Logarithmic Sobolev, Isoperimetry and Transport Inequalities on Graphs

Transcription

1 Acta Mathematica Sinica, English Series Oct., 016, Vol. 3, No. 10, pp Published online: September 15, 016 DOI: /s Acta Mathematica Sinica, English Series Springer-Verlag Berlin Heidelberg & The Editorial Office of AMS 016 Logarithmic Sobolev, Isoperimetry and Transport Inequalities on Graphs Yu Tao MA School of Mathematical Sciences & Lab. Math. Com. Sys., Beijing Normal University, Beijing , P.R.China mayt@bnu.edu.cn Ran WANG School of Mathematics and Statistics, Wuhan University, Wuhan 43007, P. R. China and School of Mathematical Sciences, University of Science and Technology of China, Hefei 3006, P.R.China wangran@ustc.edu.cn Li Ming WU Institute of Applied Math., Chinese Academy of Sciences, Beijing , P. R. China and Laboratoire de Math. CNRS-UMR 660, Université Blaise Pascal, Aubière, France Li-Ming.Wu@math.univ-bpclermont.fr Abstract In this paper, we study some functional inequalities (such as Poincaré inequality, logarithmic Sobolev inequality, generalized Cheeger isoperimetric inequality, transportation-information inequality and transportation-entropy inequality) for reversible nearest-neighbor Markov processes on connected finite graphs by means of (random) path method. We provide estimates of the involved constants. Keywords Graph, log-sobolev inequality, isoperimetry, transport inequality MR(010) Subject Classification 60E15, 05C81, 39B7 1 Introduction Let G =(V,E) be a finite connected graph with vertex set V andorientededgessete, which is a symmetric subset of V \{(x, x) :x V }. If (x, y) E, wecallthatx, y are adjacent, denoted by x y. For any x V and any function f : V R, consider the operator Lf(x) = q(x, y)(f(y) f(x)), (1.1) y V where q(x, y) is the jump rate from x to y, which is non-negative and q(x, y) > 0 if and only if x y. Received May 18, 015, accepted April 6, 016 Supported by NSFC (Grant Nos , , and ) and 985 Projects and the Fundamental Research Funds for the Central Universities

2 1 Ma Y. T. et al. Let (X t ) be the Markov process generated by L, defined on (Ω, (F t ) t 0, (P x ) x V ). We always assume the reversibility condition, i.e., there is some probability measure μ satisfying the detailed balance condition: Q(x, y) :=μ(x)q(x, y) =μ(y)q(y, x), (x, y) E. (1.) Equivalently, the operator L is self-adjoint on L (μ), that is, f, Lg μ = Lf,g μ = 1 (f(x) f(y)) (g(x) g(y)) Q(x, y) x,y = 1 D e fd e gq(e) =:E(f,g), where D e f := f(y) f(x) fore =(x, y) E. When q(x, y) =1/d x with d x the degree of x (the number of neighbors y x), L becomes the Laplacian Δ on the graph. In that case μ(x) =d x / E and Q(e) =1/ E. Define the variance of f with respect to (w.r.t. for short) μ Var μ (f) :=μ((f μ(f)) ), and the entropy of f w.r.t. μ Ent μ (f ):=μ(f log f ) μ(f )logμ(f ). We say that μ satisfies a Poincaré inequality if there exists a constant λ>0 such that for all f L (μ), Var μ (f) λe(f,f), (1.3) μ satisfies a log-sobolev inequality if there exists a constant α>0 such that for all f L (μ), Ent μ (f ) αe(f,f). (1.4) The optimal constants λ and α in (1.3) and (1.4) are called respectively the Poincaré constant and the log-sobolev constant of μ, which are denoted by c P and c LS respectively. It is well known that c P c LS, see [5]. The Poincaré inequality and logarithmic Sobolev inequality play a crucial role in the analysis of the behaviour of the process. To study of the Poincaré constant, Jerrum and Sinclair [19] introduced the path combinatoric method in theoretic computer science, which is further developed by Diaconis and Stroock [13], Fill [15], Sinclair [9], Chen [7], and so on. The logarithmic Sobolev inequality in the discrete setting was also studied by many authors, such as Diaconis and Saloff-Coste [1], Roberto [4], Lee and Yau [], Chen [8], Chen and Sheu [6], Chen et al. [5], and so on. The reader is referred to the books of Saloff-Coste [7] and Chen [9] for further information. The main purpose of this paper is to study the logarithmic Sobolev inequality, the generalized Cheeger isoperimetric inequality and the transport inequality. The remainder of this paper is organized as follows: in the next section, we focus on the logarithmic Sobolev inequality, and the third section is devoted to the transportation-information inequality and the generalized Cheeger isoperimetric inequality. In the last section, some examples are discussed and the estimates of involved constants are given.

