Belief propagation, robust reconstruction and optimal recovery of block models

Introduction

Stochastic block models were introduced more than 30 years ago in order to study the problem of community detection in random graphs. In these models, the nodes in a graph are divided into two or more communities, and then the edges of the graph are drawn independently at random, with probabilities depending on which communities the edge lies between. In its simplest incarnation—which we will study here—the model has $n$ vertices divided into two classes of approximately equal size, and two parameters: $a/n$ is the probability that each within-class edge will appear, and $b/n$ is the probability that each between-class edge will appear. Since their \hyperref[sec1]Introduction, a large body of literature has been written about stochastic block models, and a multitude of efficient algorithms have been developed for the problem of inferring the underlying communities from the graph structure. To name a few, we now have algorithms based on maximum-likelihood methods , belief propagation , spectral methods , modularity maximization and a number of combinatorial methods .

Early work on the stochastic block model mainly focused on fairly dense graphs: Dyer and Frieze ; Snijders and Nowicki ; and Condon and Karp all gave algorithms that will correctly recover the exact communities in a graph from the stochastic block model, but only when $a$ and $b$ are polynomial in $n$. In a substantial improvement, McSherry gave a spectral algorithm that succeeds when $a$ and $b$ are logarithmic in $n$; this had been anticipated previously by Boppana , but his proof was incomplete. McSherry’s parameter range was later equalled by Bickel and Chen using an algorithm based on modularity maximization.

We also note that related but different problems of planted coloring were studied in Blum and Spencer in the dense case, and Alon and Kahale in the sparse case.

The $O(\log n)$ barrier is important because if the average degree of a block model is logarithmic or larger, it is possible to exactly recover the communities with high probability as $n\to\infty$. On the other hand, if the average degree is less than logarithmic then some fairly straightforward probabilistic arguments show that it is not possible to completely recover the communities. When the average degree is constant, as it will be in this work, then one cannot get more than a constant fraction of the labels correct.

Despite these apparent difficulties, there are important practical reasons for considering block models with constant average degree. Indeed, many real networks are very sparse. For example, Leskovec et al. and Strogatz collected and studied a vast collection of large network datasets, many of which had millions of nodes, but most of which had an average degree of no more than 20; for instance, the LinkedIn network studied by Leskovec et al. had approximately seven million nodes, but only 30 million edges. Moreover, the very fact that sparse block models are impossible to infer exactly may be taken as an argument for studying them: in real networks one does not expect to recover the communities with perfect accuracy, and so it makes sense to study models in which this is not possible either.

Although sparse graphs are immensely important, there is not yet much known about very sparse stochastic block models. In particular, there is a gap between what is known for block models with a constant average degree and those with an average degree that grows with the size of the graph. Until recently, there was only one algorithm—due to , and based on spectral methods—which was guaranteed to do anything at all in the constant-degree regime, in the sense that it produced communities which have a better-than-50% overlap with the true communities.

Despite the lack of rigorous results, a beautiful conjectural picture has recently emerged, supported by simulations and deep but nonrigorous physical intuition. We are referring specifically to work of Decelle et al. , who conjectured the existence of a threshold, below which is it not possible to find the communities better than by guessing randomly. In the case of two communities of equal size, they pinpointed the location of the conjectured threshold. This threshold has since been rigorously confirmed; a sharp lower bound on its location was given by the authors , while sharp upper bounds were given independently by Massoulié and by the authors .

2 Our results: Optimal reconstruction

Given that it is not possible to completely recover the communities in a sparse block model, it is natural to ask how accurately one may recover them. In , we gave an upper bound on the recovery accuracy; here, we will show that that bound is tight—at least, when the signal to noise ratio is sufficiently high—by giving an algorithm which performs as well as the upper bound. Our main result may be stated informally as follows.An extended abstract stating the results of the current paper appeared in the proceedings of COLT 2014 (where it won the best paper award).

Let $p_{G}(a,b)$ be the highest asymptotic accuracy that any algorithm can achieve in reconstructing communities of the block model with parameters $a$ and $b$. We provide an algorithm that achieves accuracy of $p_{G}(a,b)$ with probability tending to 1 as $n\to\infty$, provided that $(a-b)^{2}/(a+b)$ is sufficiently large.

To put Theorem 1.1 into the context of earlier work by the authors and Massoulié, those works showed that $p_{G}(a,b)>1/2$ if and only if $(a-b)^{2}>2(a+b)$; in the case that $p_{G}(a,b)>1/2$, they also provided algorithms whose accuracy was bounded away from $1/2$. However, those algorithms were not guaranteed (and are not expected) to have optimal accuracy, only nontrivial accuracy. In other words, previous results have shown that for every value of $a,b$ such that $(a-b)^{2}>2(a+b)$ there exists an algorithm that recovers (with high probability) a fraction $q(a,b)>1/2$ of the nodes correctly. Our results provide an algorithm that [when $(a-b)^{2}>C(a+b)$ for a large constant $C$] recovers the optimal fraction of nodes $p_{G}(a,b)$ in the sense that it is information theoretically impossible for any other algorithms to recover a bigger fraction.

Our new algorithm, which is based on belief propagation, is essentially an algorithm for locally improving an initial guess at the communities. In our current analysis, the initial guess is provided by a previous algorithm of the authors , which we use as a black box. We should mention that standard belief propagation with random uniform initial messages and without our modifications and also without a good initial guess, is also conjectured to have optimal accuracy . However, at the moment, we do not know of any approach to analyze the vanilla version of BP for this problem.

As a major part of our analysis, we prove a result about broadcast processes on trees that may be of independent interest. Specifically, we prove that if the signal-to-noise ratio of the broadcast process is sufficiently high, then adding extra noise at the leaves of a large tree does not hurt our ability to guess the label of the root given the labels of the leaves. In other words, we show that for a certain model on trees, belief propagation initialized with arbitrarily noisy messages converges to the optimal solution as the height of the tree tends to infinity. We prove our result for regular trees and Galton–Watson trees with Poisson offspring, but we conjecture that it also holds for general trees, and even if the signal-to-noise ratio is low.

