AN EIGENVALUE REPRESENTATION FOR RANDOM WALK HITTING TIMES AND ITS APPLICATION TO THE ROOK GRAPH

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1 AN EIGENVALUE REPRESENTATION FOR RANDOM WALK HITTING TIMES AND ITS APPLICATION TO THE ROOK GRAPH HOWARD LEVINSON Abstract Give a aperiodic radom walk o a fiite graph, a expressio will be derived for the hittig times i terms of the eigevalues of the trasitio matrix The process of diagoalizig the trasitio matrix ad its associated fudametal matrix will be discussed These results will the be applied to a radom walk o a rook graph Lastly, a cover time boud depedig o the hittig times will be proved Keywords Fudametal matrix, trasitio matrix eigevalues, radom walk, hittig times, cover times, rook graph The Fudametal Matrix Cosider a aperiodic radom walk X k o a fiite graph with odes By the covergece theorem for fiite Markov chais [2], the associated trasitio matrix P satisfies lim k P k = P, where (P ij = π j, the jth compoet of the uique statioary distributio π That is, P = π π 2 π π π 2 π Note that P ca be rewritte as eπ T, where e is the colum vector cosistig of all oes We ow defie the fudametal matrix Z by Z = (I (P P We will use the fudametal matrix to derive a expressio for the expected hittig time from ode i to ode j, E i T j Let H be the hittig time matrix for P, that is h ij = E i T j Also defie the retur time matrix R by r ij = E i τ j, where we defie τ j = mi{k : X k = j} Note that for i = j, h ii = 0 ad r ii = /π i, ad for i j, h ij = r ij The followig lemmas will prove some importat properties about P ad Z that will be used i further aalysis Lemma (from [] Commutativity of P ad P Oe has the idetity, P P = P P = P P = P Proof As P is stochastic, P e = e, hece P P = P eπ T = eπ T = P Likewise, P P = eπ T P = eπ T = P Lastly, we have (P P ij =π π j + + π π j =π j (π + + π = π j = (P ij 200 Mathematics Subject Classificatio Primary 05C8

2 2 HOWARD LEVINSON Lemma 2 (from [] Z is well defied ad, Z = I + (P P = (P P = Proof First, equality betwee the secod two expressios is proved For, ( (P P = ( k P k P k k k=0 ( =P + ( k P k Therefore, =0 k= =0 =P + P (( = P P (P P = I + (P P = I + (P P Now let A = P P The, = = (I A(I + A + + A = I A = I + P P Lettig o each side, we have ( (I A A = I, =0 which shows that Z is well defied as I (P P is ivertible, ad its iverse is precisely as desired We are ow ready to prove our mai theorems relatig the fudametal matrix to hittig times I the followig we will let J = ee T ad D be a diagoal matrix with d ii = π i The followig table summarizes our matrix otatio up to this poit Symbol Name Defiitio P Trasitio Matrix p ij P Statioary Matrix eπ T Z Fudametal Matrix (I (P P H Hittig Time Matrix h ij = E i T j R Retur Time Matrix r ij = E i τ j J Oes Matrix ee T D Diagoal Matrix d ij = π i δ ij Theorem 3 (from [7] The fudametal matrix ca be writte as Z = (J + P H HD

3 AN EIGENVALUE REPRESENTATION FOR RANDOM WALK HITTING TIMES 3 Proof From first step aalysis, we obtai r ij = + k j p ik r kj = + k p ik h kj, as we take oe step ad the must retur to j from there Therefore, as H agrees with R except o the diagoal, we have h ij = r ij δ ij r ii = + k p ik h kj δ ij π i Rewritig this i matrix otatio we obtai, successively H =J + P H D (I P H =J D ( Z(I P H =Z(J D Usig Lemma 2 we have, ( Z(I P =I + (P P P + (P P = =2 =I + P P P = I P, ad, ( ( ZJ = Zee T = I + (P P ee T = e + (e e e T = ee T = = Simplifyig ( with these facts we obtai our result, (I P H = J ZD Z = (J + P H HD I particular, this theorem gives us that (2 z ij = π j ( + π k h kj h ij = π j ( + E π T j E i T j k= Usig this, it is simple to derive a direct relatioship betwee Z ad H which will lay the foudatio for the results i Sectio 3 Theorem 4 (from [7] The expected hittig time matrix H has etries give by Proof By (2 we have h ij = E i T j = z jj z ij π j z ij =π j ( + E π T j E i T j z jj =π j ( + E π T j Subtractig the first lie from the secod proves the theorem

