The Tightness of the Kesten Stigum Reconstruction Bound of Symmetric Model with Multiple Mutations

Transcription

1 J Stat Phys 08 70: The Tightness of the Kesten Stigum Reconstruction Bound of Symmetric Model with Multiple Mutations Wenjian Liu Sreenivasa Rao Jammalamadaka Ning Ning Received: 6 January 07 / Accepted: 30 November 07 / Published online: 7 December 07 Springer Science+Business Media, LLC, part of Springer Nature 07 Abstract It is well known that reconstruction problems, as the interdisciplinary subject, have been studied in numerous contexts including statistical physics, information theory and computational biology, to name a few We consider a q-state symmetric model, with two categories of q states in each category, and 3 transition probabilities: the probability to remain in the same state, the probability to change states but remain in the same category, and the probability to change categories We construct a nonlinear second-order dynamical system based on this model and show that the Kesten Stigum reconstruction bound is not tight when q 4 Keywords Kesten Stigum reconstruction bound Markov random fields on trees Distributional recursion Dynamical system Mathematics Subject Classification 60K35 8B6 8B0 Introduction Preliminaries We start with the following broadcasting process that stands as a discrete, irreducible, aperiodic, and reversible Markov chain Let T V, E,ρbe a tree with nodes V, edges E and B Wenjian Liu wjliu@qcccunyedu Sreenivasa Rao Jammalamadaka rao@pstatucsbedu Ning Ning ning@pstatucsbedu Department of Mathematics and Computer Science, Queensborough Community College, The City University of New York, New York City, NY, USA Department of Statistics and Applied Probability, University of California, Santa Barbara, CA, USA

2 68 W Liu et al root ρ V Each edge of the tree acts as a channel on a finite characters set C, whose elements are configurations on T, denoted by σ We set a probability transition matrix M M ij as the noisy communication channel on each edge The state of the root ρ, denoted by σ ρ,is chosen according to an initial distribution π on C, and then propagated in the tree as follows: for each vertex v having u as its parent, the spin at v is defined according to the probabilities Pσ v j σ u i M ij with i, j C Roughly speaking, reconstruction is to answer the question that considering all the symbols received at the vertices of the nth generation, does this configuration contain non-vanishing information transmitted by the root, as n goes to? In this paper, we will restrict our attention to d-ary trees, ie the infinite rooted tree where every vertex has exactly d offspring every vertex has degree d + except the root which has degree d Let σn denote the spins at distance n from the root and let σ i n denote σn conditioned on σ ρ i Consider a characters set C C C, consisting of two categories C {,,q} and C {q +,,q} with q, and the state of the root ρ is chosen according to the uniform distribution on C Moreover, a q q probability transition matrix M M ij q q is defined as follows: p 0 if i j, M ij p if i j and i, j are in the same category, if i j and i, j are in different categories, p where p 0, p and p are all nonnegative, such that p 0 + q p + qp It can be verified that the eigenvalues of M are λ p 0 p, λ p 0 + q p qp,and λ 3 p 0 +q p +qp Therefore, there are two candidates λ and λ for the second largest eigenvalue in absolute value, say, λ, which plays a crucial role in the reconstruction problem Definition The reconstruction problem for the infinite tree T is solvable if for some i, j C, lim sup d TV σ i n, σ j n > 0 n where d TV is the total variation distance When the lim sup is 0, we say the model has non-reconstruction on T Background Beyond the basic interest in determining the reconstruction threshold of a Markov random field in probability, this problem is relevant to statistical physics, biology Daskalakis et al [], Mossel [33], and information theory Bhamidi et al [5], Evans et al [5], where one is interested in computing the information capacity of the tree network Most closely related to the origins of this work, for spin systems in statistical physics, the threshold for reconstruction is equivalent to the threshold for extremality of the infinite-volume Gibbs measure induced by free-boundary conditions, see Georgh [7] The reconstruction threshold also has an important effect in the efficiency of the Glauber dynamics on trees and random graphs It is well known that when the model is reconstructible, the mixing time for the Glauber dynamics on trees is n +Ω, while it is slower than at higher temperature when the mixing time is On log n The corresponding bound is tight for the Ising model, namely, the mixing time is On log n when dλ < In Martinelli et al [30], this result is extended to the log Sobolev