We should point out that spectral algorithms—which, due to their efficiency, are very popular algorithms for this model—empirically do not perform as well as BP on very sparse graphs (see, e.g., ). This is despite the recent appearance of two new spectral algorithms, due to and , which were specifically designed for clustering sparse block models. The algorithm of is particularly relevant here, because it was derived by linearizing belief propagation; empirically, it performs well all the way to the impossibility threshold, although not quite as well as BP. Intuitively, the linear aspects of spectral algorithms (i.e., the fact that they can be implemented—via the power method—using local linear updates) explain why they cannot achieve optimal performance. Indeed, since the optimal local updates (those given by BP) are nonlinear, any method based on linear updates will be suboptimal.

3 Dramatis personae

Before defining everything carefully, we briefly introduce the three main objects and their relationships.

The block model detection problem is the problem of detecting communities in a sparse stochastic block model.

In the tree reconstruction problem, there is a two-color branching process in which every node has some children of its own color and some children of the other color. We observe the family tree of this process and also all of the colors in some generation; the goal is to guess the color of the original node.

The robust tree reconstruction problem is like the tree reconstruction problem, except that instead of observing exactly the colors in some generation, our observations contain some noise.

The two tree problems are related to the block model problem because a neighborhood in the stochastic block model looks like a random tree from one of the tree problems. This connection was proved in , who also showed that tree reconstruction is “easier” than the block model detection (in a sense that we will make precise later). The current work has two main steps: we show that block model detection is “easier” than robust tree reconstruction, and we show that—for a certain range of parameters—robust tree reconstruction is exactly as hard as tree reconstruction.

Definitions and main results

In this article, we restrict the stochastic block model to the case of two classes with roughly equal size.

The block model on $n$ nodes is constructed by first labeling each node $+$ or $-$ with equal probability independently. Then each edge is included in the graph independently, with probability $a/n$ if its endpoints have the same label and $b/n$ otherwise. Here, $a$ and $b$ are two positive parameters. We write $\mathcal{G}(n,a/n,b/n)$ for this distribution of (labeled) graphs.

For us, $a$ and $b$ will be fixed, while $n$ tends to infinity. More generally, one may consider the case where $a$ and $b$ may be allowed to grow with $n$. As conjectured by , the relationship between $(a-b)^{2}$ and $(a+b)$ turns out to be of critical importance for the reconstructability of the block model.

For the block model with parameters $a$ and $b$, it holds that:

If $(a-b)^{2}<2(a+b)$ then the node labels cannot be inferred from the unlabeled graph with better than $50\%$ accuracy (which could also be done just by random guessing).

If $(a-b)^{2}>2(a+b)$ then it is possible to infer the labels with better than $50\%$ accuracy.

2 Broadcasting on trees

Our study of optimal reconstruction accuracy is based on the local structure of $\mathcal{G}(n,a/n,b/n)$, which requires the notion of the broadcast process on a tree.

Given a parameter $\eta\neq 1/2$ in $$and a tree$T$, the broadcast process on$T$is a two-state Markov process$\{\sigma_{u}:u\in T\}$defined as follows: let$\sigma_{\rho}$be$+$or$-$with probability$\frac{1}{2}$. Then, for each$u$such that$\sigma_{u}$is defined, independently for every$v\in L_{1}(u)$let$\sigma_{v}=\sigma_{u}$with probability$1-\eta$and$\sigma_{v}=-\sigma_{\rho}$ otherwise.

This broadcast process has been extensively studied, where the major question is whether the labels of vertices far from the root of the tree give any information on the label of the root. For general trees, this question was answered definitively by Evans et al. , after many other contributions including . The complete statement of the theorem requires the notion of branching number, which we would prefer not to define here (see ). For our purposes, it suffices to know that a $d$-ary tree has branching number $d$ and that a Poisson branching process tree with mean $d>1$ has branching number $d$ (almost surely, and conditioned on nonextinction).

Let $\theta=1-2\eta$ and $d$ be the branching number of $T$. Then

in probability as $k\to\infty$ if and only if $d\theta^{2}\leq 1$.

The theorem implies in particular that if $d\theta^{2}>1$ then for every $k$ there is an algorithm which guesses $\sigma_{\rho}$ given $\sigma_{L_{k}(\rho)}$, and which succeeds with probability bounded away from $1/2$. If $d\theta^{2}\leq 1$ there is no such algorithm.

3 Robust reconstruction on trees

Janson and Mossel considered a version of the tree broadcast process that has extra noise at the leaves.

Given a broadcast process $\sigma$ on a tree $T$ and a parameter $\delta\in[0,1/2)$, the noisy broadcast process on $T$ is the process $\{\tau_{u}:u\in T\}$ defined by independently taking $\tau_{u}=-\sigma_{u}$ with probability $\delta$ and $\tau_{u}=\sigma_{u}$ otherwise.

We observe that the noise present in $\sigma$ and the noise present in $\tau$ have qualitatively different roles, since the noise present in $\sigma$ propagates down the tree while the noise present in $\tau$ does not. Janson and Mossel showed that the range of parameters for which $\sigma_{\rho}$ may be nontrivially reconstructed from $\sigma_{L_{k}}$ is the same as the range for which $\sigma_{\rho}$ may be nontrivially reconstructed from $\tau_{L_{k}}$. In other words, additional noise at the leaves has no effect on whether the root’s signal propagates arbitrarily far. One of our main results is a quantitative version of this statement (Theorem 2.11): we show that for a certain range of parameters, the presence of noise at the leaves does not even affect the accuracy with which the root can be reconstructed.

4 The block model and broadcasting on trees

The connection between the community reconstruction problem on a graph and the root reconstruction problem on a tree was first pointed out in and made rigorous in . The basic idea is the following:

A neighborhood in $G$ looks like a Galton–Watson tree with offspring distribution $\operatorname{Pois}((a+b)/2)$ [which almost surely has branching number $d=(a+b)/2$].

The labels on the neighborhood look as though they came from a broadcast process with parameter $\eta=\frac{b}{a+b}$.

With these parameters, $\theta^{2}d=\frac{(a-b)^{2}}{2(a+b)}$, and so the conjectured threshold for community reconstruction is the same as the proven threshold for tree reconstruction.

