Lecture 6 – Local limits

I am aiming to write a short post about each lecture in my ongoing course on Random Graphs. Details and logistics for the course can be found here.

By this point of the course, we’ve studied several aspects of the Erdos-Renyi random graph, especially in the sparse setting $G(n,\frac{\lambda}{n})$. We’ve also taken a lengthy detour to revise Galton-Watson trees, with a particular focus on the case of Poisson offspring distribution.

This is deliberate. Note that a given vertex v of $G(n,\frac{\lambda}{n})$ has some number of neighbours distributed as $\mathrm{Bin}(n-1,\frac{\lambda}{n})\stackrel{d}\approx\mathrm{Po}(\lambda)$, and the same approximation remains valid as we explore the graph (for example in a breadth-first fashion) either until we have seen a large number of vertices, or unless some ultra-pathological event happens, such as a vertex having degree n/3.

In any case, we are motivated by the notion that the local structure of $G(n,\frac{\lambda}{n})$ is well-approximated by the Galton-Watson tree with $\mathrm{Po}(\lambda)$ offspring, and in this lecture and the next we try to make this notion precise, and discuss some consequences when we can show that this form of convergence occurs.

Deterministic graphs

Throughout, we will be interested in rooted graphs, since by definition we have to choose a root vertex whose local neighbourhood is to be studied. Usually, we will study a sequence of rooted graphs $(G_n,\rho_n)$, where the vertex set of $G_n$ is [n], or certainly increasing in n (as in the first example).

For some rooted graph $(G,\rho)$, we say such a sequence $(G_n,\rho_n)$ converges to $(G,\rho)$ locally if for all radii $r\ge 1$, we have $B_r^{G_n}(\rho_n)\simeq B_r^G(\rho)$. In words, the neighbourhood around $\rho_n$ in $G_n$ is the same up to radius r as the neighbourhood around $\rho$ in $G$, so long as n is large enough (for given r).

This is best illustrated by an example, such as $T_n$, the binary tree to depth n.

If we take $\rho_n$ to be the usual root, then the trees are nested, and converge locally to the infinite binary tree $T_\infty$. Slightly less obviously, if we take $\rho_n$ to be one of the leaves, then the trees are still nested (up to labelling – ie in the sense of isomorphisms of rooted trees), and converge locally to the canopy tree, defined by a copy of $\mathbb{Z}_{\ge 0}$ with nearest-neighbour edges, and where each vertex $n\ge 1$ is connected to the root of a disjoint copy of $T_{n-1}$, as shown below:

Things get more interesting when the root is chosen randomly, for example, uniformly at random, as this encodes more global information about the graphs $G_n$. In the case where the $G_n$ are vertex-transitive, then if we only care about rooted graphs up to isomorphism, then it doesn’t matter how we choose the root.

Otherwise, we say that $G_n$ converges in the local weak sense to $(G,\rho)$ if, for all $r\ge 1$ and for all rooted graphs $(H,\rho_H)$,

$\mathbb{P}\left( B^{G_n}_r(\rho_n)\simeq (H,\rho_H) \right) \longrightarrow \mathbb{P}\left( B_r^G(\rho)\simeq H\right),$

as $n\rightarrow\infty$.

Alternatively, one can phrase this as a result about convergence of rooted-graph-valued distributions.

A simple non-transitive example is $G_n\simeq P_n$, the path of length n. Then, the r-neighbourhood of a vertex is isomorphic to $P_{2r}$unless that vertex is within graph-distance (r-1) of one of the leaves of $G_n$. As $n\rightarrow\infty$, the proportion of such vertices vanishes, and so, $\mathbb{P}\left( B^{P_n}_r(\rho_n)\simeq P_{2r}\right)\rightarrow 1$, from which we conclude the unsurprising result that $P_{n}$ converges in the local weak sense to $\mathbb{Z}$. (Which is vertex-transitive, so it doesn’t matter where we select the root.)