4 4 HOWARD LEVINSON 2 The Eigesystem of the Fudametal Matrix We will ow ivestigate the eigesystems of the matrices discussed i Sectio to further expad o the hittig time formula give by Theorem 4 (as outlied i [] ad [4] Give ay aperiodic trasitio matrix P, cojugatig P by D /2 trasforms P ito a symmetric matrix This is because (P ij = (D /2 P D /2 ij = is a symmetric fuctio of i ad j diagoalizable i the form d(i d(j p ij = d(i d(j d(i = d(id(j As this ew matrix P is symmetric, it is P = W ΛW T, where Λ is the diagoal matrix cosistig of the eigevalues of P, λ,, λ, ad W is the associated matrix of orthogoal eigevectors (colums w,, w Substitutio shows that λ = is a eigevalue of P with associated eigevector w = ( π(,, π( Now let, (3 u i = D /2 w i ad v i = D /2 w i As P ad P are similar, they have the same eigevalues, ad the calculatio P v i = P D /2 w i = λ i D /2 w i = λ i v i shows that v i is a right eigevector of P correspodig to λ i A similar calculatio shows that the u i s are the left eigevectors of P Therefore, P = V ΛU T, or i particular, as (3 implies that u = Dv, (4 P = V ΛV T D A ice check shows that substitutig w = ( π(,, π( ito (3 gives us correspodig to λ = the right eigevector v = e (which is true as P is stochastic ad the left eigevector u = π (the defiitio of statioary measure Moreover, as P is aperiodic, we ca coclude by the Perro-Frobeius Theorem that λ i < for all i (see [] Therefore, the diagoalizatio of P expressed i (4 ca be rewritte as P =eπ T + λ j v j vj T D Therefore as P k = V Λ k V T D, =P + (5 P k = P + j=2 λ j v j vj T D j=2 λ k j v j vj T D, j=2 ad usig the fact that λ i < we obtai the Markov chai covergece theorem for aperiodic radom walks o graphs, lim P k = P k

5 AN EIGENVALUE REPRESENTATION FOR RANDOM WALK HITTING TIMES 5 Remark 5 The represetatio (5 gives a ituitive approach to the mixig rate, or the rate of covergece of P k to the statioary distributio Let λ = max 2 i λ i The the rate of covergece of the secod term i (5 to 0 oly depeds o the magitude of λ Now we are ready to relate our diagoalizatio of P to the diagoalizatio of the fudametal matrix Theorem 6 (from [7] Let P be the trasitio matrix for a aperiodic radom walk, expressed as i (4 The the fudametal matrix Z ca be diagoalized i the followig form, Z = V ΛV T D where λ =, ad for i >, λ i = ( λ i Proof As W T W = I, I = W T W = U T D /2 D /2 V = U T V = V T DV Usig this ad Z = I (P P, we obtai where V T DZ V =V T DV V T DP V + V T DP V =I Λ + V T DP V, V T DP V =(π,, v T D T (π,, π T (e,, v =(π,, v T D T (e, 0,, 0 = diag(, 0,, 0, where diag(, 0,, 0 is the diagoal matrix with the give etries o the diagoal Therefore we have or equivaletly, V T DZ V =I Λ + diag(, 0,, 0 =diag(, λ 2,, λ, Z = ( (V T D diag(, λ 2,, λ V =V diag(, ( λ 2,, ( λ V T D = V ΛV T D Usig this diagoalizatio of the fudametal matrix ad Theorem 4, we ca obtai a eigevalue formula for the expected hittig times Corollary 7 (6 h ij = E i T j = λ k (vjk 2 v ik v jk k=2 Proof From theorem 4, we kow that h ij = (z jj z ij /π j Usig the previous result, we ca calculate that ( z jj = π j λ k vjk 2, k= ( z ij = π j λ k v ik v jk k=