This local approximation can be formalized as convergence locally on average, a type of local weak convergence defined in . We should mention that in the case of more than two communities (i.e., in the case that the broadcast process has more than two states) then the picture becomes rather more complicated, and much less is known; see for some conjectures.

5 Reconstruction probabilities on trees and graphs

Note that Theorem 2.4 only answers the question of whether one can achieve asymptotic reconstruction accuracy better than $1/2$. Here, we will be interested in more detailed information about the actual accuracy of reconstruction, both on trees and on graphs.

Note that in the tree reconstruction problem, the optimal estimator of $\sigma_{\rho}$ given $\sigma_{L_{k}(\rho)}$ is easy to write down: it is simply the sign of $X_{\rho,k}:=2\operatorname{Pr}(\sigma_{\rho}=+\mid\sigma_{L_{k}(\rho)})-1$. Compared to the trivial procedure of guessing $\sigma_{\rho}$ completely at random, this estimator has an expected gain of

Let $T$ be an infinite Galton–Watson tree with $\operatorname{Pois}((a+b)/2)$ offspring distribution, and $\eta=\frac{b}{a+b}$. Consider the broadcast process on the tree with parameter $\eta$ and define

In words, $p_{T}(a,b)$ is the probability of correctly inferring $\sigma_{\rho}$ given the “labels at infinity”.

Note that by Theorem 2.4, $p_{T}(a,b)>1/2$ if and only if $(a-b)^{2}>2(a+b)$.

We remark that the limit in Definition 2.6 always exists because the right-hand side is nonincreasing in $k$. To see this, it helps to write $p_{T}(a,b)$ in a different way: let $\mu_{k}^{+}$ be the distribution of $\sigma_{L_{k}(\rho)}$ given $\sigma_{\rho}=+$ and let $\mu_{k}^{-}$ be the distribution of $\sigma_{L_{k}(\rho)}$ given $\sigma_{\rho}=-$. Then

Hence, if we set $\nu_{k}^{+}$ to be the distribution of $\{\sigma_{L_{k^{\prime}}}(\rho):k^{\prime}\geq k\}$ and similarly for $\nu_{k}^{-}$, we have

Now the right-hand side is clearly nonincreasing in $k$, because $\nu_{k+1}^{+}$ can be obtained from $\nu_{k}$ by marginalization.

One of the main results of is that the graph reconstruction problem is at least as hard as the tree reconstruction problem in the sense that for any community-detection algorithm, the asymptotic accuracy of that algorithm is bounded by $p_{T}(a,b)$.

Let $(G,\sigma)$ be a labeled graph on $n$ nodes. If $f$ is a function that takes a graph and returns a labeling of it, we write

for the accuracy of $f$ in recovering the labels $\sigma$. For $\varepsilon>0$, let

where the first supremum ranges over all functions $f$, and the probability is taken over $(G,\sigma)\sim\mathcal{G}(n,a/n,b/n)$. Let

where the limit exists because $p_{G,n,\varepsilon}(a,b)$ is monotonic in $\varepsilon$.

One should think of $p_{G}(a,b)$ as the optimal fraction of nodes that can be reconstructed correctly by any algorithm (not necessarily efficient) that only gets to observe an unlabeled graph. More precisely, for any algorithm and any $p>p_{G}(a,b)$, the algorithm’s probability of achieving accuracy $p$ or higher converges to zero as $n$ grows. Note that the symmetry between the $+$ and $-$ is reflected in the definition of $\operatorname{acc}$ (e.g., in the appearance of the constant $1/2$), and also that $\operatorname{acc}$ is defined to be large if $f$ gets most labels incorrect (because there is no way for an algorithm to break the symmetry between $+$ and $-$).

An immediate corollary of the analysis of implies that graph reconstruction is always at most as accurate as tree reconstruction.

We remark that Theorem 2.8 is not stated explicitly in ; because the authors were only interested in the case $(a-b)^{2}\leq 2(a+b)$, the claimed result was that $(a-b)^{2}\leq 2(a+b)$ implies $p_{G}(a,b)=\frac{1}{2}$. However, a cursory examination of the proof of , Theorem 1, reveals that the claim was proven in two stages: first, they prove via a coupling argument that $p_{G}(a,b)\leq p_{T}(a,b)$ and then they apply Theorem 2.4 to show that $(a-b)^{2}\leq 2(a+b)$ implies $p_{T}(a,b)=\frac{1}{2}$.

6 Our results

In this paper, we consider the high signal-to-noise case, namely the case that $(a-b)^{2}$ is significantly larger than $2(a+b)$. In this regime, we give an algorithm (Algorithm 1) which achieves an accuracy of $p_{T}(a,b)$.

There exists a constant $C$ such that if $(a-b)^{2}\geq C(a+b)$ then

Moreover, there is a polynomial time algorithm such that for all such $a,b$ and every $\varepsilon>0$, with probability tending to one as $n\to\infty$, the algorithm reconstructs the labels with accuracy $p_{G}(a,b)-\varepsilon$.

We will assume for simplicity that our algorithm is given the parameters $a$ and $b$. This is a minor assumption because $a$ and $b$ can be estimated from the data to arbitrary accuracy , Theorem 3.

A key ingredient of Theorem 2.9’s proof is a procedure for amplifying a clustering that is a slightly better than a random guess to obtain optimal clustering. In order to discuss this procedure, we define the problem of “robust reconstruction” on trees.

Consider the noisy tree broadcast process with parameters $\eta=\frac{a}{a+b}$ and $\delta\in[0,1/2)$ on a Galton–Watson tree with offspring distribution $\operatorname{Pois}((a+b)/2)$. We define the robust reconstruction accuracy as

Our main technical result is that when $a-b$ is large enough then in fact the extra noise does not have any effect on the reconstruction probability.

There exists a constant $C$ such that if $(a-b)^{2}\geq C(a+b)$ then

We conjecture that the robust reconstruction accuracy is independent of $\delta$ for any parameters, and also for more general trees; however, our proof does not naturally extend to cover these cases.

7 Algorithmic amplification and robust reconstruction

Combining Theorem 2.12 with Theorems 2.8 and 2.11 proves Theorem 2.9. We remark that Theorem 2.12 easily extends to other versions of the block model (i.e., models with more clusters or unbalanced classes); however, Theorem 2.11 does not. In particular, Theorem 2.9 may not hold for general block models. In fact, one fascinating conjecture of says that for general block models, computational hardness enters the picture (whereas it does not play any role in our current work).

