheretheconfigurationstartedat−isalways“below”theconfigurationstartedat+)satisfiesthisby
showingthatforeachvinpolynomialtimethetwoconfigurationsatvcoupledexceptwithprobability
n−1/(2e).
• In order to establish the later fact, it suffices to show that running the dynamics on B(v,alogn)
starting at all + and all + boundary conditions and the dynamics starting at all − and all − will
coupleatv exceptwithprobability n−1/(2e)withinpolynomial time.
• The final fact then follows from the fact that the dynamics inside B(v,alogn) have polynomial
mixingtimeandthatthestationarydistributionsinB(v, 1log n)given+and−boundaryconditions
2 d
agreeatv withprobability atleast1−n−1/(4e).
Wenotethatthedecayofcorrelation ontheself-avoiding treedefinedbyWeitzthatweprovehereallowsa
different sampling scheme from the target distribution. Indeed, this decay of correlation implies that given
anyassignment toasubset ofthevertices S andanyv 6∈ S wemaycalculate using theWeitztreeofradius
alogninpolynomial timetheconditional probability thatσ(v) = +uptoanadditive errorofn−1/100. It
is easy to see that this allow sampling the distribution in polynomial time. More specifically, consider the
followingalgorithm from[27].
Algorithm1.11 Fix a radius parameter L and label the vertices v ,...,v . Then the algorithm approxi-
1 n
matelysamplesfromP(σ)byassigning thespinsofv sequentially. Repeatingfrom1 ≤ i ≤ n:
i
• Instepiconstruct TL (v ),thetreeofself-avoiding walkstruncated atdistance Lfromv .
SAW i i
• Calculate
pi = PTSLAW(σvi = +|σ{v1,...,vi−1},τA−Vi−1).
(The boundary conditions at the tree can be chosen arbitrarily; in particular, one may calculate p
i
withnoboundary conditions).
• Fixσ = X whereX isarandomvariable withp = P(X = +)= 1−P(X = −).
vi vi vi i vi vi
Thenweprovethat:
Theorem1.12 LetGbearandomgraphdistributed asG(n,d/n). When
1
tanh(β) < ,
d
for any γ > 0 there exist constants r = r(d,β,γ) and C = C(d,β,γ) such that with high proba-
bility Algorithm 1.11, with parameter rlogn, has running time O(nC) and output distribution Q with
d (P,Q) < n−γ. Theresultholdsforthehomogeneous model(1)andfortheinhomogeneous model(2).
TV
8
Theorem1.13 LetG = (V,E) be any graph on n vertices satisfying the following properties. Thereexist
a> 0,0 < b< ∞suchthatforallv ∈ V,
|V (v,alogn)|≤ balogn (6)
TSAW(v)
whereV (v,r) = {u∈ T (v) :d(u,v) ≤ r}. When
TSAW(v) SAW
1
tanh(β) < ,
b
for any γ > 0 there exist constants r = r(a,b,β,γ) and C = C(a,b,β,γ) such that Algorithm 1.11, with
parameter rlogn, has running time O(nC) and output distribution Q with d (P,Q) < n−γ. The result
TV
holdsforthehomogeneous model(1)andfortheinhomogeneous model(2).
1.7 Acknowledgment
E.M. thanks Andrea Montanari and Alistair Sinclair for interesting related discussions. The authors thank
JinshanZhangforpointing outanerrorinaprevious draft.
2 Proofs
2.1 Relaxationtimeon Sparseand GaltonWatsonTrees
Recall thatthe local neighborhood ofavertex inG(n,d/n)looks like abranching process tree. Inthe first
step of the proof we bound the relaxation time on a tree generated by a Galton-Watson branching process.
Moregenerally, weshowthattreesthatarenottoodensehavepolynomialmixingtime.
Definition2.1 LetT beafiniterootedtree. Wedefinem(T) = max d wherethemaximumistaken
Γ v∈Γ v
overallsimplepathsΓemanating fromtherootandd isthedegreeofnodev.
v
P
Theorem2.2 Letτ betherelaxation timeofthecontinuous timeGibbsSampleronT where0 ≤ β ≤ β
u,v
foralluandv andgivenarbitrary boundary conditions andexternal field. Then
τ ≤ exp(4βm(T)).
Proof:
Weproceedbyinductiononmwithasimilarargumenttotheoneusedin[18]foraregulartree. Notethatif
m = 0theclaimholdstruesinceτ = 1. Forthegeneral case,letv betherootofT,anddenoteitschildren
by u ,...,u and denote the subtree of the descendants of u by Ti. Now let T′ be the tree obtained by
1 k i
removing the k edges from v to the u , let P′ be theIsing model on T′ and let τ′ bethe relaxation timeon
i
T′. Byequation (3)wehavethat
max P(σ)/P′(σ)
τ/τ′ ≤ σ ≤ exp(4βk). (7)
min Q(σ,τ)/Q′(σ,τ)
σ,τ
Now we divide T′ into k + 1 blocks {{v},{T1},...,{Tk}}. Since these blocks are not connected to
each other the block dynamics is simply the product chain. Each block updates at rate 1 and therefore the
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relaxation time of the block dynamics is simply 1. By applying Proposition1.3 we get that the relaxation
timeonT′ issimplythemaximumoftherelaxation timesontheblocks,
τ′ ≤ max{1,τi}.
where τi is the relaxation time on Ti. Note that by the definition of m, it follows that the value of m for
eachofthesubtreesTi satisfiesm(Ti) ≤ m−k,andthereforeforalliitholdsthatτi ≤ exp(4β(m−k)).
Thisthenimpliesby(7)thatτ ≤ exp(4βm)asneeded. (cid:4)
2.2 Someproperties ofGaltonWatsonTrees
Here we prove a couple of useful properties for Galton Watson trees that will be used below. We let T be
the tree generated by a Galton-Watson branching process with offspring distribution N such that for all t,
Eexp(tN) < ∞ and such that E(N) = d. Of particular interest to us would be the Poisson distribution
withmeandwhichhas
Eexp(tN)= exp(d(et −1)).
WeletT denotethefirstrlevelsofT. WeletM(r)denotethevalueofmforT(r)andτ(r)thesupremum
r
of the relaxation times of the continuous time Gibbs Sampler on T(r) over any boundary conditions and
externalfieldsassumingthatβ = supβ . WedenotebyZ thenumberofdescendants atlevelr.
u,v r
Theorem2.3 Undertheassumptions abovewehave:
• Thereexistsapositivefunction c(t)suchthatforalltandallr:
E[exp(tM(r))] ≤ exp(c(t)r).
• ThenEτ(r) ≤ C(β)r forsomeC(β) < ∞depending onβ = supβ only.
u,v
• IfN isthePoissondistribution withmeandthenforallt > 0,
supE[exp(tZ d−r)] < ∞.
r
r
Proof: LetK denote the degree of the root of T and for 1 ≤ i ≤ K let M (r−1) denote the value of m
r i
forthesub-treeofT rootedatthei’thchild. Then:
r
E[exp(tM(r))] = E[max(1, max exp(t(M (r−1)+K)))]
i
1≤i≤K
K
≤ E[(1+exp(tK)) exp(tM (r−1))]
i
i=1
X
= E[(1+Kexp(tK))]E[exp(tM(r−1))].
andsotheresultfollowsbyinduction provided thatc(t)islargeenough sothat
exp(c(t)) ≥ E(1+Kexp(tK)).
Forthesecond statementofthetheorem,notethatbytheprevious theoremwehavethat
Eτ(r) ≤ E[exp(4βM(r))],
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