6 6 HOWARD LEVINSON The desired result is the obtaied by oticig that as v = e ad v k = for all k, λ ( v 2 j v i v j = ( = 0, so this first term does ot eed to be added to the sum i (6 3 Applicatio to the Rook Graph We will ow ivestigate the hittig times of a radom walk o a rook graph usig oly the methods discussed i the previous sectios The rook graph represets all of the legal moves of a rook o a chessboard, which is ay distace i a horizotal or vertical directio (see Figure More specifically, cosider a chessboard graph cosistig of the 2 odes, {(i, j i, j } The edges coect ay two odes that agree i oe coordiate, ad hece the graph is regular of degree 2 2 Figure Legal Moves o the Rook Graph The trasitio matrix for the radom walk o the rook graph is give by P (i,j(k,l = { 2 2 if i = k or j = l 0 else While this otatio is useful coceptually, we will give the odes a atural orderig for our calculatios (, will be the first ode, ad we will cotiue orderig to the right util reachig the edge (, The (, 2 will be ode ( +, ad the cotiue i this fashio Now the trasitio matrix R ca be writte as the 2 2 block matrix A I I R = I A 2 2 I, I I A

7 AN EIGENVALUE REPRESENTATION FOR RANDOM WALK HITTING TIMES 7 where I is the idetity matrix ad A is the matrix give by 0 A = 0 0 We will ow fid the eigevalues of R First, ote that R ca be writte as (7 R = 2 2 [(A I + (I A ], where is the Kroecker product of two matrices We will use this form to easily fid the eigevalues ad eigevectors of R Lemma 8 (from [3] Let A, B R have full sets of eigevalues λ i, µ i, respectively The (I A + (B I has 2 eigevalues give by λ + µ,, λ + µ, λ 2 + µ,, λ 2 + µ,, λ + µ Additioally, if x,, x ad y,, y are liearly idepedet sets of eigevectors for the λ i, µ i respectively, the y j x i are liearly idepedet eigevectors for λ i + µ j Proof By the biliearity, associativity, ad the mixed product property of the Kroecker product, [(I A + (B I ](y j x i =(y j Ax i + (By j x i =(y j λ i x i + (µ j y j x i =(λ i + µ j (y j x i Therefore usig (7 ad Lemma 8, we ca obtai the eigevalues of R directly from the eigevalues of A Lemma 9 A has eigevalues λ = ad for 2 i, λ i = The correspodig eigevector for λ is x = e, ad for 2 i, we have the liearly idepedet set of eigevectors 0 0 X = {x 2,, x } = 0,,, Proof A = ee T I Therefore, A e = ee T e I e = e e = ( e For each x X, e T x = 0 Therefore A x = x, completig the proof As a direct applicatio of the two previous lemmas, we obtai the followig result Theorem 0 R has the followig eigevalues:

8 8 HOWARD LEVINSON ( λ = with eigevector x x = e 2 (multiplicity (2 λ 2 = ( 2/(2 2 with eigevectors {x x i, x i x 2 i } (multiplicity 2( (3 λ 3 = /( with eigevectors {x i x j 2 i, j } (multiplicity ( 2 Note that to esure that λ i < for i >, we will assume that > 2 We have almost diagoalized R ito the form R = V ΛV, where V is the eigevector matrix with colums give by V = (x x, x x 2,, x x, x 2 x,, x x, ad Λ is the diagoal matrix of eigevalues Thus we are oly left with calculatig V Let y = (/,, / ad for 2 j, ( j + y j =, j +, j } {{ },, j }{{ } j terms j+ terms The a direct calculatio shows that V = (y y, y y 2,, y y, y 2 y,, y y T Our desired result is to derive a hittig time formula for the rook radom walk usig (6 However, while our choice of eigevectors is atural, we ca ot use (6 directly as our eigevectors are ot orthogoal, hece V DV T Tweakig the equatio, we still obtai for the rook graph (8 h ij = E i T j = 2 π j k=2 λ k v kj (v jk v ik, where 2( ( λ k =,, or, ad π j = 2 By the symmetries of the rook graph, we oly eed to cosider two cases These are if the distace from ode i to ode j is, ad if the distace from ode i to ode j is 2 Hece, we will oly calculate h,2 ad h,(+2 Usig (8 to calculate h,2, ay eigevectors of R such that their first two etries agree will have o cotributio For ay, there are oly four eigevectors that do ot have this property This is because for j 3, x j has a 0 as its first etry, hece for ay i, at least the first etries of x j x i are 0 Likewise, for j 4, x j has a 0 as its first two etries, thus for ay i, x i x j has 0 for its first two etries Lastly, as x = e, for ay i, x i x agrees i its first two etries Therefore, we are left with oly four eigevectors, x x 2, x x 3, x 2 x 2, ad x 2 x 3 The same reasoig applied to h,(+2 shows that we oly eed to cosider the seve eigevectors x x 2, x x 3, x 2 x, x 2 x 3, x 3 x, x 3 x 2, ad x 3 x 3 These symmetries ad the chose diagoalizatio of R allow us to reduce a calculatio ivolvig 2 terms to a maageable calculatio idepedet of to fid a hittig time formula We ow arrive at our hittig time formulas for the rook graph