8 Algorithm outline

Before getting into the technical details, let us give an outline of our algorithm: for every node $u$, we remove a neighborhood (whose radius $r$ is slowly increasing with $n$) of $u$ from the graph $G$. We then run a black-box community-detection algorithm on what remains of $G$. This is guaranteed to produce some communities which are correlated with the true ones, but they may not be optimally accurate. Then we return the neighborhood of $u$ to $G$, and we consider the inferred communities on the boundary of that neighborhood. Now, the neighborhood of $u$ is like a tree, and the true labels on its boundary are distributed like $\sigma_{L_{r}(u)}$. The inferred labels on the boundary are hence distributed like $\tau_{L_{r}(u)}$ for some $0\leq\delta<\frac{1}{2}$, and so we can guess the label of $u$ from them using robust tree reconstruction. (In the previous sentence, we are implicitly claiming that the errors made by the black-box algorithm are independent of the neighborhood of $u$. This is because the edges in the neighborhood of $u$ are independent of the edges in the rest of the graph, a fact that we will justify more carefully later.) Since robust tree reconstruction succeeds with probability $p_{T}$ regardless of $\delta$, our algorithm attains this optimal accuracy even if the black-box algorithm does not.

To see the connection between our algorithm and belief propagation, note that finding the optimal estimator for the tree reconstruction problem requires computing $\operatorname{Pr}(\sigma_{u}\mid\tau_{L_{r}(u)})$. On a tree, the standard algorithm for solving this is exactly belief propagation. In other words, our algorithm consists of multiple local applications of belief propagation. Although we believe that a single global run of belief propagation would attain the same performance, these local instances are easier to analyze.

Finally, a word about notation. Throughout this article, we will use the letters $C$ and $c$ to denote positive constants whose value may change from line to line. We will also write statements like “for all $k\geq K(\theta,\delta)\dots$” as abbreviations for statements like “for every $\theta$ and $\delta$ there exists $K$ such that for all $k\geq K\dots.$”

Robust reconstruction on regular trees

Our main effort is devoted to proving Theorem 2.11. Since the proof is quite involved, we begin with a somewhat easier case of regular trees which already contains the main ideas of the proof. The adaptation to the case of Poisson random trees will be carried in Section 4.

First, we need to define the reconstruction and robust reconstruction probabilities for regular trees. Their definitions are analogous to Definitions 2.6 and 2.10.

Let $\sigma$ be distributed according to the broadcast process with parameter $\eta$ on an infinite $d$-ary tree. Let $\tau$ be distributed according to the noisy broadcast process with parameters $\eta$ and $\delta$ on the same tree. We define

We may also define the noisy magnetization $Y$:

There exists a constant $C$ such that if $\theta^{2}d>C$ and $\delta<\frac{1}{2}$ then

Our main method for proving Theorem 3.3 (and also Theorem 2.11) is by studying certain recursions. Indeed, Bayes’ rule implies the following recurrence for $X$ (see, e.g., ):

The same reasoning that gives (3) also shows that (3) also holds when every instance of $X$ is replaced by $Y$. Since our entire analysis is based on the recurrence (3), the only meaningful (for us) difference between $X$ and $Y$ is that their initial conditions are different: $X_{u,0}=\pm 1$ while $Y_{u,0}=\pm(1-2\delta)$. In fact, we will see later that Theorem 3.3 also holds for some more general estimators $Y$ satisfying (3).

2 The simple majority method

Our first step in proving Theorem 3.3 is to show that when $\theta^{2}d$ is large, then both the exact reconstruction and the noisy reconstruction do quite well. While it is possible to do so by studying the recursion (3), such an analysis is actually quite delicate. Instead, we will show this by studying a completely different estimator: the one which is equal to the most common label among $\sigma_{L_{k}(\rho)}$. This estimator is easy to analyze, and it performs quite well; since the estimator based on the sign of $X_{\rho,k}$ is optimal, it performs even better. The study of the simple majority estimator is quite old, having essentially appeared in the paper of Kesten and Stigum ; however, we include most of the details for the sake of completeness.

The second moment calculation uses the recursive structure of the tree. The argument is not new, but we include it for completeness.

We decompose the variance of $S_{k}$ by conditioning on the first level of the tree:

Now, $S_{\rho,k}=\sum_{u\in L_{1}}S_{u,k-1}$, and $S_{u,k-1}$ are i.i.d. under $\operatorname{Pr}^{+}$. Thus, the first term of (4) decomposes into a sum of variances:

Plugging this back into (4), we get the recursion

Since $\operatorname{Var}^{+}S_{\rho,0}=0$, we solve this recursion to obtain

where the last equality follows from (3.2) and the fact that $\operatorname{Var}(aX)=a^{2}\operatorname{Var}(X)$.

Taking $k\to\infty$ in Lemmas 3.4 and 3.5, we see that if $\theta^{2}d>1$ then

We will use this fact repeatedly, so let us summarize in a lemma.

There is a constant $C$ such that if $\theta^{2}d>1$ and $k\geq K(\delta)$ then

By Markov’s inequality, we find that $X_{u,k}$ is large with high probability:

There is a constant $C$ such that for all $k\geq K(\delta)$ and all $t>0$

As we will see, Lemma 3.6 and the recursion (3) are really the only properties of $Y$ that we will use. Hence, from now on $Y_{u,k}$ need not be defined by (3.1). Rather, we will make the following assumptions on $Y_{u,k}$.

There is a $K=K(\delta)$ such that for all $k\geq K$, the following hold: {longlist}[3.]

$Y_{u,k+1}=\frac{\prod_{i\in\mathcal{C}(u)}(1+\theta Y_{ui,k})-\prod_{i\in\mathcal{C}(u)}(1-\theta Y_{ui,k})}{\prod_{i\in\mathcal{C}(u)}(1+\theta Y_{ui,k})+\prod_{i\in\mathcal{C}(u)}(1-\theta Y_{ui,k})}$.