3 The Tightness of the Kesten Stigum Reconstruction Bound 69 constant and it is also shown that for measures on trees, a super-linear decay of point-toset correlations implies an Ω spectral gap for the Glauber dynamics with free boundary conditions A similar transition takes place in the colouring model as shown in Tetali et al [46] Sharp bounds of this type are not known for the hardcore model, although it is conjectured that the Glauber dynamics should again be On log n in the non-reconstruction regime For any channelm, it is well known that the reconstruction problem is connected closely to λ, the second largest eigenvalue in absolute value of M An important general bound was obtained by Kesten and Stigum [,]: the reconstruction problem is solvable if d λ > λ may be a compleumber, which is known as the Kesten Stigum bound On the other hand, for larger noise d λ < one may wonder whether reconstruction is possible, by exploiting the whole set of symbols received at the nth generation, through a clever use of the correlations between the symbols received on the leaves The answer depends on the channel For the binary symmetric channel, it was shown in Bleher et al [7] that the reconstruction problem is solvable if and only if dλ > For all other channels, however, it would be a little challenging to prove the non-reconstructibility Mossel [3,34] showed that the Kesten Stigum bound is not the bound for reconstruction in the binary asymmetric model with sufficiently large asymmetry or in the Potts model with sufficiently many characters, which sheds the light on exploring the tightness of the Kesten Stigum bound The first exact reconstruction threshold in roughly a decade, was obtained by Borgs et al [8], in which the authors displayed a delicate analysis of the moment recursion on a weighted version of the magnetization, and thus achieved a breakthrough result A particularly important example is provided by q-state symmetric channels, ie Potts models in the terminology of statistical mechanics, with the transition matrix p 0 p p p p 0 p M p p p 0 and λ p 0 p This model was completely investigated by Sly [43] bymeansofthe recursive structure of the tree, and more importantly, Sly showed that non-reconstruction is equivalent to lim n 0, where EPσ ρ σ n q Thusthekeyideawas to analyze the recursion relationship between and + This work then went on to engage the refined recursive equations of vector-valued distributions and concentration analyses, to confirm much of the picture conjectured earlier by Mézard and Montanari [3] Inspired by the popular K80 model proposed by Kimura [3], which distinguishes between transitions and transversions, we analyze the case that transition matrix has two mutation classes with q states in each class Improved flexibility comes along with increased complexity, which is mainly due to the fact that the additional class of mutation complicates the discussion of the second largest eigenvalue in absolute value However, by introducing additional auxiliary quantities y n and z n besides defined in Sect, we succeed in investigating the tightness of the Kesten Stigum bound 3 Applications The reconstruction problem arises naturally in many fields including statistical physics, where the Ising model and the Potts model are popular and have been studied extensively from

4 60 W Liu et al different angles, see [,0,3,4,6,9,0,7 9,36,38,40,4,44,45,47,48] In this article, we focus on the reconstruction threshold on trees, which plays an important role in the dynamic phase transitions in certain glassy systems subject to random constraints For random colorings on the Erdös Rényi random graph with average connectivity d, Achlioptas and Coja-Oghlan [] proved that there is a phase transition, from the situation that most of the mass is contained in one giant component, to the case that the space of solutions breaks into exponentially many smaller clusters This phase transition has been proved corresponding to known bounds on the reconstruction threshold for proper colorings on trees, see eg Mossel and Peres [35], Semerjian [39]andSly[4] In computational biology, the broadcast model is the main model for the evolution of base pairs of DNA Phylogenetic reconstruction is a major task of systematic biology, which is to construct the ancestry tree of a collection of species, given the information of present species The corresponding reconstruction threshold answers the question whether the ancestral DNA information can be reconstructed from a known phylogenetic tree This threshold is also crucial to determine the number of samples required, in the sense that, only enumerations of each type of spin at the leaves are collected, regardless of their positions on the leaves Interested readers on Phylogenetic tree reconstruction are referred to Roch [37] and Daskalakis et al [] The popular K80 model [3], has some obvious advantages over other models in Phylogeny reconstruction, which is favored by both Akaike Information Criterion and Bayesian Information Criterion see Sect in Cadotte and Davies [9] The K80 model distinguishes between transitions A G, ie from purine to purine, or C T, ie from pyrimidine to pyrimidine and transversions from purine to pyrimidine or vice versa Inspired by this and related literatures, we analyze the case that the transition matrix has two mutation classes and q states in each class We believe that the q-state symmetric Potts model as a generalization of -state symmetric Ising model, cannot fully represent the spirit of the classical -state symmetric Ising model in terms of dichotomy, and this is one of the areas this work can contribute to A tree is a connected undirected graph with no simple circuits In other words, an undirected graph is a tree if and only if there is a unique simple path between any two of its vertices The theory that the reconstruction threshold on trees corresponds to the reconstruction threshold on locally treelike graphs, is verified in Gerschenfeld and Montanari [8] The strong and increasing interest in the study of the properties of social networks, is a result of the rapid and global emergence of online social networks and their meteoric adoption by millions of Internet users When it comes to Socio psychological mechanisms of generation and dissemination of network, our model s advantage in providing more flexibility to mimic psychological behaviors is obvious For example, our model and the construction threshold can be used to effectively identify community effect in social networks and customer loyalty in marketing research, especially for different firms or organizations who want to promote their products or philosophies In this sense, many possible extensions can be made on research on graph structures with psychological factors involved, such as the work by Liu, Ying and Shakkottai [4] on analyzing the formation and propagation of opinions across networks by an Ising model based approach, the work by Bisconti et al [6] on reconstruction of a real world social network using the Potts model and Loopy Belief Propagation, etc 4 Main Results and Proof Sketch Because non-reconstruction happens at most d λ, without loss of generality, it would be convenient to presume / d λ in the following context