3 Logarithmic Sobolev, Isoperimetry and Transport Inequalities on Graphs 13 Logarithmic Sobolev Inequality.1 Length Functions, Random Paths Apathγ xy from x to y is a family of edges {e 1,...,e n },wheree k =(x k 1,x k ) E, such that x 0 = x, x n = y. It is said to have no circle if all x k,k =0,...,n, are different. A positive function w : E (0, + ) defined on the edge set E is called length function, ifw(x, y) =w(y, x) for any e =(x, y) E. Given the length function w, thew-length of a path γ xy from x to y is defined by γ xy w := w(e), e γ xy and the distance associated with w is ρ w (x, y) :=min γ xy w. γ xy When w 1, ρ w =: ρ 1 is the natural graph distance on V. Diaconis and Stroock [13] showed that the Poincaré constant c P satisfies that c P max 1 γxy (e) γ xy 1/Q μ(x)μ(y) (.1) for any collection of paths {γ xy : x, y V }, where γ xy 1/Q is the length γ xy w with w(e) = 1/Q(e), a quite natural distance associated with the Markov process. Furthermore, by using the length functions, the estimate (.1) can be improved to be 1 c P max 1 γxy (e) γ xy w μ(x)μ(y), (.) Q(e)w(e) which is sharp for birth-death processes (see Kahale [0] or Chen [7]). Now for any x, y V different, let γ xy be a random (maybe deterministic) path without circle from x to y. By convention, we set γ xx = and denote by E γ the expectation w.r.t. {γ xy : x, y V }.. Logarithmic Sobolev Inequality Theorem.1 For any length function w and any edge e, let L w,e (x) :=E γ 1 γxy (e) γ xy w μ(y). y V (.3) The logarithmic Sobolev constant c LS is bounded by c LS inf 1 ( Entμ (L w,e )+μ(l w,e )log(e +1) ), w Q(e)w(e) (.4) where inf w is taken over all length functions w on E and e is the Euler constant. The upper bound in (.4) gives us a very practical criterion for the logarithmic Sobolev inequality and the estimate above is based on the following weighted Poincaré inequality, which is a slight generalization of (.). Lemma. (Weighted Poincaré inequality) Let ϕ be a nonnegative function on V. Then for any length function w, (f(x) μ(f)) ϕ(x)μ(x) c(ϕ, w)e(f,f), f : V R, (.5) x V

4 14 Ma Y. T. et al. where c(ϕ, w) :=max =max Q(e)w(e) Eγ Q(e)w(e) 1 γxy (e) γ xy w ϕ(x)μ(x)μ(y) L w,e (x)ϕ(x)μ(x). x V When ϕ 1, our constant c(ϕ, w) is the twice of the quantity on the right-hand side of (.). Proof For any fixed realization of random path {γ xy : x, y V }, (f(x) μ(f)) ϕ(x)μ(x) x V = ( (f(x) f(y))μ(y)) ϕ(x)μ(x) x V y V = ( μ(y) ) D e f ϕ(x)μ(x) x V y V e γ xy ( ) D e f ϕ(x)μ(x)μ(y) e γ xy ( )( ) 1 w(e) w(e) (D ef) ϕ(x)μ(x)μ(y) e γ xy e γ xy = 1 Q(e)(D e f) 1 γxy (e) γ xy w ϕ(x)μ(x)μ(y), Q(e)w(e) where the Cauchy Schwarz inequality is applied twice. Taking the expectation E γ w.r.t. the randomness of γ, we get the desired result. Now recall two important lemmas: the first one is due to Rothaus [6] and the second is given by Barthe Roberto []. Lemma.3 For any real function f on V and any constant a R, Lemma.4 For any real function f on V, Ent μ (f ) Ent μ ( (f a) ) +μ ( (f a) ). Ent μ (f )+μ(f ) sup { μ(f ϕ): ϕ 0,μ(e ϕ ) e +1 }. Consequently, by Donsker Varadhan s variational formula (see [14]), we have μ(f ϕ) μ(f )logμ(e ϕ ) Ent μ (f ), ϕ : V R, sup { μ(f ϕ): ϕ 0,μ(e ϕ ) e +1 } Ent μ (f )+μ(f )log(e +1). (.6) Proof of Theorem.1 By Lemmas.3 and.4, we have Ent μ (f ( ) Ent μ (f μ(f)) ) +μ ( (f μ(f)) ) { } sup (f(x) μ(f)) ϕ(x)μ(x) : ϕ 0,μ(e ϕ ) e +1 x V