The distribution of $Y_{u,k}$ given $\sigma_{u}=+$ is equal to the distribution of $-Y_{u,k}$ given $\sigma_{u}=-$.

We will prove Theorem 3.3 under Assumption 3.1. Note that part 2 above immediately implies

Also, part 3 implies that Lemma 3.7 holds for $Y$.

3 The recursion for small \texorpdfstringθ𝜃\thetat​h​e​t​a𝑡ℎ𝑒𝑡𝑎theta

Our proof of Theorem 3.3 proceeds in two cases, with two different analyses. In the first case, we suppose that $\theta$ is small (i.e., smaller than a fixed, small constant). In this case, we proceed by Taylor-expanding the recursion (3) in $\theta$. For the rest of this section, we will assume that $X$ and $Y$ satisfy parts 1 and 2 of Assumption 3.1, and that $x_{k},y_{k}\geq 5/6$ for $k\geq K(\delta)$. This restriction will allow us to reuse most of the argument in the Galton–Watson case (where part 3 of Assumption 3.1 fails to hold, but we nevertheless have $x_{k},y_{k}\geq 5/6$).

There are absolute constants $C$ and $\theta^{*}>0$ such that if $d\theta^{2}\geq C$ and $\theta\leq\theta^{*}$ then for all $k\geq K(\theta,d,\delta)$,

In proving Proposition 3.8, the first step is to replace the right-hand side of (3) with something easier to work with; in particular, we would like to have something without $X$ in the denominator. For this, we note that

Hence, if $a=\prod_{i}(1+\theta X_{ui,k})$, $b=\prod_{i}(1-\theta X_{ui,k})$, and $a^{\prime}$ and $b^{\prime}$ are the same quantities with $Y$ replacing $X$, then

Using Taylor’s theorem, the right-hand side can be bounded in terms of $|(b/a)^{p}-(b^{\prime}/a^{\prime})^{p}|$ for some $0<p<1$ of our choice.

Let $f(x)=\frac{1}{1+x}$ and $g(x)=x^{p}$. By the fundamental theorem of calculus, the proof would follow from the inequality $|f^{\prime}(x)|\leq p^{-1}g^{\prime}(x)$. Now, $|f^{\prime}(x)|=\frac{1}{(1+x)^{2}}$ and $g^{\prime}(x)=px^{p-1}$. When $x\geq 1$, we have $|f^{\prime}(x)|\leq x^{-2}\leq x^{p-1}$, while if $x\leq 1$ then $|f^{\prime}(x)|\leq 1\leq x^{p-1}$.

As an immediate consequence of Lemma 3.9 (for $p=1/4$) and (8),

Next, we present a general bound on the second moment of differences of products. Of course, we have in mind the example $A_{i}=(\frac{1-\theta X_{ui,k}}{1+\theta X_{ui,k}})^{1/4}$ and similarly for $B_{i}$ and $Y_{i}$.

Let $(A_{1},B_{1}),\dots,(A_{d},B_{d})$ be i.i.d. copies of $(A,B)$. Then

By a second-order Taylor expansion, any twice differentiable $f$ satisfies $f(x)+f(y)\leq 2f((x+y)/2)+\frac{1}{4}(x-y)^{2}\max_{z}f^{\prime\prime}(z)$, where the maximum ranges over $z$ between $x$ and $y$. Applying this for $f(x)=x^{d}$ yields

and the same expression with $Y$ instead of $X$.

There is a constant $\theta^{*}>0$ such that if $x_{k},y_{k}\geq 5/6$ then

First, note that for sufficiently small $x$,

Now, if $\theta^{*}$ is sufficiently small then we may apply this with $x=\theta X_{ui,k}$, obtaining

Recalling the assumption that $x_{k}\geq 5/6$, we have

The same argument applies to $B_{i}$, but using $Y_{i}$ instead of $X_{i}$.

(recalling in the last line that $\theta=1-2\eta$).

There is a $\theta^{*}>0$ such that if $\theta<\theta^{*}$ then

The result follows because $1-\theta^{2}$ and $1+\theta^{2}$ can be made arbitrarily close to 1 by taking $\theta^{*}$ small enough.

There is a $\theta^{*}>0$ such that for all $\theta<\theta^{*}$,

By (3.4) and (3.4) (and analogously with $A$ replaced by $B$), we have

5 Combining the estimates to complete the proof

Next, we combine Lemma 3.10 with the estimates provided in Lemmas 3.11 and 3.13.

There is some constant $\theta^{*}>0$ such that the following holds. Suppose that $X$ and $Y$ satisfy parts 1 and 2 of Assumption 3.1 and that $x_{k},y_{k}\geq 5/6$ for $k\geq K(\delta)$. If $u$ has $d\geq 4$ children and $\theta\leq\theta^{*}$ then for $k\geq K(\delta)$,

Taking the square of (9) and taking the expectation on both sides, we have

Conditioned on $\sigma_{u}$, the pairs $(A_{i},B_{i})$ are i.i.d. and so Lemma 3.10 implies that

Now, if $\theta^{*}$ is sufficiently small then the function $x\mapsto(\frac{1-\theta x}{1+\theta x})^{1/4}$ has derivative at most $\theta$ for $x\in$. Hence,

provided that $\theta^{*}$ is sufficiently small. Define

By Lemma 3.11 and the assumption that $x_{k},y_{k}\geq 5/6$, if $\theta^{*}$ is sufficiently small, then $m\leq 1-\theta^{2}/5\leq\exp(-\theta^{2}/5)$. Moreover, Lemma 3.13 implies that $(a-b)^{2}\leq 9\theta^{4}z$. Plugging these and (3.5) back into (3.5), we have

Proof of Proposition 3.8 If $\theta^{2}d$ is sufficiently large, then Lemma 3.6 implies that $x_{k},y_{k}\geq 5/6$ for $k\geq K(\delta)$; hence, the conditions of Lemma 3.14 are satisfied. Finally, if $d\theta^{2}$ is large enough then the right-hand side in Lemma 3.14 is at most $\frac{1}{2}$.

6 The recursion for large \texorpdfstringθ𝜃\thetat​h​e​t​a𝑡ℎ𝑒𝑡𝑎theta

To handle the case in which $\theta$ is not small, we require a different argument. In this case, we study the derivatives of the recurrence, obtaining the following result.