9 AN EIGENVALUE REPRESENTATION FOR RANDOM WALK HITTING TIMES 9 Theorem Let i, j be two odes o the rook graph R, with > 2 (a If d(i, j =, the h ij = 2 (b If d(i, j = 2, the h ij = Proof The proof of (a will be give, ad (b is a idetical calculatio By the discussio above, we oly eed to calculate h,2 ad ca reduce (8 to a sum of oly four terms Therefore, h,2 = 2 ( 2( (y y 2 (2 [ ] (x x 2 (2 (x x 2 ( + + [ ] (y 2 y 3 (2 (x 2 x 3 (2 (x 2 x 3 ( ( = 2 2( + = 2 ( 2 ( 2 2( ( 2 + ( 2 + ( 2 2 ( ( ( ( 2 2 ( 3 Geeralized Biased Radom Walk o Rook Graph We will ow geeralize the radom walk o the rook graph to allow a bias towards visitig either horizotal or vertical eighbors We itroduce a parameter α > 0, such that the radom walk is α times more likely to visit a horizotal eighbor tha a vertical oe The trasitio matrix for this biased radom walk is give by α ( (α+ if i = k ˆP (i,j(k,l = ( (α+ if j = l 0 else Therefore, usig the same otatio as earlier, we have ˆR = ( (α + [(A I + (I αa ] Applyig Lemmas 8 ad 9 to our expressio for ˆR, we see that ˆR has the followig eigedecompositio Theorem 2 ˆ R has the same eigevectors as R with the followig eigevalues: ( ˆλ = with eigevector x x = e 2 (multiplicity (2 ˆλ 2 = ( α/[( (α + ] with eigevectors {x x i 2 i } (multiplicity ( (3 ˆλ 3 = (α( /[( (α + ] with eigevectors {x i x 2 i } (multiplicity ( (4 ˆλ 4 = /( with eigevectors {x i x j 2 i, j } (multiplicity ( 2

10 0 HOWARD LEVINSON Moreover, the related fudametal matrix eigevalues are give by λ =, λ2 = ( (α +, λ3 = α As ˆR has the same eigevectors as R, ˆR = V ˆΛV ( (α +, ad λ 4 = where ˆΛ is the diagoal matrix of eigevalues of ˆR Moreover, as ˆR is a doubly stochastic matrix, its statioary distributio is still give by π = (/ 2,, / 2 Therefore we ca apply (8 directly to calculate the hittig times for ˆR For this biased radom walk, we eed to cosider three cases: d(i, j = with i ad j i the same row, d(i, j = with i ad j i the same colum, ad d(i, j = 2 Hece, we oly eed to calculate h,2, h,+, ad h,+2 Calculatig h,2 ad h,+2 is idetical to our calculatio before, except substitutig i the ew eigevalues λ give i Theorem 2 Similar reasoig shows that we ca reduce the calculatio of h,+ by cosiderig oly the four eigevectors x 2 x, x 2 x 2, x 3 x, ad x 3 x 2 These calculatios yield the followig result Theorem 3 Let i, j be two odes o the rook graph ˆR, with > 2 (a If d(i, j = with i ad j i the same row, the h ij = α [α2 + ( α ] (b If d(i, j = with i ad j i the same row, the h ij = 2 + ( α α (c If d(i, j = 2, the h ij = α [α2 + (α 2 α + α 2 ] A ice check by settig α = shows that these results coicide with the hittig time results for R As metioed i Remark 5, the mixig rate is the rate of covergece of powers of the trasitio matrix P k to the statioary matrix P This rate of covergece oly depeds o the magitude of λ = max 2 i ˆλ i For large eough, as ˆλ 4 O(/ ad ˆλ 2, ˆλ 3 O(, the mixig rate will be determied by max{ˆλ 2, ˆλ 3 } To study the mixig rate of ˆR, we will fix ad set the eigevalues as fuctios of α, that is ˆλ 2 (α = α ( (α + ad ˆλ3 (α = Calculatig the derivatives of these fuctios, we see that ˆλ ( ( 2 2(α = [( (α + ] 2 < 0 ad ˆλ ( + ( 2 3(α = [( (α + ] 2 > 0 α( ( (α + Therefore, ˆλ 2 is a decreasig fuctio of α, ad ˆλ 3 is a icreasig fuctio of α, ad as ˆλ 2 ( = ˆλ 3 (, if α λ (α = λ ( = ˆλ 2 ( = ˆλ 3 ( Thus, perhaps as ituitively expected, the rook radom walk coverges to the statioary distributio quickest whe there is o bias preset