5 Logarithmic Sobolev, Isoperimetry and Transport Inequalities on Graphs 15 sup c(ϕ, w) E(f,f), ϕ 0,μ(e ϕ ) e +1 where the last inequality follows from (.5). Moreover by (.5) and (.6), we have sup c(ϕ, w) =max ϕ 0,μ(e ϕ ) e +1 max which implies (.4). The proof is complete. Q(e)w(e) Q(e)w(e) sup μ(l w,e ϕ) ϕ 0,μ(e ϕ ) e +1 ( Entμ (L w,e )+μ(l w,e )log(e +1) ), 3 Transportation Inequalities In this section, we shall establish the transportation-information inequality W 1 I and as a corollary, the transportation-entropy inequality W 1 H. For this purpose, let us introduce some notations. 3.1 Wasserstein Distance, Entropy and Information Given a metric ρ on V, the Lipschitzian norm of a function g is denoted by g Lip(ρ). Let M 1 (V ) be the space of all probability measures on V. For any ν, μ M 1 (V ), recall that (i) The Wasserstein distance W 1,ρ (ν, μ) associated with ρ is defined as W 1,ρ (ν, μ) =inf ρ(x, y)π(dx, dy), π V where π runs over all couplings of (ν, μ), i.e., probability measures on V such that π(a V )= ν(a) andπ(v A) =μ(a) for all Borel subsets A of V.Ifρ(x, y) =1 x y is the discrete metric, W 1,ρ (ν, μ) = 1 ν μ TV where ν TV =sup f 1 ν(f) is the total variation of a signed measure ν. (ii) The relative entropy of ν w.r.t. μ is given by ν(x) x V ν(x)log, if ν μ; H(ν μ) = μ(x) +, otherwise. (iii) Fisher Donsker Varadhan information of a probability ν = h μ w.r.t. μ is defined by I(ν μ) := 1 (h(x) h(y)) Q(x, y) = 1 (D e h) Q(e), where D e h = h(y) h(x) for the oriented edge e =(x, y) E. 3. Transportation-information Inequality Guillin et al. [17] introduced the following transportation-information inequality for the given metric ρ, W1,ρ(ν, μ) c G I(ν μ), ν M 1 (V ), (3.1) where c G is the best constant. In [17], it is proved that (3.1) is equivalent to the following Gaussian concentration inequality: for all probabilities ν μ and ρ-lipschitzian function g on

6 16 Ma Y. T. et al. V, P ν ( 1 t t 0 ) g(x s )ds > μ(g)+r dν dμ L { exp tr c G g Lip(ρ) }, t, r > 0. (3.) Here (X t ) is the Markov process generated by L, defined on some probability space (Ω, F, P) with initial distribution ν. So c G is also called the Gaussian concentration constant for (X t ) (w.r.t. the metric ρ). The reader is referred to the book of Villani [30] for optimal transport, transport inequalities and related bibliographies. Theorem 3.1 The transportation-information inequality (3.1) holds with c G K := inf K(w), (3.3) w where the infimum is taken over all length functions w and the geometric constant K(w) is given by K(w) =max 1 Q(e)w(e) E γ 1 γxy (e)ρ (x, y) γ xy w μ(x)μ(y). (3.4) Remark 3. When ρ(x, y) = 1 x y (the discrete metric), K coincides with the quantity in (.). By Guillin et al. [17, Theorem 3.1], the transportation-information inequality w.r.t. thediscretemetricandthepoincaré inequality are equivalent: c P 8 c G c P. If we apply this result together with (.), we obtain only c G K. Since c P = K for birthdeath processes (see [7, 0]), we get c G K/8. In other words, our estimate of c G is of correct order. Proof of Theorem 3.1 For each probability measure ν = h μ, by Kantorovich Robinstein s identity (see [30]) and the Cauchy Schwarz inequality, W 1,ρ (ν, μ) = sup g(x) ( h (x) 1 ) μ(x) g Lip(ρ) 1 x V = 1 sup (g(x) g(y)) ( h (x) h (y) ) μ(x)μ(y) g Lip(ρ) 1 1 (g(x) g(y)) (h(x) h(y)) μ(x)μ(y) sup g Lip(ρ) 1 (h(x)+h(y)) μ(x)μ(y) x,y ρ (x, y)(h(x) h(y)) μ(x)μ(y). For any fixed random path {γ xy = γ xy (ω) :x, y V } and the length function w, wehaveby the Cauchy Schwarz inequality, ρ (x, y)(h(x) h(y)) μ(x)μ(y) x,y