The binary trees offer a slightly richer perspective. Let $\mathcal{L}_n$ be the set of leaves of $T_n$, and we claim that when $\rho_n$ is chosen uniformly from the vertices of $T_n$, then $d_{T_n}(\rho_n,\mathcal{L}_n)$ converges in distribution. Indeed, $\mathbb{P}\left( d_{T_n}(\rho_n,\mathcal{L}_n)=k\right) = \frac{2^{n-k}}{2^{n+1}-1}$, whenever $n\ge k$, and so the given distance converges in distribution to the Geometric distribution with parameter 1/2 supported on {0,1,2,…}.

This induces a random local weak limit, namely the canopy tree, rooted at one of the vertices we denoted by $\mathbb{Z}_{\ge 0}$, with the choice of this vertex given by Geometric(1/2). Continue reading

Discontinuous Phase Transitions

Yesterday, Demeter Kiss from Cambridge gave a seminar in Oxford about a model for self-destructive percolation on $\mathbb{Z}^2$ that had implications for the (non-)existence of an infinite-parameter forest fire model on the same lattice. I enjoyed talking about this and his recent work on the related model of frozen percolation on $\mathbb{Z}^2$. Considering these models in the lattice setting present a whole range of interesting geometric challenges that are not present in the mean-field case that has mainly occupied my research direction so far.

The afternoon’s discussion included lots of open problems about percolation. Several of these are based around continuity of the phase transition, so I thought I would write a quite post about some simple examples of this, and one example where it does not hold.

A helpful base example is bond percolation on the lattice $\mathbb{Z}^2$. Here, we specify some probability p in [0,1], and we declare edges of the lattice open with probability p, independently of each other. We then consider the graph induced by the open edges. We say that percolation occurs if the origin is contained in an infinite open component. The terminology arises from the interpretation as fluid being added at the origin and flowing down open edges. We define $\theta(p)$ to be the probability that the origin is in an infinite component when the parameter is p. By translation-invariance, we can get some sort of 0-1 law, to conclude that there is an infinite component somewhere in the system with probability either 0 or 1, depending on whether $\theta(p)$ is positive or zero. Indeed, we can further show that if it is positive, then with probability 1 there is a unique infinite component.

We define the critical probability $p_c:= \inf\{\theta(p)>0\}$. A question worth asking is then, what is $\theta(p_c)$? In some examples, we can find $p_c$, but we cannot prove that $\theta(p)$ is continuous around $p_c$. In the case of $\mathbb{Z}^2$ this is known, and it is known from work of Kesten that $p_c=1/2$. See below for a plot of $\theta(p)$ in this setting (obtained from this blog, though possibly originating elsewhere).

The aim is to find an example where we do not have such a continuous phase transition. The original work on frozen percolation took place on trees, and one of Kiss’s results is confirms that these show qualitatively different phenomena to the same process on the lattice. In some sense, trees lie halfway between a lattice and a mean-field model, since there is often some independence when we look down the tree from a given generation, if it is well-defined to use such language.

Anyway, first we consider percolation on an infinite regular rooted k-ary tree. This means we have a root, which has k children, each of which in turn has k children, and so on. As before we consider bond percolation with parameter p. In this setting, we have a language to describe the resulting open component of the root. The offspring distribution of any vertex in the open component is given by Bin(k,p) independently of everything else, so we can view this component as the realisation of a Galton-Watson tree with this offspring distribution. This distribution has finite mean kp, and so we can state explicitly when the survival probability is positive. This happens when the mean is greater than 1, ie p>1/k.

For our actual example, we will consider the survival probability, but the technicalities are easier to explain if we look at the extinction probability, now using the language of branching processes. Suppose the offspring distribution has pgf given by

$f(x)=p_0+p_1x+p_2x^2+\ldots.$

Then the extinction probability q satisfies f(q)=q. I want to pause to consider what happens if this equation has multiple solutions. Indeed, in most interesting cases it will have multiple solutions, since f(1) will always be 1 if it is a non-defective offspring distribution. It is typically cited that: the extinction probability q is the smallest solution to this equation. I want to discuss why that is the case.