5 The Tightness of the Kesten Stigum Reconstruction Bound 6 Main Theorem Assume 0 < λ λ Whenq 4, for every d the Kesten Stigum bound is not tight, ie the reconstruction is solvable for some λ even if dλ < The ideas and techniques used to prove the Main Theorem can be seen as the following One standard to classify reconstruction and nonreconstruction is to analyze the quantity : the probability of giving a correct guess of the root given the spins σn at distance n from the root, minus the probability of guessing the root randomly which is q in this case Nonreconstruction means that the mutual information between the root and the spins at distance n goes to 0 as n tends to infinity It can be established that the nonreconstruction is equivalent to lim n 0 Our analysis is similar to Borgs et al [8], Chayes et al [] in the context of spin glasses, and Sly [43] However, the two classes of mutation complicates the discussion of λ, the second largest eigenvalue in absolute value of the transition matrix, which makes the problem much more challenging In this case, it is necessary to consider the corresponding quantities similar to, viz wrong guess but right group y n, and wrong guess and even wrong group z n In Sect, we investigate the properties and relations between, y n and z n By these preliminary results, we focus on the analysis of and z n in the sequel In order to research the reconstruction, according to the Markov random field property, we establish the distributional recursion and moment recursion, by analyzing the recursive relation between the nth and the n + th generations structure of the tree Furthermore, we display that the interactions between spins become very weak, if they are sufficiently far away from each other Therefore, we can obtain a nonlinear dynamical system If is small, we are able to develop the concentration analysis and achieve the approximation to the dynamical system: + dλ + dλ dλ z n + dd 4qλ 4 z n z n+ dλ z n dd q q λ4 + z n 4qλ 4 z n qq 5 q λ 4 + z n 4qλ λ + z n z n Finally, we investigate the stability of the system and then establish the threshold of q relevant to the reconstruction When q 4, even if dλ < forsomeλ, will not converge to 0 and hence there is reconstruction beyond the Kesten Stigum bound More detailed definitions and interpretations can be seen in the next section Second Order Recursion Relation Notations Let u,,u d be the children of ρ and T v be the subtree of descendants of v TFurthermore, if we set d, as the graph-metric distance on T, denote the nth level of the tree by L n {v V : dρ, v n} and then let σ j n be the spins on L n T u j For a configuration A on L n, define the posterior function f n i, A Pσ ρ i σn A

6 6 W Liu et al By the recursive nature of the tree for a configuration A on Ln + T u j,thereisan equivalent form f n i, A Pσ u j i σ j n + A Now for any i q, define a collection of random variables X i n f n i,σn to describe the posterior probability of state i at the root given the random configuration σn of the leaves, and analogously, X n f n,σ n, X n f n,σ n, X 3 n f n q +,σ n By symmetry, the collections { f n i,σ n : i q} and { f n i,σ n : q + i q} are exchangeable respectively; in addition, f n j,σ i n is distributed as f n j,σ i n D X n if i j, X n if i j are in the same category, X 3 n if i j are in different categories Finally, denote the first and second central moments of X n, X n and X 3 n,which would be the principal quantities in our analysis, as E X n, y n E X n, z n E X 3 n, q q q and u n E X n, v n E X n, w n E X 3 n q q q Preliminaries For any i,, q and nonnegative n Z, it is concluded from the symmetric property ofthetreethat EX i n q is always true Lemma For any n N {0}, wehave E q X i n 0, z n 0, and + z n 0 q Proof First, by Bayes rule, we have + q f n, APσ n A σ ρ q A A qex n and 0 E q X i n q q EX i n q q Pσ n A fn, A EX i n + q

7 The Tightness of the Kesten Stigum Reconstruction Bound 63 { } q Next, we consider the covariance matrix of random variables X i n q and express covariances in terms of, y n and z n Similarly, we obtain y n + q q A Pσ n A f n, A f n, A qex nx n, so for any i < i in the same category, it is concluded from the symmetric property of the tree that E X i n X i n E X X y n q q q q q Similarly, if i and i are from different categories, we have E X i n X i n z n q q q Therefore, the covariance matrix is given by Σ X n q y n q y n q z n q z n q z n q y n y q n z n z n z q q q n q y q n z n z n z q q q n q y n x q n z n z n z q q q n q z n z q n y n y q q q n q z n z q n y n y q q q n q z n q z n q y n q y n q q q q whose eigenvalues are 0, +q y n qz n q and y n q It is well known that the covariance matrix of a multivariate probability distribution is always positive semi-definite, which implies that all eigenvalues are nonnegative, say, + q y n qz n 0and y n 0 It suffices to complete the proof, by these results and the fact of + q y n + qz n 0 Lemma For any n N {0}, the following hold: i u n + q v n + qw n ; ii E X n q X n q v n + y n q ; iii E X n q X 3 n q w n + z n q ; iv E X n q X 3 n q w n q z n v E f n q +,σ n q f n q,σ n q vi E f n,σ n q f n q,σ n q qq y n q ; w n q v n q z n q + z n q ; qw n q q

8 64 W Liu et al Proof By the total probability formula and using Lemma, we can prove i as follows: E q q j q j X i n q E q + qe q [ E X i n σ ρ j q X n q ] X 3 n q E X n q +qe X 3 n q Pσ ρ j + q E X n q + q E X n q u n + q v n + qw n Applying the same technique, we obtain EX nx n Pσ ρ σn APσ ρ σn APσn A A σ ρ [Pσ ρ σn A] Pσ n A σ ρ A E X n and hence ii follows: E X n X n q q Similarly, iii turns out to be true due to EX nx 3 n E E X + y n q q X 3 n v n + y n q The statement iv, v and vi can be handled in the same way, using the symmetry, EX nx 3 n Pσ ρ σn APσ ρ q + σn AP σn A A σ ρ Pσ ρ σn APσ ρ σn AP σn A A σ ρ q + Pσ ρ q + σn APσ ρ q σn AP σn A