7 Logarithmic Sobolev, Isoperimetry and Transport Inequalities on Graphs 17 = ( ) ρ (x, y) D e h μ(x)μ(y) x,y e γ x,y ( )( ρ (x, y)μ(x)μ(y) (D e h) 1 ) w(e) w(e) x,y e γ xy e γ xy = (D e h) 1 Q(e) 1 γx,y (e)ρ (x, y) γ xy w μ(x)μ(y). Q(e)w(e) Taking first the expectation E γ and then the maximum of the last term over all oriented edges e, wegetc G K(w). The proof is complete. Corollary 3.3 Assume that there exists some constant M>0such that 1 ρ (x, y)q(x, y) M. (3.5) sup x V y x Then the following transportation-entropy inequality holds W 1,ρ(ν, μ) KMH(ν μ), ν M 1 (V ), (3.6) or equivalently for any Lipschitzian function g, ( λ e λ(g μ(g)) MK dμ exp 4 Proof g Lip(ρ) ), λ R. (3.7) The transportation-entropy inequality (3.6) follows from the transportation information inequality (3.3) under the condition (3.5), by Guillin et al. [18, Theorem 4.]. The equivalence between (3.6) and the Gaussian concentration (3.7) is the famous Bobkov Götze s characterization in [4]. Corollary 3.4 For the Laplacian L =Δon the connected graph G =(V,E), we have, for the graph metric ρ 1, c G K d bd 3, E where d =max x V d x, D is the diameter of G and Proof b =max {ρ 1 -shortest paths γ : e γ}. (3.8) e Choose γ xy distributed uniformly on all shortest paths from x to y and w =1,wesee that K is bounded from above by (noting that Q(e) =1/ E and μ(x) =d x / E ) max ( ) E Eγ 1 γxy (e)ρ 1 (x, y) 3 d d bd 3. E E The proof is complete. 3.3 Generalized Cheeger Isoperimetric Inequality Consider the following generalized Cheeger isoperimetric inequality W 1,ρ (fμ,μ) c I D e f Q(e), ν = fμ M 1 (V ), (3.9) where c I is the best constant, called as Cheeger constant w.r.t. the metric ρ.

8 18 Ma Y. T. et al. Define the geometric constant κ Theorem 3.5 Proof V }, κ := max It holds that 1 Q(e) Eγ c I κ. 1 γxy (e)ρ(x, y)μ(x)μ(y). (3.10) By Kantorovich Robinstein s identity, we have, for any fixed random path {γ xy : x, y W 1,ρ (fμ,μ)= sup g(x)(f(x) 1)μ(x) g Lip(ρ) 1 x V = 1 sup (g(x) g(y)) (f(x) f(y)) μ(x)μ(y) g Lip(ρ) 1 1 ρ(x, y) D e f μ(x)μ(y) e:e γ xy = 1 1 D e f Q(e) 1 γxy (e)ρ(x, y)μ(x)μ(y). Q(e) Taking the expectation E γ, we obtain the desired result. Corollary 3.6 (Weighted L 1 -Poincaré inequality) for any function f on V, V Given a positive function ϕ on V, we have, f μ(f) ϕdμ κ D e f Q(e), (3.11) where κ is given by (3.10) with ρ(x, y) =1 x y (ϕ(x)+ϕ(y)),x,y V. Proof Considering (f c 1 )/c if necessary, we assume without loss of generality that f>0 and μ(f) = 1. In that case, for the metric ρ(x, y) =1 x y (ϕ(x)+ϕ(y)), it is known that W 1,ρ (fμ,μ)= ϕ(ν μ) TV = f 1 ϕdμ. By Theorem 3.5, we get (3.11). The proof is complete. Remark 3.7 Taking ϕ 1 and considering the corresponding geometric constant κ, (3.11) is equivalent to (by Bobkov and Houdré [3, Theorem 1.1]) μ(a)μ(a c ) κ Q(e), A V, where e A A := {e =(x, y) E : x A, y A c } is the boundary of A. Thus(3.11) implies the standard Cheeger inequality μ(a) κ Q(e), A V such that μ(a) 1/, e A which has an equivalent functional version as: for every function f on V, f(x) med μ (f) μ(x) κ D e f Q(e), (3.1) x V