To approach this, we have to consider what extinction means. It is the limit in the event sense of the events {we are extinct after n generations}. Let the probabilities of these events be $q_n$, so $q_0=0$. Then by a straightforward coupling argument, we must have

$0=q_0\le q_1\le q_2 \le\ldots\le q:= \lim q_n \le 1.$

But, by the same generating function argument as before, $q_{n+1}=f(q_n)\ge q_n$. So if we split [0,1] into regions A where $f(x)\ge x$ and B where $f(x), all the $(q_n)$s must occur in the former, and so since it is closed, their limit must be in A also. Note that if f(x) intersects x lots of times, then region A is not necessarily connected. In the diagram below, in moving from $q_n$ to $q_{n+1}$ we might jump across part of B.

This is bad, as we are trying to prove that q is the right boundary of the connected component of A containing 0. But this cannot happen, as f is monotonic. So if one of the roots of f(x)=x in between the hypothesised $q_n is called z, then $f(q_n)< f(z)=z < q_{n+1}$, a contradiction.

Ok, so now we are ready to consider our counterexample to continuity over the percolation threshold. See references for a link to the original source of this example. We have to choose a slightly more complicated event than mere survival or extinction. We consider bond percolation as before on the infinite ternary tree, where every vertex has precisely 3 offspring. Our percolation event is now that the root is the root of an infinite binary tree. That is, the root has at least two children, each of which have at least two children, each of which, and so on.

If we set this probability equal to q, and the probability of an edge being open equal to p, then we have the recurrence:

$q=3p^2(1-p)q^2+p^3[3q^2(1-q)+q^3].$

The first term corresponds to the root having two open edges to offspring, and the second to the root having all three open edges to offspring. After manipulating, we end up with

$q\left[2p^3q^2-3p^2q+1\right]=0.$

We are therefore interested in roots of the quadratic lying between 0 and 1. The discriminant can be evaluated as

$\Delta=p^3(9p-8),$

and so there are no real roots where p<8/9. But when p=8/9, we have a repeated root at q=27/32, which is obviously not zero!

This equation is qualitatively different to the previous one for the extinction probability of a Galton-Watson tree. There, we had a quadratic, with one root at 1. As we varied p, the other root moved continuously from greater than one to less than one, so it passed through 1, giving continuity at the critical probability. Here, we have a cubic, again with one root at 1. But now the other roots are complex for small p, meaning that the local minimum of the cubic lies above the x-axis. As p gets to the critical value, it the local minimum passes below the x-axis, and suddenly we have a repeated root, not at zero.

I would like to have a neat probabilistic heuristic for this result, without having to make reference to generating functions. At the moment, the best I can come up with is to say that the original problem is simple, in the sense that the critical probability is as small as it could be while still making sense in expectation. To be concrete, when the mean of the offspring generation is less than 1, the expected size of the nth generation tends to zero, so there certainly could not be positive probability of having an infinite component.

Whereas in the binary tree example, we only require p=2/3 to have, in expectation, the right number of open edges to theoretically allow an infinite binary tree. If we think of percolation as a dynamic process by coupling in p, essentially as we move from p=2/3 to p=8/9 we need to add enough edges near the origin to be able to take advantage of the high density of edges available far from the origin. The probability of this working given you start from n vertices grows much faster (as n grows) than in the original problem, so you might expect a faster transition.

This is so content-free I’m reluctant even to call it a heuristic. I would be very interested to hear of any more convincing argument for this phenomenon!

REFERENCES

Dekking, Pakes – On family trees and subtrees of simple branching processes (link)

The Contour Process

As I explained in my previous post, I haven’t been reading around as much as I would generally like to recently. A few days in London staying with my parents and catching up with some friends has therefore been a good chance to get back into the habit of leafing through papers and Pitman’s book among other things.