11 AN EIGENVALUE REPRESENTATION FOR RANDOM WALK HITTING TIMES 4 Cover Times A iterestig questio related to hittig times is the expected time to visit every ode i a graph, kow as the cover time We will ow discuss a theorem (from [5] ad proof ideas from [6] that gives upper ad lower bouds for the time take for a Markov chai to visit a fixed collectio of states A, referred to as the cover time C Give A = {a 0, a,, a }, the bouds will be give i terms of the extremal hittig times h (A, h (A defied by: h (A = mi i,j,i j E i(t j h (A = max i,j,i j E i(t j Give a permutatio β = (β(,, β( S, defie τ β (i to be the first time whe {a β(,, a β(i } have all bee visited For simplicity, as β will be fixed, τ β (i will be deoted more cocisely as τ(i This defies τ(i to be a stoppig time with respect to the filtratio {F k = σ(x 0,, X k, k 0} We ow defie R i = τ(i τ(i to be the time to visit a β(i from X τ(i If X k = a β(i for some k < τ(i, the R i = 0 Thus the set {R i 0} F i, ad we have the followig lemma Lemma 4 (from [5] P (R i 0 = /i Proof Let (b,, b be the orderig i which the a i s are visited, b i b j That is, for all i < j there exists k such that X k = a bi ad for all m such that X m = a bj, k < m This orderig is oly depedet o the Markov chai X, amely beig idepedet of β The probability that R i 0 ca be rewritte as the probability that the coordiate where β(i appears i (b,, b is further to the right the the coordiate where β(i appears i (β(,, β(i As β is a radom permutatio, this probability is /i We are ow ready to prove bouds o the cover time depedig o the extreme hittig times Theorem 5 (from [5] Let (X, 0 be a Markov chai with fiite state space Let C be the time take for the Markov chai to visit a fixed collectio of states A = {a 0, a,, a } Let X 0 = a 0 The for H, the th harmoic umber, h (AH EC h (AH Proof Let β = (β(,, β( be a elemet of the permutatio group S The, EC = E[τ( τ(0] = ER i = E[E(R i F i ], where E(R i F i = (R i 0E Xτ(i T aβ(i by the strog Markov property But h (A E Xτ(i T aβ(i h (A by defiitio, givig us i= i= (R i 0h (A E(R i F i (R i 0h (A Takig expectatios ad the summig from to gives us our result, h (AH EC h (AH

12 2 HOWARD LEVINSON As we kow a exact formula for the hittig times of a radom walk o a rook graph, we ca eatly apply this theorem to our results from Sectio 3 Because we kow both the miimum ad maximum hittig time, ( 2 H EC ( 2 + 2H, or geerally that the cover time of R is of order 2 log Similarly, the cover time of the biased rook graph ˆR is also of order 2 log Ackowledgmet A special thaks to Rya Rogers for his careful editig ad costructive feedback Refereces [] Pierre Bremaud Markov Chais: Gibbs Fields, Mote Carlo Simulatio, ad Queues, chapter 6, pages [2] Rick Durrett Probability: Theory ad Examples, chapter 6 Cambridge Uiversity Press, fourth editio, 200 [3] Ala J Laub Matrix Aalysis For Scietists Ad Egieers, chapter 3, page 4 Society for Idustrial ad Applied Mathematics, Philadelphia, PA, USA, 2004 [4] L Lovász Radom walks o graphs: a survey I Combiatorics, Paul Erdős is eighty, Vol 2 (Keszthely, 993, volume 2 of Bolyai Soc Math Stud, pages Jáos Bolyai Math Soc, Budapest, 996 [5] Peter Matthews Coverig problems for Browia motio o spheres A Probab, 6(:89 99, 988 [6] Amie M Sbihi Coverig times for radom walks o graphs PhD thesis, 990 Thesis (PhD McGill Uiversity (Caada [7] Christiae Takacs O the fudametal matrix of fiite state Markov chais, its eigesystem ad its relatio to hittig times Math Pao, 7(2:83 93, 2006