9 Logarithmic Sobolev, Isoperimetry and Transport Inequalities on Graphs 19 where med μ (f) is the median of f under μ. The Cheeger inequality (3.1) with the geometric constant κ is due to Diaconis and Stroock [13], whose idea goes back to Jerrum and Sinclair [19]. Thus, Corollary 3.6 slightly improves theirs in this particular case. The Cheeger isoperimetric inequality for general jump processes is studied by Chen and Wang [10]. Remark 3.8 If G =(V,E) is a tree, i.e., there is only one path without circle from x to y for any two different vertices x, y, then the geometric constant κ becomes optimal for two types of metrics: (a) ρ(x, y) =1 x y (ϕ(x)+ϕ(y)) (then the constant κ in the weighted L 1 -Poincaré inequality above is optimal in the case of trees); (b) ρ(x, y) =ρ w (x, y), the distance induced by some length function w. The optimality of κ for those two types of metrics in the case of trees is established by Liu Ma Wu [3], in a completely different way. The usual Cheeger inequality exhibits the relationship between the isoperimetry and the Poincaré inequality (see [1, 19, 1]). Now we present the relationship between the generalized Cheeger isoperimetric inequality and the Gaussian concentration. Corollary 3.9 Assume that y:y x q(x, y) B for all x V.Then c G κ B. Proof This is due to [18]. But for the self-completeness, we still present its proof. By the generalized Cheeger isoperimetric inequality in Theorem 3.5 and the Cauchy Schwarz inequality, we have, for any probability measure ν = fμ, W 1,ρ (fμ,μ) κ μ(x)q(x, y) f(x) f(y) x y κ 1 I(ν μ) μ(x)q(x, y)( f(x)+ f(y)) x y κ I(ν μ) μ(x)q(x, y)f(x) x y κ I(ν μ) B μ(x)f(x) =κ BI(ν μ), x V where the desired result follows. Corollary 3.10 For the Laplacian L =Δon the connected graph G =(V,E), we have, for the graph metric ρ 1, κ d bd, E where d,d,b are given in Corollary 3.4. Proof Choosing γ xy distributed uniformly on all shortest paths from x to y and w =1,since

10 130 Ma Y. T. et al. Q(e) =1/ E and μ(x) =d x / E, the geometric constant κ is bounded from above by max ( ) d E Eγ 1 γxy (e)ρ 1 (x, y) d bd. E E The proof is complete. 4 Several Examples and Graphs with Symmetry 4.1 Several Examples Webeginwithababy-model. Example 4.1 (Complete graph) Let G =(V,E) be a complete graph with n different vertices, i.e., for any different x, y V,(x, y) E (n of course). Consider the Laplacian L =Δand the graph metric ρ 1 which is now ρ 1 (x, y) =1 x y. Hence μ is the uniform distribution on V and Q(e) =1/ E =1/[n(n 1)]. In such case the Dirichlet form is given by E(f,f) = 1 (D e f) Q(e) = n n 1 Var μ(f), where Var μ (f) =μ(f ) μ(f) is the variance of f w.r.t. μ. Soc P = n 1 n. In this example, we take γ xy = {(x, y)} as random path and the length function w 1. For the logarithmic Sobolev constant c LS, notice that for any fixed edge e =(x 0,y 0 ), e γ xy if and only if x = x 0 and y = y 0 ;andl w,e (x) =1 x=x0 /n. Thus by Theorem.1, c LS E ( Ent μ (L w,e )+μ(l w,e )log(e +1) ) [ 1 = n(n 1) n log 1 n 1 n log 1 n + 1 ] n log(e +1) ( = 1 n) 1 [log n +log(e +1) ]. (4.1) Comparing with the optimal logarithmic Sobolev constant c LS = n 1 n log(n 1) for complete graph (see [1, Corollary A.5]), the estimate (4.1) has the correct order log n. Now we turn to bound the Gaussian concentration constant c G by the geometric quantity K in (3.4), w.r.t. the graph metric ρ 1.Wehave K = max (x,y) E 1 μ(x)μ(y) = E Q(x, y) n = n 1 n. Then by Theorem 3.1, for any μ-probability density f, W1,ρ 1 (fμ,μ)= 1 ( ) f 1 dμ n 1 4 n E( f, f) = Var μ ( f). V But by [17, Theorem 3.1], the corresponding optimal constant c G = n 1 n. Consequently, we have K = n 1 n = c G K.