This morning’s post should be a relatively short one. I’m going to define the contour process, a function of a (random or deterministic) tree, related to the exploration process which I have mentioned a few times previously. I will then use this to prove a simple but cute result equating in distribution the sizes of two different branching processes via a direct bijection.

The Contour Process

To start with, we have to have a root, and from that root we label the tree with a depth-first labelling. An example of this is given below. It is helpful at this stage to conceive this process as an explorer walking on the tree, and turning back on themselves only when there is no option to visit a vertex they haven’t already seen. So in the example tree shown, the depth-first exploration visits vertex V_2 exactly four times. Note that with this description, it is clear that the exploration traverses every edge exactly twice, and so the length of the sequence is 2n-1, where n is the number of vertices in the tree since obviously, we start and end at the root.

Another common interpretation of this depth-first exploration is to take some planar realisation of the tree. (Note trees are always planar – proof via induction after removing a leaf.) Then if you treat the tree as a hedge and starting at the root walk along, following the outer boundary with your right hand, this exactly recreates the process.

The height of a tree at a particular vertex is simply the graph distance between that vertex and the root. So when we move from one vertex to an adjacent vertex, the height must increase or decrease by 1.

The contour process is the sequence of heights seen along the depth-first exploration. It is therefore a sequence:

$0=h_0,h_1,\ldots,h_{2n-1}=0,\quad h_i\geq 0,$

and such that $|h_{i+1}-h_i|=1$.

Note that though the contour process uniquely determines the tree structure, the choice of depth-first labelling is a priori non-canonical. For example, in the display above, V_3 might have been explored before V_2. Normally this is resolved by taking the suitable vertex with the smallest label in the original tree to be next. It makes little difference to any analysis to choose the ordering of descendents of some vertex in a depth-first labelling randomly. Note that this explains why it is rather hard to recover Cayley’s theorem about the number of rooted trees on n vertices from this characterisation. Although the number of suitable contour functions is possible to calculate, we would require a complicated multiplicative correction for labelling if we wanted to recover the number of trees.

The only real observation about the uses of the contour process at this stage is that it is not in general a random walk with IID increments for a Galton-Watson branching process. This equivalence is what made the exploration process so useful. In particular, it made it straightforward, at least heuristically, to see why large trees might have a limit interpretation through Brownian excursions. If for example, the offspring distribution is bounded above, say by M, then the contour process certainly cannot be a random walk, as if we have visited a particular vertex exactly M+1 times, then it cannot have another descendent, and so we must return closer to the root at the next step.

I want to mention that in fact Aldous showed his results on scaling limits towards the Continuum Random Tree through the contour process rather than the exploration process. However, I don’t want to say any more about that right now.

A Neat Equivalence

What I do want to talk about is the following distribution on the positive integers. This comes up in Balazs Rath and Balint Toth’s work on forest-fires on the complete graph that I have been reading about recently. The role of this distribution is a conjectured equilibrium distribution for component size in a version of the Erdos-Renyi process where components are deleted (or ‘struck by lightning’) at a rate tuned so that giant components ‘just’ never emerge.

This distribution has the possibly useful property that it is the distribution of the total population size in a Galton-Watson process with Geom(1/2) offspring distribution. It is also the distribution of the total number of leaves in a critical binary branching process, where every vertex has either two descendents or zero descendents, each with probability 1/2. Note that both of these tree processes are critical, as the expected number of offspring is 1 in each case. This is a good start, as it suggests that the relevant equilibrium distribution should also have the power-law tail that is found in these critical branching processes. This would confirm that the forest-fire model exhibits self-organised criticality.

Anyway, as a sanity check, I tried to find a reason why, ignoring the forest-fires for now, these two distributions should be the same. One can argue using generating functions, but there is also the following nice bijective argument.