9 The Tightness of the Kesten Stigum Reconstruction Bound 65 To obtain EX nx 3 n, recall that z n + q EX 3 n which implies that E X n q A σ ρ E f n q +,σ n f n q,σ n 3 E f n q +,σ n q f n i,σ n EX nx 3 n + q EX nx 3 n +EX 3 + q E f n q +,σ n f n q,σ n EX nx 3 n + q EX nx 3 n +EX 3, X 3 n w n q q Thus, 3 together with 4gives E f n q +,σ n f n q,σ n q q As the preceding discussion, consider z n qq y n q 4 w n q z n q and thus y n + q q E f n,σ n f i,σ n EX n + q E f n,σ n f n q,σ n +qex nx 3 n, E f n,σ n q q q y n + y n + q f n q,σ n q q EX n qex nx 3 n + 4q v n q z n q + qw n q q 3 Means and Covariances of Y ij Define Y ij n f n i,σ j n +, and it is apparent that the random vectors Y ij q are independent, for j,,d, by the symmetries of the model The central moments of Y ij wouldplayakeyroleinfurther

10 66 W Liu et al analysis, and therefore it is necessary to figure them out in the first place For each j d, we rely on the total probability formula to conclude: i when i, ii for i q, E Y ij n q E Y j n p 0 E X n q q + q p E X n q + qp E X 3 n q λ + λ λ z n ; p E X n q +[p 0 + q p ]E + qp E X 3 n q X n q λ q λ + q λ z n ; q iii for q + i q, by means of the identity q Y ijn, it follows immediately that E Y ij n q E Y ij n λ z n ; q q q iv resembling the discussion of i, ii and iii, it is further concluded that when i, E Y j n + λ λ + λ u n + λ λ w n ; q q v for i q, E Y ij n q q + λ q + λ λ qq q u n λ + q λ w n ; q vi for q + i q, E Y ij n λ + λ w n ; q q vii for i q, E Y j n Y ij n q + λ λ z n q q qq q λ q u n q + λ λ w n ; q

11 The Tightness of the Kesten Stigum Reconstruction Bound 67 viii for q + i q, E Y j n Y ij n λ q q q + z n q + λ w n ; ix for < i < i q, E Y i j n q Y i j n q [ q + λ + q λ qq q qq λ + ] x for < i q < i q, E Y i j n Y i j n q q xi for q + i < i q, E Y i j n Y i j n q q 4 Distributional Recursion z n q q q u n + q + λ + q λ q q w n ; λ qq + z n q λ q w n; λ qq z n q λ q w n The key method of this paper is to analyze the relation between X n, X 3 n and X n+, X 3 n + using the recursive structure of the tree Take A σ n + and then the following relations follow from the Markov random field property: and X n + f n+,σ n + Z q Z i where A for i q, X 3 n + f n+ q +,σ n + Z q+ q Z, i Z i Z i n d j [ + qp 0 p Y ij q + qp p d j l i q Y lj q [ + qp 0 p Y ij q

12 68 W Liu et al B for q + i q, Z i Z i n qp p d j d j q+ l q [ + qλ Y ij q + λ λ q+ l q Y lj q Y lj, q [ + qp 0 p Y ij q + qp p d j q+ l i q Y lj q [ + qp 0 p Y ij q qp p d j l q [ + qλ Y ij q + λ λ l q Y lj q Y lj q To continue the proof, it is necessary to firstly reveal some identities concerning Z i n Lemma 3 For any nonnegative n Z and i q, we have EZ nz i n EZ i n, and given any i q < q + i q, we have EZ i nz i n EZ q+ nz q n Proof When i, the result is trivial If i q, for any configurations A A,,A d on the n + th level, where A j denote the spins on L n+ T u j,we have Pσ n + A Z i A q dj Pσ j n + A j Pσ ρ i σn + A

13 The Tightness of the Kesten Stigum Reconstruction Bound 69 By the symmetry of the tree, we have EZ Z i q Pσ n + A dj Pσ ρ σn + A A Pσ j n + A j Pσ ρ i σn + APσ n + A σ ρ q Pσ n + A dj P σ ρ σn + A Pσ j n + A j A Pσ n + A σ ρ i q Pσ n + A dj P σ ρ i σn + A Pσ j n + A j A Pσ n + A σ ρ EZi Similarly, given arbitrary i q < i q, EZ i Z i q Pσ n + A dj Pσ ρ σn + A A Pσ j n + A j Pσ ρ i σn + APσ n + A σ ρ i q Pσ n + A dj Pσ ρ q + σn + A A Pσ j n + A j Pσ ρ q σn + APσ n + A σ ρ EZ q+ Z q Next approximate the means and variances of monomials of Z i by expanding them using Taylor series The following relations hold, where the symbol O q emphasizes that the corresponding constant depends only on q i When i, EZ + d [ qλ + qλ λ z ] n dd [ + qλ + qλ λ z ] n + Oq xn 3 ; ii For i q, [ ] EZ i + d qλ qλ q q + qλ z n [ ] dd + qλ qλ q q + qλ z n + O q xn 3 ; iii For q + i q, EZ i + dqλ z n + dd qλ z n + O q xn 3