11 Logarithmic Sobolev, Isoperimetry and Transport Inequalities on Graphs 131 For the generalized Cheeger isoperimetric inequality in Theorem 3.5 w.r.t. ρ 1,wehave or equivalently, x V c I κ = K n 1 n f(x) μ(f) μ(x) n 1 n D e f Q(e). This inequality becomes equality for the indicator function 1 A. Hence c I = κ, i.e., our generalized Cheeger isoperimetric inequality in Theorem 3.5 is optimal in this example. Example 4. (Star) Consider a star G =(V,E) with a central vertex v 0 and n outside vertices {v i : i = 1,...,n} connecting only with v 0. For the Laplacian L = Δ, we have μ(v 0 )= 1,μ(v k)= 1 1 n, 1 k n and Q(e) = n for every edge e. Itisknownthatc P =1(c.f. [13]). Taking the length function w 1 in (.4), we obtain by calculus ( 3 c LS 1 ) log[n(e +1)]. n Applying the logarithmic Sobolev inequality to f =1 vk, 1 k n, we get c LS log(n)/. Clearly, for large n, we have the correct order log n. For the geometric constants K and κ associated with the graph distance ρ 1,takingγ xy as the unique path from x to y without circle, we have Considering f(v) =n1 [v=v1 ], we have K 9 4 n and κ = 3 1 n. W 1 (fμ,μ)= 3 1 n, D e f Q(e) =, c I = κ = 3 1 n, i.e., the geometric quantity κ as an upper bound of c I is optimal. Example 4.3 (Trees) Consider the full binary tree of depth d. For d 1, such a tree has d+1 1 vertices, d+1 edges and the maximum degree is 3. Consider the Markov chain arising from nearest neighbor random walk on this tree. The longest path is of length d and the value of b defined in (3.8) is ( d 1) d. By Corollary 3.4, we have K 18 d d 3. For κ w.r.t. the graph distance ρ 1, by (3.10), Theorem 4.1 and Corollary 7.1 in [3], κ =(d 3) d +3 9 d d 1, which shows that Corollary 3.10 offers a good upper bound. 4. Graphs with Symmetry In this section, we shall consider various graphs with symmetry. See Chung [11] for examples and properties of symmetric graphs. For a graph G =(V,E), an automorphism f : V V is

12 13 Ma Y. T. et al. one-to-one mapping which preserves edges, i.e., for any x, y V,wehave(x, y) E if and only if (f(x),f(y)) E. For any oriented edge e =(x, y) E, consider the opposite oriented edge e := (y, x) and the non-oriented edge e 0 := {x, y}. PutE 0 := {e 0 : e E}, the set of all non-oriented edges Edge-transitive Graph A graph G is called edge-transitive if for any two non-oriented edges {x, y} and {x,y },there is an automorphism f such that {f(x),f(y)} = {x,y }. Corollary 4.4 Assume that G is edge-transitive. For the Laplacian L =Δand the graph metric ρ 1, we have c I κ E [ ρ 1(X, Y ) ] and c G κ (E[ρ 1(X, Y )]), where the law of (X, Y ) is μ μ and μ is the uniform measure on V. Proof We consider a random (ordered) pair of vertices (X, Y ), chosen according to μ μ. Now given (X, Y ), we choose randomly a shortest path γ XY between X and Y (uniformly chosen over all possible shortest paths from X to Y ). Notice that for w 1, ρ = ρ 1, 1 h(e) := E γ 1 γxy (e)ρ(x, y)μ(x)μ(y) = E Eρ 1 (X, Y )1 γxy (e) Q(e)w(e) satisfies h(e) =h( e )fore E (that is true on any graph). Then h canberegardedasalength function on E 0. Now by the edge-transitivity, h(e) does not depend on e, so we get by averaging over all edges e, Therefore h(e) = 1 E E E [ρ 1 (X, Y )1 γxy (e)] = E [ ρ 1(X, Y ) ]. κ max h(e) =E [ ρ 1(X, Y ) ]. Since y:y x q(x, y) 1 for all x V,wehavec G κ by Corollary 3.9. The proof is complete. Remark 4.5 From the result above, we may wonder whether on edge transitive graphs the correct order in diameter D of c G is D 4, and that of c I is D, which is indeed true. For example, for the Laplacian on the circle Z p := Z/pZ, c P is of order D. Taking the eigenfunction h corresponding to λ 1 =1/c P with h Lip(ρ1 ) =1,weseethatVar μ (h) isoforderd,too. By the central limit theorem, 1 t h(x s )ds t converges weakly to the normal law N(0,σ (h)), where the limit variance is 0 σ (h) = ( Δ) 1 h, h μ =c P Var μ (h),