For any $0<\theta^{*}<1$, there is some $d^{*}=d^{*}(\theta^{*})$ such that for all $\theta\geq\theta^{*}$, $d\geq d^{*}$, and $k\geq K(\theta,d,\delta)$,

Combined with Proposition 3.8, this proves Theorem 3.3. Indeed, to complete the choices of parameters we first take $\theta^{*}$ to be the universal constant in Proposition 3.8. Then let $d^{*}=d^{*}(\theta^{*})$ be given by Proposition 3.15 (note that $d^{*}$ is also a universal constant). Finally, choose $C$ to be the maximum of $d^{*}$ and the $C$ from Proposition 3.8. Now, if $\theta^{2}d\geq C$ then either $\theta\leq\theta^{*}$ in which case Proposition 3.8 applies, or $\theta\geq\theta^{*}$ in which case $\theta\leq 1$ implies that $d\geq C\geq d^{*}$ and so Proposition 3.15 applies. In either case, we deduce Theorem 3.3.

Then the recurrence (3) may be written as $X_{u,k+1}=g(X_{u1,k},\dots,X_{ud,k})$. We will also abbreviate $(X_{u1,k},\dots,X_{ud,k})$ by $X_{L_{1}(u),k}$, so that we may write $X_{u,k+1}=g(X_{L_{1}(u),k})$.

Define $g_{1}(x)=\prod_{i=1}^{d}(1+\theta x_{i})$ and $g_{2}(x)=\prod_{i=1}^{d}(1-\theta x_{i})$ so that $g$ can be written as $g=\frac{g_{1}-g_{2}}{g_{1}+g_{2}}$. Since $\frac{\partial g_{1}}{\partial x_{i}}=\theta\frac{g_{1}}{1+\theta x_{i}}$ and $\frac{\partial g_{2}}{\partial x_{i}}=-\theta\frac{g_{2}}{1-\theta x_{i}}$, we have

If $|x_{i}|\leq 1$, then $g_{1}$ and $g_{2}$ are both positive, so $\frac{g_{1}g_{2}}{(g_{1}+g_{2})^{2}}\leq\frac{g_{1}g_{2}}{g_{1}^{2}}=\frac{g_{2}}{g_{1}}$; of course, we also have the symmetric bound $\frac{g_{1}g_{2}}{(g_{1}+g_{2})^{2}}\leq\frac{g_{1}}{g_{2}}$. Define

The point is that if $\sigma_{u}=+$ then for most $v\in L_{1}(u)$, $X_{v,k}$ will be close to 1 and so $h_{i}^{+}(X_{L_{1}(u),k})$ will be small. On the other hand, if $\sigma_{u}=-$ then for most $v\in L_{1}(u)$, $X_{v,k}$ will be close to $-1$ and so $h_{i}^{-}(X_{L_{1}(u),k})$ will be small.

Note that $h_{i}^{+}$ is convex on $^{d}$ because it is the tensor product of nonnegative, convex functions. Hence, for any $x,y\in^{d}$ and any $0<\lambda<1$,

Applied for $x=X_{L_{1}(u),k}=(X_{u1,k},\dots,X_{ud,k})$ and $y=Y_{L_{1}(u),k}=(Y_{u1,k},\dots,\break Y_{ud,k})$, this yields

Note that the two terms on the right-hand side of (3.6) are dependent on one another. Hence, it will be convenient to bound $h_{i}^{+}(X_{L_{1}(u),k})$ by something that does not depend on $X_{ui}$. To that end, note that for $|x_{i}|\leq 1$, we have $1+\theta x_{i}\geq 1-\theta=2\eta$, and so

Since $m_{i}(x)$ does not depend on $x_{i}$, it follows that $m_{i}(X_{L_{1}(u),k})$ is independent of $X_{ui,k}$ given $\sigma_{u}$ (and similarly with $Y$ instead of $X$). Hence, (3.6) implies that

For any $0<\theta^{*}<1$, there is some $d^{*}=d^{*}(\theta^{*})$ and some $\lambda=\lambda(\theta^{*})<1$ such that for all $\theta\geq\theta^{*}$, $d\geq d^{*}$ and $k\geq K(\theta,d,\delta)$,

The proof of Lemma 3.16 is straightforward but tedious, and we postpone it until the \hyperref[app]Appendix. Instead, we will now prove Proposition 3.15.

Proof of Proposition 3.15 By Lemma 3.16, and the definition (19) of $m_{i}$, it follows that

In particular, if $d^{*}(\theta^{*})$ is sufficiently large then $d\lambda^{d-5}\leq 1/4$ for all $d\geq d^{*}$. The same argument applies with $Y$ replacing $X$, and hence

Reconstruction accuracy on Galton–Watson trees

As in Section 3, we let $\{\sigma_{u}:u\in T\}$ be distributed as the two-state broadcast process on $T$ with parameter $\eta$, and let $\{\tau_{u}:u\in T\}$ be the noisy version, with parameter $\delta$. We recall the magnetization

Note that unlike in Section 3, $X_{u,k}$ now depends on both the randomness of the tree and the randomness of $\sigma$. Hence, $x_{k}$ now averages over both the randomness of the tree and the randomness of $\sigma$.

We recall that $X$ satisfies the recursion (3). As in Section 3, we will let $\{Y_{u,k}\}$ be any collection of random variables which satisfies the same recursion (for large enough $k$), and for which $Y_{u,k}$ is a good estimator of $\sigma_{u}$ given $\sigma_{L_{k}(u)}$.

There is a $K=K(\delta)$ and a constant $C$ such that for all $k\geq K$, the following hold: {longlist}[2.]

$Y_{u,k+1}=\frac{\prod_{i\in\mathcal{C}(u)}(1+\theta Y_{ui,k})-\prod_{i\in\mathcal{C}(u)}(1-\theta Y_{ui,k})}{\prod_{i\in\mathcal{C}(u)}(1+\theta Y_{ui,k})+\prod_{i\in\mathcal{C}(u)}(1-\theta Y_{ui,k})}$.

The distribution of $Y_{u,k}$ given $\sigma_{u}=+$ is equal to the distribution of $-Y_{u,k}$ given $\sigma_{u}=-$.