14 630 W Liu et al Consider covariances of Z i By Lemma 3, it is known that EZ Z i EZi, therefore we obtain the following results: a when i, where Π E b for i q, EZ + dπ + dd Π + O qxn 3, + qλ Y j + λ λ q 6qλ + 6qλ λ z n + 4q λ 3 +q λ λ λ w n x n ; q EZ i + dπ + q+ l q u n x n q dd Π + O qxn 3, where Π E + qλ Y ij + λ λ q qq 3 qq 3 λ q + λ q 6qλ 4q 3λ + q 3λ λ w n x n ; q q c for q + i q, where Π 3 E EZ i + dπ 3 + q+ l q z n 4q q λ3 dd Π3 + O qxn 3, + qλ Y ij + λ λ q qλ + qλ + λ z n + 4q λ λ d for i < i q, EZ i Z i EZ Z q + dπ 4 + q+ l q w n x n ; q Y lj q Y lj q u n x n q Y lj q dd Π4 + O qxn 3,

15 The Tightness of the Kesten Stigum Reconstruction Bound 63 where Π 4 E + qλ Y j q + λ λ + qλ Y qj + λ λ q 6qλ q e for q + i q, where Π 5 E 6qλ q + 6qλ +4q 6λ + 3q 6λ q q λ w n q EZ Z i EZ Z q+ + dπ qλ Y j q q+ l q q+ l q 8q λ 3 z n + q q ; + λ λ Y lj q Y lj q u n x n q dd Π5 + O qxn 3, q+ l q + qλ Y q+ j + λ λ q qλ q + qλ q + qλ 5 Main Expansion of + and z n+ l q z n 4q q λ λ Y lj q Y lj q w n q In this section, we aim to figure out the second order recursive relation between + and z n+, by virtue of the following identity a s + r a s ar s + r a s s + r 5 Specifically, taking a Z, s q and r q Z i q,5 yields + + q E Z q Z i z n+ + q E Z q+ q Z i E Z q E Z +E E Z q+ q Z q Z i q Z i q q q Z i q q ; 6 E Z q q+ Z i q q +E Z q+ q Z i q q Z i q 7

16 63 W Liu et al Finally, plugging the results of Sect 4 into 6 and7 and taking substitutions of X n + z n and Z n z n, we obtain a two dimensional recursive formula of the linear diagonal canonical form: where and X n+ dλ X n + dd Z n+ dλ Z n + dd Z R x E q R z E qq 3 q λ 4 X n + 4qλ λ X nz n + R x + R z + V x 8 q q λ4 X n 4qλ4 Z n R z + V z Z i Zq+ q q Z i q q Z i q q, q Z i q q, V x, V z C V xn u n q + w n q + with C V a constant depending on q only 3 Proof of the Main Theorem If the reconstruction problem is solvable, then σn contains significant information on the root variable This may be expressed in several equivalent ways Mossel [3], Proposition 4 Lemma 4 The nonreconstruction is equivalent to lim 0 n In order to study the stability of dynamical system 8, we expect R x, R z and V x, V z to be just small perturbations, for example, of the order oxn It is known that fixed finite different vertices far away from the root can affect the root little, based on which, it is possible Z to explore further the concentration analysis We can verify that q Z i and Z q+ q Z i are both sufficiently around q, and thus are able to bound R x and R z in 8 Lemma 5 Assume min{ λ, λ } >ϱ,forsomeϱ>0 For any ε>0, thereexistn Nq,εand δ δq,ε,ϱ>0, such that if n N and δ, then R x εx n and R z εx n Proof For any η>0and i q, applying Cauchy-Schwartz inequality, E Z q Z i q q Z i q E q Z i q q q E q Z i q Z q q Z i q

17 The Tightness of the Kesten Stigum Reconstruction Bound 633 q ηe Z i q q ; q +E Z i q q I ηe + q Z i q q E q Z i q 4 q 4 Z q Z i Z q Z i / P q η q >η Z q Z i / q >η We can derive from the calculation for distributional recursion that q E Z i q q C qxn and E q Z i q 4 q 4 C q Similar to Lemma of Sly [43] and Lemma 43 of Liu and Ning [5], there exist C 3 C 3 q,η,ϱand N Nq,η, such that when n N, Z P q Z i q >η C 3 xn 6 Thus there exists C 4 C 4 q,η,ϱ, such that R x E Z q Z q Z i q i q q ηc xn Finally, it suffices to take C η ε/, and then if δ,wehaver x εxn Similar analysis gives R z εxn Prior to establishing the concentration results involving V x and V z, we need to firstly show that the value of does not drop too fast to be non-reconstruction Lemma 6 For any ϱ>0, there exists a constant γ γq,ϱ>0, such that for all n, if min{ λ, λ } >ϱ + γ, Proof Similarlyto, for a configuration A on T u Ln+, define the posterior function g n+, A Pσ ρ σ n + A q + p 0 f n, A q q +p f n i, A q i +p q iq+ f n i, A q