13 Logarithmic Sobolev, Isoperimetry and Transport Inequalities on Graphs 133 which is of order D 4. But from the Gaussian concentration inequality, we always have c G σ (h). In other words, c G is at least of order D 4. We leave to the reader for verifying that the correct order of c I is D. Example 4.6 (Circle Z p ) Let p Z and consider the integers mod p as p points around a circle. For x and y in Z p,chooseγ xy as the shorter of the two paths from x to y. Forthismodel μ(x) =1/p (x Z p ),Q(e) =1/(p) for any edge e. Itiswellknownthatc P =(1 cos π p ) 1, which is also the logarithmic Sobolev constant c LS when p is even (c.f. [6]). Taking the length function w 1 in (.4), we obtain by careful calculation, log(3(e +1)) (p +1)(p +), p is even; c LS 1 log(3(e +1)) (p +1)(p +)(1+3/p), p is odd, (4.) 1 ( 5 log(3(e +1)) 1 cos π ) 1. p This, together with c LS c P =(1 cos π p ) 1, offers a two sided estimate for c LS with factor 5 log(3(e +1)). Since the circle Z p is edge transitive, by Corollary 4.4, we have c I κ p 1 + o(p ); c G κ p4 1 + o(p4 ). For this model, Sammer and Tetali [8] proved that the best constant in the transportationentropy inequality (3.6) is p 48 + o(p ). Then by Corollary 3.3, we have p o(p4 ) c G p4 1 + o(p4 ). Thus the correct order of c G is p 4, as mentioned in Remark Vertex-transitive Graph A graph G is vertex-transitive if for any two vertices u and v, there is an automorphism f such that f(u) =v. The automorphism group defines an equivalent relation on the edges of G. Two undirected edges e 0 1, e 0 are equivalent if and only if there is an automorphism mapping e 0 1 to e 0. We can consider equivalent classes of undirected edges, denoted by E 0 1,...,E 0 s. The index of G is defined as index(g) =max i E 0 E 0 i. Clearly E 0 i V,i =1,...,s, 1 index(g) d, where d = d x for any x V, the degree of the graph G. See[11]. For any edge e such that e 0 Ei 0, i =1,...,s, E μ μ [ ] 1 ρ 1 (X, Y )1 [e γxy ] = Ei 0 E μ μ [ ] ρ 1 (X, Y )1 [e γxy ] e:e 0 Ei 0 index(g) E μ μ [ ρ E 1(X, Y ) ].

14 134 Ma Y. T. et al. For any two vertices y, y V, there is an automorphism such that f(y) =y. Since the stationary distribution μ 1/ E and ρ 1 (x, f(y)) = ρ 1 (f 1 (x),y) for any x V,wehave E μ [ ρ 1(X, y ) ] = E μ [ ρ 1(X, f(y)) ] = E μ [ ρ 1(f 1 (X),y) ] = E μ [ ρ 1(X, y) ]. Then for any fixed vertex v 0 V, E μ μ [ ρ 1(X, Y ) ] = E μ [ ρ 1(X, v 0 ) ]. Therefore, we have proved the following estimate of the Gaussian constant c G in (3.1). Corollary 4.7 Let Δ be the Laplacian operator on a vertex transitive graph G. For any fixed vertex v 0 V and the graph metric ρ 1, we have c I κ index(g)e μ [ρ 1(X, v 0 )] de μ [ρ 1(X, v 0 )] and where c G K κ, K index(g)e μ [ρ 4 1(X, v 0 )] de μ [ρ 4 1(X, v 0 )]. The proof of the bound on K is similar, omitted here Distance Transitive Graph AgraphG is distance transitive if for any two pairs of vertices {x, y} and {x,y } with ρ 1 (x, y) = ρ 1 (x,y ), there is an automorphism mapping x to x and y to y. The distance transitive graph is both edge-transitive and vertex-transitive. Then index(g) = 1. The estimate in Corollary 4.7 holds for distance transitive graphs, that is, for any fixed vertex v 0 V, c G ( E μ [ ρ 1(X, v 0 ) ]) and c I E μ [ ρ 1(X, v 0 ) ]. (4.3) Example 4.8 The vertex set V consists of all the subsets of k elements in {1,,..., n} (1 k n is fixed). Define a metric on V by d(x, y) =k x y. The edge set is given by {(x, y) V : d(x, y) =1}. This is a distance transitive graph. The Markov process generated by the Laplacian Δ on G is known as the Bernoulli Laplace diffusion model. See Lee and Yau [] for the estimate of the logarithmic Sobolev constant, Gao and Quastel [16] for the exponential decay rate of entropy. By (4.3), we have for the graph metric ρ 1 = d, c I κ and c G κ, where κ ( 1 min{k,n k} ( k n j k) j j=0 )( n k j ). Acknowledgements The authors are grateful to the anonymous referees for constructive comments and corrections.