We focus first on the critical Geometric branching process. We examine its contour function. As explained above, the contour process is not in general a random walk with IID increments. However, for this particular case, it is. The geometric distribution should be viewed as the family of discrete memoryless distributions.

This is useful for the contour process. Note that if we are at vertex V for the (m+1)th time, that is we have already explored m of the edges out of V, then the probability that there is at least one further edge is 1/2, independently of the history of the exploration, as the offspring distribution is Geometric(1/2), which we can easily think of as adding edges one at a time based on independent fair coin tosses until we see a tail for example. The contour process for this random tree is therefore a simple symmetric random walk on Z. Note that this will hit -1 at some point, and the associated contour process is the RW up to the final time it hits 0 before hitting -1. We can check that this obeys the clear rule that with probability 1/2 the tree is a single vertex.

Now we consider the other model, the Galton-Watson process with critical binary branching mechanism. We should consider the exploration process. Recall that the increments in this process are given by the offspring distribution minus one. So this random sequence also behaves as a simple symmetric random walk on Z, again stopped when we hit -1.

To complete the bijective argument, we have to relate leaves in the binary process to vertices in the geometric one. A vertex is a leaf if it has no offspring, so the number of leaves is the number of times before the hitting time of -1 that the exploration process decreases by 1. (*)

Similarly for the contour process. Note that there is bijection between the set of vertices that aren’t the root and the set of edges. The contour process explores every edge exactly twice, once giving an increase of 1 and once giving a decrease of 1. So there is a bijection between the times that the contour process decreases by 1 and the non-root vertices. But the contour process was defined only up to the time we return to the root. This is fine if we know in advance how large the tree is, but we don’t know which return to the root is the final return to the root. So if we extend the random walk to the first time it hits -1, the portion up until the last increment is the contour process, and the final increment must be a decrease by 1, hence there is a bijection between the number of vertices in the Geom(1/2) G-W tree and the number of times that the contour process decreases by 1 before the hitting time of -1. Comparing with (*) gives the result.

Local Limits

In several previous posts, I have talked about scaling limits of various random graphs. Typically in this situation we are interested in convergence of large-scale properties of the graph as the size grows to some limit. These properties will normally be metric in flavour: diameter, component size and so on. To describe convergence of these properties, we divide by the relevant scale, which will often be some simple function of n. If we are looking to find an actual limit object, this is even more important. This is rather similar to describing properties of centred random walks. There, if we run the walk for time n, we have to rescale by $\frac{1}{\sqrt{n}}$ to see the fluctuations on a finite positive scale.

One of the best examples is Aldous’ Continuum Random Tree which we can view as the limit of a Galton-Watson tree conditioned to have total size n, as n tends to infinity. Because of the exploration process or contour process interpretation, where these functions behave rather like a random walk, the correct scaling in this context is again $\frac{1}{\sqrt{n}}$. The point about this convergence is that it is realised entirely as a convergence of some function that represents the tree. For each finite n, it is clear that the tree with n vertices is a graph, but this is neither clear nor true for the limit object. Although it does indeed have no cycles, if nothing else, if the CRT were a graph it would have [0,1] as vertex set and then would be highly non-obvious how to define the edges.

Local limits aim to give convergence towards a (discrete) infinite graph. The sort of properties we are looking for are now local properties such as degrees and correlations of degrees. These don’t require knowledge of the whole graph, only of some finite subset. First consider the possibility that the sequence of deterministic graphs has the property:

$G_1\leq G_2\leq G_3\leq\ldots$

where $\leq$ denotes an induced subgraph. Then it is relatively clear what the limit should be, as it is well-defined to take a union. This won’t work directly for a limit of random graphs, because the above relation in probability doesn’t even really make sense if we have a different probability space for each finite graph. This is a general clue that we should be looking to use convergence in distribution rather than anything stronger.