18 634 W Liu et al and then q + λ f n, A q + λ λ q q iq+ f n i, A q Eg n+,σ n + q + λ E Y n q + λ λ qe q Y q+ n q q + λ + λ λ z n The estimator that chooses a state with probability f n+ i,σ n+ correctly reconstructs the root with probability q +λ +λ λ z n Apparently this probability must be less than the maximum-likelihood estimator Mézard and Montanari [3] Therefore, the following inequalities hold: q + λ + λ λ z n E max X i n + i q q + E max X i n + / i q q q + E X i n + / q q + x/ n+ On one hand, if λ λ, then it is concluded from + z n 0 in Lemma that λ λ + λ λ xn + z n λ + λ λ zn x / n+ On the other hand, if λ λ then λ x / n+,sincez n 0 To sum up, we always have min{λ,λ } x / n+ Next choose ε ϱ It can be concluded from 8, Lemma 5, as well as the inequalities u n q, w n q, 3 that there exists a δ δq,ε>0, such that, when <δ, one has + d min{λ,λ } ε d ϱ ϱ On the contrary, if δ, one has + min{λ,λ } ϱ 4 δ Finally, taking γ min{ϱ,ϱ 4 δ} completes the proof The following lemma improves the result of Lemma by establishing the strict positivity of the sum of and z n

19 The Tightness of the Kesten Stigum Reconstruction Bound 635 Lemma 7 Assume λ 0 For any nonnegative n Z, we always have + z n > 0 Proof In Lemma we have proved that + z n 0, so here it suffices to exclude the equality Now let us apply reductio ad absurdum and assume + z n 0forsomen N It follows that for i j in the same configuration set, one has EX i n X j n EXi n EX i nx j n + z n q 0 Therefore, X n X n X q n and X q+ n X q+ n X q n as, that is, for any configuration combination A on the nth level, we always have Pσ ρ σn A Pσ ρ σn A Denote the leftmost vertex on the nth level by v n, and it follows Pσ ρ σ vn Pσ ρ σ vn Define the transition matrices at distance s by U s M s,, V s M s,, and W s M s,q+, and then it is convenient to figure out the iterative formulae for them U s p 0 U s + q p V s + qp W s V s p U s +[p 0 + q p ]V s + qp W s W s p U s + q p V s +[p 0 + q p ]W s To evaluate this three order recursive system, starting with the difference of the first two equation U s V s λ U s V s, and then in light of the initial conditions U 0 andv 0 W 0 0, it follows that U s V s λ s 3 Finally, from the reversible property of the channel, we can conclude that λ n U n V n Pσ ρ σ vn Pσ ρ σ vn 0, ie, λ 0, a contradiction to the assumption of λ 0 The following result provides the crucial concentration estimates of u n q and w n when is small Lemma 8 Assume λ >ϱ, and λ λ or λ / λ κfor some κ> For any ε>0,thereexistn Nq,κ,εand δ δq,κ,ϱ,ε>0, such that if n N and δ, one has u n q <ε and w n q <ε q,

20 636 W Liu et al Proof Applying the identity 5, we have Z q q Z i u n+ E 4q E 6q 4 E + 6q 4 E q Z i Z q q Z i Z q Z q Z i q q Z i q Z i q Z i q Z i 4q 4q The first expectation of 33 will contribute the major terms of the expansion: E Z q Z i q EZ q q EZ Z i q + 4q E q Z i q dqλ + dqλ λ z n + 4dq λ 3 u n x n q + dq λ λ λ w n x n + O q xn q Similarly, we can bound both the second and third terms of 33byO q xn : E Z q Z i q Z i 4q O q xn q, 33 and E q Z i 4q O q xn Note that all the O q terms in the context only depend on q Substituting these bounds into 33 gives u n+ x n+ q + dλ3 u n x n + 3dλ q λ λ w n x n + O q xn, 34 q

21 The Tightness of the Kesten Stigum Reconstruction Bound 637 by means of + dλ + dλ λ z n + O q xn Moreover, the similar expansion of w n+ would be and thus w n+ 4q EZ q+ + O q x n + q + dλ λ w n+ + q dλ λ + w n x n + O q xn q, wn x + O n q 35 q + Next display the discussion in the X OZ plane First consider the case of λ / λ κfor κ> In a small neighborhood of 0, 0, sincedλ <κ d λ dλ < andx n > 0, the discrete trajectories approach to the origin point tangentially to the X -axis, if is small enough for some n see Bernussou [4] for reference Besides, the conclusion of Lemma 7 excludes the trajectory along Z-axis Then for some M >, there exist absolute constants N N q,κ,m and δ δ q,κ,m, such that if n N and δ,wehave X n MZ n and MM + dλ + O q xn >0, where the remainder term O q xn comes from the expansion of + Consequently, it follows + z n X n M+ M, which yields, in connection with z n 0 in Lemma, + dλ + dλ λ z n + O q x n M M dλ M dλ M M+ dλ + O q xn 36 The second case taken into account is λ λ Inviewof/ dλ dλ, there also exist absolute constants N N q, M and δ δ q, M, such that if n N and δ, one has + dλ + O q x n For fixed k, it is known from 8that M M dλ + dλ X n + dλ Z n Cqx n, M and then there exists δ 3 δ 3 q,κ,m, k <min{δ,δ }, such that if <δ 3 then one has +l < δ 3,forany l k Therefore, for any positive integer k, 35 yields w n+k +k q dλ λ +k wn+k +k + O q +k +k +k q +k dλ λ k k l +l +l wn q + R, dλ where dλ λ k k l +l dλ x λ k M n+l M dλ k M k M λ