With probability at least $1-e^{-cd}$ over $T$,

Note that Assumption 4.1 is the same as Assumption 3.1 except for part 3. Indeed, the change in part 3 between Assumption 3.1 and Assumption 4.1 points to the main change, and biggest challenge, in extending our previous argument to Galton–Watson trees: unlike for a regular tree, there is always some chance that a Galton–Watson tree will go extinct, or that it will be thinner and more spindly than expected. In this case, we will not be able to reconstruct the broadcast process as well as we might want, even as $\eta\to 0$.

In any case, in order to prove Theorem 2.11 it suffices to prove that $Y$ satisfies part 3 of Assumption 4.1 as well as the following theorem.

The first step toward extending Theorem 3.3 to the Galton–Watson case is to show that the magnetization of each node tends to be large.

There is a universal constant $c>0$ such that for all $k\geq K(\theta,d,\delta)$,

and similarly for $Y_{\rho,k}$. Hence, $x_{k},y_{k}\geq 1-\frac{8\eta}{\theta^{2}d}-2e^{-cd}$.

Note that the proposition implies that $Y$ satisfies part $3$ of Assumption 4.1.

In the regular case, the proof of Lemma 3.6 was based on the fact that a simple majority vote at the leaves estimates the root well. Here, we will follow Evans et al. by using a weighted majority vote. For this, we will need to use the terminology of electrical networks, in particular the notion of effective conductance and effective resistance. An \hyperref[sec1]Introduction to these concepts may be found in ; the essential properties that we will need are that conductances add over parallel paths, while resistances add over consecutive paths.

There exist weights $w(u)$ such that if $R_{k}=\sum_{v\in L_{k}(\rho)}w(v)\sigma_{v}$ and $S_{k}=(1-2\delta)^{-1}\sum_{v\in L_{k}(\rho)}w(v)\tau_{v}$ then

We mention that $w(v)$ in Theorem 4.3 is proportional to the unit current flow from $\rho$ to $v$; for our work, however, we only need to know that it exists and that it can be easily computed.

Consider the estimators $\operatorname{sgn}(R_{k})$ and $\operatorname{sgn}(S_{k})$ for $\sigma_{\rho}$. By Chebyshev’s inequality,

There is a universal constant $c>0$ such that for all $k\geq K(\theta,d,\delta)$,

Now, the first $k$ levels of a Galton–Watson tree consist of a root with $\operatorname{Pois}(d)$ independent subtrees of $k-1$ levels each. For each child $i$, the conductance between $i$ and $L_{k-1}(i)$ is distributed like $\theta^{2}Z_{i}$ (the factor $\theta^{2}$ arises because at each level of the tree the conductances are multiplied by an extra factor of $\theta^{2}$). Since the edge between $\rho$ and $i$ has conductance $\theta^{2}(1-\theta^{2})^{-1}$, the conductance between $\rho$ and $L_{k-1}(i)$ is distributed like

Recall that $\operatorname{Pr}(Z_{i}\geq\alpha_{k-1})\geq 1/2$ and $\alpha_{k-1}\leq(4\eta)^{-1}$. Hence, $\alpha_{k-1}/(4\eta\alpha_{k-1}+1)\geq\alpha_{k-1}/2$, and so

Now, there is a universal constant $c>0$ such that $\operatorname{Pr}(\operatorname{Pois}(d/2)\leq d/4)\leq e^{-cd}$; hence

Now Proposition 4.2 follows directly from Theorem 4.3 and Lemma 4.4.

2 The small-\texorpdfstringθ𝜃\thetat​h​e​t​a𝑡ℎ𝑒𝑡𝑎theta case

The proof of Proposition 3.8 extends fairly easily to the Galton–Watson case. The weakening of Lemma 3.6 to Proposition 4.2 makes hardly any difference because the proof of Proposition 3.8 only needed $x_{k}\geq 1/2$.

Consider the broadcast process on a Poisson Galton–Watson tree. Then there are absolute constants $C$ and $\theta^{*}>0$ such that if $d\theta^{2}\geq C$ and $\theta\leq\theta^{*}$ then for all $k\geq K(\theta,d,\delta)$,

Let $D$ be the number of children of $u$, so that $D\sim\operatorname{Pois}(d)$. If $\theta^{2}d$ is sufficiently large then Proposition 4.2 implies that $x_{k},y_{k}\geq 5/6$ and so applying Lemma 3.14 conditioned on $D$ yields

In particular, the right-hand side is smaller than $z/2$ if $\theta^{2}d$ is sufficiently large.

3 The large-\texorpdfstringθ𝜃\thetat​h​e​t​a𝑡ℎ𝑒𝑡𝑎theta case

We now give an analogue of Proposition 3.15 in the Galton–Watson case.

For any $0<\theta^{*}<1$, there is some $d^{*}=d^{*}(\theta^{*})$ such that for the broadcast process on the Poisson mean $d$ tree it holds that for all $\theta\geq\theta^{*}$, $d\geq d^{*}$, and $k\geq K(\theta,d,\delta)$,

This completes the proof of Theorem 4.1 (by the same argument that followed Proposition 3.15).

Our eventual goal is to prove Proposition 3.15 by a similar analysis of the partial derivatives of $g$ that led to the proof of Proposition 3.15. In this section, however, we will deal with one case where the derivatives of $g$ cannot be controlled well. First, we introduce a parameter $\varepsilon=\varepsilon(d)>0$ that will be specified later. Next, fix a vertex $u$ and let $\Omega$ be the event that all children $i$ of $u$ satisfy $|X_{ui,k}-Y_{ui,k}|\leq\varepsilon$. On $\Omega$, we will analyze derivatives of $g$; off $\Omega$ we have the following lemma (recalling that $D$ is the number of children of $u$).