15 Logarithmic Sobolev, Isoperimetry and Transport Inequalities on Graphs 135 References [1] Alon, N., Milman, V.: λ 1, isoperimetric inequalities for graphs, and superconcentrators. J. Combin. Theory, Ser. B., 38, (1985) [] Barthe, F., Roberto, C.: Sobolev inequalities for probability measures on the real line. Studia Math., 159(3): (003) [3] Bobkov, S., Houdré, C.: Some connections between isoperimetric and Sobolev-type inequalities. Mem. Amer. Math. Soc., 19, 1997 [4] Bobkov, S., Götze, F.: Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal., 163, 1 8 (1999) [5] Chen, G. Y., Liu, W.-W., Saloff-Coste, L.: The logarithmic Sobolev constant of some finite Markov chains. Ann. Fac. Sci. Toulouse Math., 17(6), (008) [6] Chen, G. Y., Sheu, Y. C.: On the log-sobolev constant for the simple random walk on the n-cycle: the even cases. J. Funct. Anal., 0, (003) [7] Chen, M. F.: Analytic proof of dual variational formula for the first eigenvalue in dimension one. Sci. China Ser. A, 4(8), (1999) [8] Chen, M. F.: Logarithmic Sobolev inequality for symmetric forms. Sci. China Ser. A, 43(6), (000) [9] Chen, M. F.: Eigenvalues, Inequalities and Ergodic Theory, Springer, London, 005 [10] Chen, M. F., Wang, F. Y.: Cheeger s inequalities for general symmetric forms and existence criteria for spectral gap. Ann. Probab., 8(1), (000) [11] Chung, F. R. K.: Spectral Graph Theory, CBMS Regional Conference Series in Mathematics, 9, Washington, American Mathematical Society, Providence, RI, 1997 [1] Diaconis, P., Saloff-Coste, L.: Logarithmic Sobolev inequalities for Markov chains. Ann. App. Probab., 6(3), (1996) [13] Diaconis, P., Stroock, D.: Geometric bounds for eigenvalues of Markov chains. Ann. App. Probab., 1(1), 36 6 (1991) [14] Donsker, M. D., Varadhan, S. R. S.: Asymptotic evaluation of certain Markov process expectations for large time. IV. Comm. Pure Appl. Math., 18, (1983) [15] Fill, J.: Eigenvalue bounds on bonvergence to stationarity for nonreversible Markov chains, with an application to the exclusion process. Ann. App. Probab., 1(1), 6 87 (1991) [16] Gao, F., Quastel, J.: Exponential decay of entropy in the random transposition and Bernoulli Laplace models. Ann. Appl. Probab., 13, (003) [17] Guillin, A., Léonard, C., Wu, L. M., et al.: Transportation-information inequalities for Markov processes. Probab. Th. Relat. Fields, 144, (009) [18] Guillin, A., Joulin, A., Léonard, C., et al.: Transportation-information inequalities for Markov processes (III). Processes with jumps, preprint, (013) [19] Jerrum, M., Sinclair, A.: Approximating the permanent. SIAM J. Computer, 18, (1989) [0] Kahale, N.: A semidefinite bound for mixing rates of Markov chains, Integer Programming and Combinatorial Optimization, Lecture Notes in Computer Science, 1084, 1996, [1] Lawler, G. F., Sokal, A. D.: Bounds on the L spectrum for Markov chains and Markov processes: a generalization of Cheeger s inequalities. Trans. Amer. Math. Soc., 309, (1988) [] Lee, T. Y., Yau, H. T.: Logarithmic Sobolev inequality for some models of random walks. Ann. Probab., 6, (1998) [3] Liu, W., Ma, Y. T., Wu, L. M.: Spectral gap, isoperimetry and concentration on trees. Sci. China Math., 59, (016) [4] Roberto, C.: A path method for the logarithmic Sobolev constant. Combin. Probab. Comput., 1, (003) [5] Rothaus, O.: Diffusion on compact Riemannian manifolds and logarithmic Sobolev inequalities. J. Funct. Anal., (1981) [6] Rothaus, O.: Analytic inequalities, isoperimetric inequalities and logarithmic Sobolev inequalities. J. Funct. Anal., 64, (1985) [7] Saloff-Coste, L.: Lectures on finite Markov chains, In Lectures on Probability Theory and Statistics: École d Été de Probabilités de St-Flour, Vol of Lecture Notes in Mathematics, Springer, Berlin, 1996

16 136 Ma Y. T. et al. [8] Sammer, M., Tetali, P.: Concentration on the discrete torus using transportation. Combin. Probab. Comput., 18, (009) [9] Sinclair, A.: Improved bounds for mixing rates of Markov chains and multicommodity flow. Combin. Probab. Comput., 1, (199) [30] Villani, C.: Optimal transport. Old and new. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338, Springer-Verlag, Berlin, 009