22 638 W Liu et al and k R Cδ 3 M M dλ i dλ λ i δ 3 M k M λ M M λ M M with C denoting the O q constant in 35 From the identity i in Lemma, we obtain 0 w n q, which implies w n q q Noting that λ λ d / /, it is possible to achieve M M λ < by choosing M 4 Therefore, it is feasible to take k kε sufficiently large and δ 4 δ 4 q,κ,k,ε δ 4 q,κ,ε<δ 3 sufficiently small to guarantee w n+k +k q <ε Finally, in view of λ >ϱ, there exists γ γq,ϱby Lemma 6 satisfying k γ k Thus choose N Nq,κ,ε,k Nq,κ,ε > max{n + k, N + k} and δ γ k δ 4,such that, if δ and n N, one has w n q dλ <ε 37 Finally, the second part of the lemma follows by plugging 37 into34 and proceeding similarly as above Proof of the Main Theorem First, consider ϱ < λ < λ,foranyfixedϱ > 0 By Lemma 4, it suffices to show that when dλ is close enough to, X n does not converge to 0 Therefore, it implies that does not converge to 0 either, considering 0 X n +z n It is convenient to make λ >ϱfixed and just λ varying, and then without loss of generality, assume dλ > +dλ +dλ / Consequently choose κ κd,λ > and thus dλ λ / λ κ As in Lemma 8, display our proof in the X OZ plane Under the condition of q 4 and 8, it is apparent that X n+ dλ X dd qq 3 n + λ 4 q X n + 4qλ λ X nz n +R x + R z + V x dλ X dd qq 3 n + λ 4 q X n R x R z C V xn u n q + w n q +, where the last inequality comes from λ d / < Then by Lemma 8 and Lemma 5, there exist N Nq,κ,ϱ and δ δq, d,κ,ϱ > 0, such that if n N and δ, then in the small neighborhood of the origin point 0, 0, wehavex n Z n and X n Meanwhile, the following estimates hold:,

23 The Tightness of the Kesten Stigum Reconstruction Bound 639 dd qq 3 λ 4 48C V q ; u n q, w n q dd qq 3 λ 4 48C V q ; R x, R z dd qq 3 λ 4 3 q 8 dd qq 3 λ 4 q X n Therefore, the quadratic term of Xn is large enough to control the remainder terms: X n+ dλ X n + dd qq 3 λ 4 q X n [ X n dλ + ] dd qq 3 λ 4 q X n 38 Take ε min{ 4 γ N,γδ} > 0, where γ γq,ϱ>0 is the constant in Lemma 6 Because q 4andε is independent of λ, we can choose λ < d / to make dλ + dd qq 3 λ 4 ε> 39 q Since x 0 q >, it is concluded that > γ n ε when n N, in addition, X N X N +Z N x N ε Now suppose X n ε for some n N Then display our discussion of X n as follows: a If X n γ ε,then X n+ + γ γ X n ε; b If ε X n γ ε,then X n γ ε δ, and thus it follows from 38and39 that + X n+ X n [ dλ + dd ] qq 3 λ 4 q X n X n ε Finally, we have X n ε for all n, by induction Consequently, it is established that the Kesten Stigum bound is not tight The second case to be considered is λ λ, under which there are two equal multipliers in this nonlinear second order point mapping and the origin point must be a star node Although the principal axis is undetermined, just by the comparison of the quadratic terms and for q 4, it is concluded that dd qq 3 λ 4 q X n + 4qλ4 X nz n dd q q λ4 X n 4qλ4 Z n dd q 7q q λ4 X n + 4qλ4 X nz n + 4qλ 4 Z n dd λ 4, and thus the decay rate of X n is much slower than Z n if is sufficiently small Therefore, in light of the preceding discussion, there still exist N Nq and δ δq, such that if n N