In the previous example, suppose the first finite graph $G_1$ consists of a single vertex v. If the limit graph (remember this is just the union, since that is well-defined) has bounded degrees, then there is some N such that $G_N$ contains all the information we might want about the limiting neighbourhood of vertex v. For some larger N, $G_N$ contains all the vertex and edges within distance r from our starting vertex v that appear in the limit graph.

This is all the motivation we require for a genuine definition. We will define our limit in terms of neighbourhoods, so we need some mechanism to choose the central vertex of such a neighbourhood. The answer is to consider rooted graphs, that it a graph with an identified vertex. We can introduce randomness by specifying a random graph, or by giving a distribution for the choice of root. If G is finite, the canonical choice is to choose the root uniformly from the set of vertices. This isn’t an option for an infinite graph, so we define the system as (G, p) where G is a (for now deterministic) graph, and p is a probability measure on V(G).

We say that the limit of finite $(G_n)$ is the random rooted infinite graph (G, p) if the neighbourhoods of $G_n$ around a randomly chosen vertex converge in distribution to the neighbourhoods of G around p. Formally, say $(G_n)[U_n]\stackrel{d}{\rightarrow} (G,p)$ if for all r>0, for any finite rooted graph (H,w), the probability that (H,w) is isomorphic to the ball of radius r in $G_n$ centred at randomly chosen $v_n$ converges to the probability that (H,w) is isomorphic to the ball of radius r around v in (G,v), where v is distributed according to measure p.

Informally, we might say that if we zoom in on an average vertex in $G_n$ for large n, the neighbourhood looks the same as the neighbourhood around the root in (G, p). We now consider three examples.

1) When we talk about approximating the component size in a sparse Erdos-Renyi random graph by a $\text{Po}(\lambda)$ branching process, this is exactly the limit sense we mean. The approximation fails if we fix n and take the neighbourhood size very large (eg radius n), but for finite neighbourhoods, or any radius growing more slowly than n, the approximation is good.

2) To emphasise why rooting the finite graphs makes a difference, consider the full binary tree with n levels (so $2^n-1$ vertices). If we fix the root, then the limit is the infinite-level binary tree, though this isn’t especially surprising or interesting.

Things get a bit more complicated if we root randomly. Remember that the motivation for random rooting is that we want to know the local structure around a vertex chosen at random in many applications. If we definitely know what vertex we are going to choose, we know the local structure a priori. Note that in an n-level binary tree, $2^{n-1}$ vertices are leaves, not counting the base of the tree, and $2^{n-2}$ are distance 1 from a leaf, and $2^{n-3}$ are distance 2 from a leaf and so on.

This gives us a precise description of the limiting local neighbourhood structure. The resulting limiting object is called the canopy tree. One picture of this can be found on page 6 of this paper. A verbal description is also possible. Consider the set of non-negative integers, arranged in the usual manner on the real line, with edges between adjacent elements. The distribution of the root will be supported on this set of vertices, corresponding to the distance from the leaves in the pre-limit graph. So we have mass 1/2 at 0, 1/4 at 1, 1/8 at 2 and so on. We then connect each vertex k to a full k-level binary tree. The resulting canopy tree looks like an infinite-level full binary tree, viewed from the leaves, which is of course a reasonable heuristic, since that is there the mass is concentrated if we randomly root.

3) In particular, the limit is not the infinite-level binary tree. The canopy tree and the infinite-level binary tree have qualitatively different properties. Simple random walk on the canopy tree is recurrent for example. In fact, a result of Benjamini and Schramm, as explained in this review by Curien, says that any local limit of uniformly bounded degree, uniformly rooted, planar graphs is recurrent for SRW. The infinite-level binary tree can be expressed as a local limit if we choose the root distribution sensibly, using large random 3-regular graphs. The previous result does not apply because the random 3-regular graphs are not almost surely planar.

REFERENCES:

– Much of this article is a paraphrase of a section of Itai Benjamini’s mini-course at the DSSA in Haifa March 2013.

– As well as the review paper linked above, these notes by David Aldous were very useful.