For any $0<\theta^{*}<1$, there exist $c,C>0$ such that if $\eta<c$, $\theta\in[\theta^{*},1)$, and $\theta^{2}d>C$ then for any $\varepsilon>0$ and $k\geq K(\theta,d,\delta)$

First, we condition on $D$; we may then write

where the equality follows because all the terms in the sum have the same distribution. Now we will condition on $X_{ui,k}$ and $Y_{ui,k}$, and we will show that on the event $\{|X_{ui,k}Y_{ui,k}|\geq\varepsilon\}$ we have

After bounding $1\leq\varepsilon^{-1/2}\sqrt{|X_{ui,k}-Y_{ui,k}|}$ on the event $\{|X_{ui,k}Y_{ui,k}|\geq\varepsilon\}$ and then integrating out $X_{ui,k}$ and $Y_{ui,k}$, the proof will be complete.

Now we prove (24). Condition on $\sigma_{u}$, and suppose without loss of generality that $\sigma_{u}=+$. If $\theta^{2}d$ is sufficiently large then Proposition 4.2 implies that (conditioned on $\sigma_{u}=+$) every child $j\neq i$ of $u$ independently satisfies

If we condition also on $D$, Hoeffding’s inequality implies that there is a constant $c>0$ such that with probability at least $e^{-cD^{2}}$, at least $3/4$ of $u$’s children $j$ satisfy $X_{uj,k}\geq 1-\eta$. The remaining children (which possibly include $i$) satisfy $X_{uj,k}\geq-1$, and so on this event

Now, $X_{u,k+1}=\frac{1-A}{1+A}\geq 1-2A$, and so we conclude that

The previous argument applies equally well with $X$ replaced by $Y$; hence, the union bound implies

On the other hand, we always have the bound $|X_{u,k+1}-Y_{u,k+1}|\leq 2$, and so

Now, if $\eta<c$ for $c$ sufficiently small, the right-hand side is bounded by $Ce^{-cD}$. This proves (24) in the case that $\sigma_{u}=+$. To complete the proof, we apply the symmetric argument conditioned on $\sigma_{u}=-$.

3.2 An analogue of Lemma \texorpdfstring3.163.16

The proof of Proposition 4.6 proceeds by analysing the derivatives of the recurrence (15). Recalling that these derivatives involve a large product, an important ingredient in the analysis is a bound on the expectation of each term. The following lemma is analogous to Lemma 3.16 in the regular case; an important difference is that Lemma 4.8 does not improve as $\eta\to 0$. In fact, as we remarked after Assumption 4.1, we cannot expect such behavior because of the possibility of extinction.

For any $0<\theta^{*}<1$, there are some $\lambda=\lambda(\theta^{*})<1$ and $d^{*}=d^{*}(\theta^{*})$ such that for all $\theta\geq\theta^{*}$, $d\geq d^{*}$ and $k\geq K(\theta,d,\delta)$,

We postpone the details of Taylor expansion and approximation to the \hyperref[app]Appendix, but we will include here one of the main ingredients of the proof of Lemma 4.8. The point is that in the Galton–Watson case (unlike the $d$-ary case) if $d$ is fixed and $\eta\to 0$ then we cannot expect $X_{\rho,k}$ to be large (i.e., close to $1$) with probability converging to 1. It turns out to be enough, however, to show that $X_{\rho,k}$ is nonnegative with probability converging to 1.

There is a constant $C$ such that if $\theta^{2}d\geq C$ then for any $k\geq K(\theta,d,\delta)$,

We will give the argument for $X$ only (the argument for $Y$ is identical). First, note that if $\eta\geq 1/12$ then (25) follows directly from Proposition 4.2 if $d^{*}$ is sufficiently large. Hence, we may assume that $\eta<1/12$. Let $p_{k}=\operatorname{Pr}^{+}(X_{\rho,k}<0)$. Then by Proposition 4.2, if $C$ is sufficiently large then $p_{k}\leq 1/12$ for $k\geq K(\delta)$.

Let $Z_{-}$ be the number of children $i$ of the root with $X_{i,k}<0$ and $Z_{+}$ be the number with $X_{i,k}\geq 1-\eta$. Consider the quantity

and note that $X_{u,k}<0$ if and only if $Z>1$. Now, $Z$ is increasing in each $X_{ui,k}$, and $Z$ only increases if we drop some terms $i$ with $X_{ui,k}\geq 0$. Hence,

Conditioned on $\sigma_{\rho}$ and $D$, $Z_{+}-Z_{-}$ is a sum of i.i.d. variables with values $1,-1$, and . Moreover, Proposition 4.2 with $d$ sufficiently large implies that the probability of $X_{i,k}\geq 1-\eta$ is at least $5/6$, while (4.3.2) implies that the probability of $X_{i,k}<0$ is at most $p_{k}+\eta\leq 1/6$. Hence, Hoeffding’s inequality implies that

for universal constants $c,C>0$. Note also that if $Z_{-}=0$ then $Z\geq 1$ and that in order to have $Z_{-}>0$, there must be some $i$ with $X_{i,k}<0$. Note also that if $Z_{+}-Z_{-}\geq D/3$ then $Z\leq 3^{D}\eta^{D/3}\leq(3/4)^{D/3}<1$. Thus, applying a union bound, Hoeffding’s inequality, and (4.3.2),

Recursing with $k$, we see that $\lim_{k\to\infty}\operatorname{Pr}^{+}(X_{\rho,k}<0)\leq\eta/2$, which implies that $\operatorname{Pr}^{+}(X_{\rho,k}<0)\leq\eta$ for sufficiently large $k$.

3.3 Analysis of the derivatives of g𝑔g

Our goal in this section is the following lemma, for which we recall that $\Omega$ is the event that all children $i$ of $u$ satisfy $|X_{ui,k}-Y_{ui,k}|\leq\varepsilon$. Let $\Omega_{i}$ be the event that $|X_{ui,k}-Y_{ui,k}|\leq\varepsilon$.

For any $0<\theta^{*}<1$, there are constants $c,C>0$ such that for all $0<\varepsilon<1/4$, all $d\geq d^{*}(\theta^{*})$, and for any $k\geq K(\theta,d,\delta)$,

We begin with an slightly improved version of (3.6): since $|X_{u,k+1}-Y_{u,k+1}|\leq 2$, we can trivially improve (3.6) to

Note that $1_{\Omega}\leq 1_{\Omega_{i}}$ for any $i$ (recall that $\Omega_{i}=\{|X_{ui,k}-Y_{ui,k}|\leq\varepsilon\}$), and so