24 640 W Liu et al and δ, wehavex n Z n and thus X n + Z n X n Then the rest discussion would be similar as the first part Acknowledgements We would like to thank two anonymous reviewers who provided us with many constructive and helpful comments References Achlioptas, D, Coja-Oghlan, A: Algorithmic barriers from phase transitions In IEEE 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 08, pp IEEE 008 Baxter, RJ: The Riemann surface of the chiral Potts model free energy function J Stat Phys, Berger, N, Kenyon, C, Mossel, E, Peres, Y: Glauber dynamics on trees and hyperbolic graphs Probab Theory Related Fields 3, Bernussou, J: Point Mapping Stability Pergamon Press, New York Bhamidi, S, Rajagopal, R, Roch, S: Network delay inference from additive metrics Random Struct Algorithms 37, Bisconti, C, Corallo, A, Fortunato, L, Gentile, A A, Massafra, A, Pell, P: Reconstruction of a real world social network using the Potts model and Loopy belief propagation Front Psychol 05 doiorg/03389/fpsyg Bleher, PM, Ruiz, J, Zagrebnov, VA: On the purity of the limiting Gibbs state for the Ising model on the Bethe lattice J Statist Phys 79, Borgs, C, Chayes, J T, Mossel, E, Roch, S: The Kesten-Stigum reconstruction bound is tight for roughly symmetric binary channels In: IEEE Comput Soc FOCS, Berkeley, CA, pp Cadotte, MW, Davies, TJ: Phylogenies in Ecology: A Guide to Concepts and Methods Princeton University Press, Princeton 06 0 Cassandro, M, Ferrari, PA, Merola, I, Presutti, E: Geometry of contours and Peierls estimates in d Ising models with long range interactions J Math Phy 465, Chayes, JT, Chayes, L, Sethna, JP, Thouless, DJ: A mean field spin glass with short-range interactions Commun Math Phys 06, Daskalakis, C, Mossel, E, Roch, S: Optimal phylogenetic reconstruction In: STOC 06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, pp ACM, New York Derrida, B, Bray, AJ, Godreche, C: Non-trivial exponents in the zero temperature dynamics of the D Ising and Potts models J Phys A 7, L Dhar, D: The relaxation to equilibrium in one-dimensional Potts models J Indian Inst Sci 753, Evans, W, Kenyon, C, Peres, Y, Schulman, LJ: Broadcasting on trees and the Ising model Ann Appl Probab 0, Ferrari, PA, Fernndez, R, Garcia, NL: Perfect simulation for interacting point processes, loss networks and Ising models Stoch Process Appl 0, Georgii, HO: Gibbs Measures and Phase Transition de Gruyter, Berlin Gerschenfeld, A, and Montanari, A: Reconstruction for models on random graphs In: 48th Annual IEEE Symposium on Foundations of Computer Science, FOCS 07 IEEE Giuliani, A, Mastropietro, V: Universal finite size corrections and the central charge in non-solvable Ising models Commun Math Phys 34, Giuliani, A, Seiringer, R: Periodic striped ground states in Ising models with competing interactions Commun Math Phys 3473, Kesten, H, Stigum, BP: Additional limit theorems for indecomposable multidimensional Galton-Watson processes Ann Math Statist 37, Kesten, H, Stigum, BP: Limit theorems for decomposable multi-dimesional Galton-Watson processes J Math Anal Appl 7, Kimura, M: A simple method for estimating evolutionary rates of base substitutions through comparative studies of nucleotide sequences J Mole Evol 6, Liu, S, Ying, L, Shakkottai, S: Influence maximization in social networks: an ising-model-based approach In 00 48th Annual Allerton Conference on Communication, Control, and Computing Allerton, pp IEEE 00

25 The Tightness of the Kesten Stigum Reconstruction Bound 64 5 Liu, W, and Ning, N: Reconstruction for the asymmetric Ising model on regular trees In: Proceedings of 3rd International Conference on Electrical, Electronics, Engineering Trends, Communication, Optimization and Sciences EEECOS IET-IEEE, pp Ma, J, Ratan, A, Raney, BJ, Suh, BB, Miller, W, Haussler, D: The infinite sites model of genome evolution Proc Natl Acad Sci USA 05, Maes, C, Redig, F, Van Moffaert, A: The restriction of the Ising model to a layer J Stat Phys 96, Maes, C, Velde, KV: The fuzzy Potts model J Phys A 85, Majumdar, SN, Dhar, D: Equivalence between the Abelian sandpile model and the q 0 limit of the Potts model Physica A 85 4, Martinelli, F, Sinclair, A, Weitz, D: Fast mixing for independent sets, colorings, and other models on trees Random Struct Algorithms 3, Mézard, M, Montanari, A: Reconstruction on trees and spin glass transition J Stat Phys 4, Mossel, E: Reconstruction on trees: beating the second eigenvalue Ann Appl Probab, Mossel, E: Phase transitions in phylogeny Trans Am Math Soc 356, Mossel, E: Survey: information flow on trees In: Graphs, Morphisms and Statistical Physics DIMACS Ser Discrete Math Theoret Comput Sci vol 63, pp American Math Soc, Providence, RI Mossel, E, Yuval, P: Information flow on trees Ann Appl Probab 33, Olejarz, J, Krapivsky, PL, Redner, S: Zero-temperature coarsening in the d Potts model J Stat Mech 0306, P Roch, S: A short proof that phylogenetic tree reconstruction by maximum likelihood is hard IEEE/ACM Trans Comput Biol Bioinform 3, Saul, L, Kardar, M: Exact integer algorithm for the two-dimensionalj Ising spin glass Phys Rev E 485, R Semerjian, G: On the freezing of variables in random constraint satisfaction problems J Stat Phys 30, Sire, C, Majumdar, SN: Coarsening in the q-state Potts model and the Ising model with globally conserved magnetization Phys Rev E 5, Sire, C, Majumdar, SN: Correlations and coarsening in the q-state Potts model Phys Rev Lett 74, Sly, A: Reconstruction of random colourings Commun Math Phys 883, Sly, A: Reconstruction for the Potts model Ann Probab 39, Spirin, V, Krapivsky, PL, Redner, S: Fate of zero-temperature Ising ferromagnets Phys Rev E 633, Spohn, H, Dmcke, R: Quantum tunneling with dissipation and the Ising model over J Stat Phys 43 4, Tetali, P, Vera, JC, Vigoda, E, Yang, L: Phase transition for the mixing time of the Glauber dynamics for coloring regular trees Ann Appl Probab, Tracy, CA: Universality class of a Fibonacci Ising model J Stat Phys 53, Weeks, JD, Gilmer, GH, Leamy, HJ: Structural transition in the Ising-model interface Phys Rev Lett 38,