Lecture 8 – Bounds in the critical window

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.

Preliminary – positive correlation, Harris inequality

I wrote about independence, association, and the FKG property a long time ago, while I was still an undergraduate taking a first course on Percolation in Cambridge. That post is here. In the lecture I discussed the special case of the FKG inequality applied in the setting of product measure setting, of which the Erdos-Renyi random graph is an example, and which is sometimes referred to as the Harris inequality.

Given two increasing events A and B, say for graphs on [n], then if \mathbb{P} is product measure on the edge set, we have

\mathbb{P}(A\cap B)\ge \mathbb{P}(A)\mathbb{P}(B).

Intuitively, since both A and B are ‘positively-correlated’ with the not-rigorous notion of ‘having more edges’, then are genuinely positively-correlated with each other. We will use this later in the post, in the form \mathbb{E}[X|A]\ge \mathbb{E}[X], whenever X is an increasing RV and A is an increasing event.

The critical window

During the course, we’ve discussed separately the key qualitative features of the random graph G(n,\frac{\lambda}{n}) in the

  • subcritical regime when \lambda<1, for which we showed that all the components are small, in the sense that \frac{1}{n}|L_1| \stackrel{\mathbb{P}}\rightarrow 0, although the same argument would also give |L_1|\le K\log n with high probability if we used stronger Chernoff bounds;
  • supercritical regime when \lambda>1, for which there is a unique giant component , ie that \frac{1}{n}|L_1|\stackrel{\mathbb{P}}\rightarrow \zeta_\lambda>0, the survival probability of a Galton-Watson branching process with Poisson(\lambda) offspring distribution. Arguing for example by a duality argument shows that with high probability all other components are small in the same sense as in the subcritical regime.

In between, of course we should study G(n,\frac{1}{n}), for which it was known that L_1\stackrel{d}\sim n^{2/3},\, L_2\stackrel{d}\sim n^{2/3},\ldots. (*) That is, the largest components are on the scale n^{2/3}, and there are lots of such critical components.

In the early work on random graphs, the story ended roughly there. But in the 80s, these questions were revived, and considerable work by Bollobas and Luczak, among many others, started investigating the critical setting in more detail. In particular, between the subcritical and the supercritical regimes, the ratio \frac{|L_2|}{|L_1|} between the sizes of the largest and second-largest components goes from ‘concentrated on 1’ to ‘concentrated on 0’. So it is reasonable to ask what finer scaling of the edge probability p(n) around \frac{1}{n} should be chosen to see this transition happen.

Critical window

In this lecture, we studied the critical window, describing sequences of probabilities of the form

p(n)=\frac{1+\lambda n^{-1/3}}{n},

where \lambda\in(-\infty,+\infty). (Obviously, this is a different use of \lambda to previous lectures.)

It turns out that as we move \lambda from -\infty to +\infty, this window gives exactly the right scaling to see the transition of \frac{|L_2|}{|L_1|} described above. Work by Bollobas and Luczak and many co-authors and others in the 80s establish a large number of results in this window, but for the purposes of this course, this can be summarised as saying that the critical window has the same scaling behaviour as p(n)=1/n, with a large number of components on the scale \sim n^{2/3} (see (*) earlier), but different scaling limits.

Note: Earlier in the course, we have discussed local limits, in particular for G(n,\lambda/n), where the local limit is a Galton-Watson branching process tree with offspring distribution \mathrm{Poisson}(\lambda). Such local properties are not sufficient to distinguish between different probabilities within the critical window. Although there are lots of critical components, it remains the case that asymptotically almost all vertices are in ‘small components’.

The precise form of the scaling limit for

\frac{1}{n^{2/3}} \left( |L_1|, |L_2|, |L_3|,\ldots \right)

as n\rightarrow\infty was shown by Aldous in 1997, by lifting a scaling limit result for the exploration process, which was discussed in this previous lecture and this one too. Since Brownian motion lies outside the assumed background for this course, we can’t discuss that, so this lecture establishes upper bounds on the correct scale of |L_1| in the critical window. Continue reading

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Lecture 7 – The giant component

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.

As we edge into the second half of the course, we are now in a position to return to the question of the phase transition between the subcritical regime \lambda<1 and the supercritical regime \lambda>1 concerning the size of the largest component L_1(G(n,\lambda/n)).

In Lecture 3, we used the exploration process to give upper bounds on the size of this largest component in the subcritical regime. In particular, we showed that

\frac{1}{n}\big| L_1(G(n,\lambda/n)) \big| \stackrel{\mathbb{P}}\rightarrow 0.

If we used slightly stronger random walk concentration estimates (Chernoff bounds rather than 2nd-moment bounds from Chebyshev’s inequality), we could in fact have shown that with high probability the size of this largest component was at most some logarithmic function of n.

In this lecture, we turn to the supercritical regime. In the previous lecture, we defined various forms of weak local limit, and asserted (without attempting the notationally-involved combinatorial calculation) that the random graph G(n,\lambda/n) converges locally weakly in probability to the Galton-Watson tree with \text{Poisson}(\lambda) offspring distribution, as we’ve used informally earlier in the course.

Of course, when \lambda>1, this branching process has strictly positive survival probability \zeta_\lambda>0. At a heuristic level, we imagine that all vertices whose local neighbourhood is ‘infinite’ are in fact part of the same giant component, which should occupy (\zeta_\lambda+o_{\mathbb{P}}(1))n vertices. In its most basic form, the result is

\frac{1}{n}\big|L_1(G(n,\lambda/n))\big|\;\stackrel{\mathbb{P}}\longrightarrow\; \zeta_\lambda,\quad \frac{1}{n}\big|L_2(G(n,\lambda/n))\big| \;\stackrel{\mathbb{P}}\longrightarrow\; 0, (*)

where the second part is a uniqueness result for the giant component.

The usual heuristic for proving this result is that all ‘large’ components must in fact be joined. For example, if there are two giant components, with sizes \approx \alpha n,\approx \beta n, then each time we add a new edge (such an argument is often called ‘sprinkling‘), the probability that these two components are joined is \approx 2ab, and so if we add lots of edges (which happens as we move from edge probability \lambda-\epsilon\mapsto \lambda ) then with high probability these two components get joined.

It is hard to make this argument rigorous, and the normal approach is to show that with high probability there are no components with sizes within a certain intermediate range (say between \Theta(\log n) and n^\alpha) and then show that all larger components are the same by a joint exploration process or a technical sprinkling argument. Cf the books of Bollobas and of Janson, Luczak, Rucinski. See also this blog post (and the next page) for a readable online version of this argument.

I can’t find any version of the following argument, which takes the weak local convergence as an assumption, in the literature, but seems appropriate to this course. It is worth noting that, as we shall see, the method is not hugely robust to adjustments in case one is, for example, seeking stronger estimates on the giant component (eg a CLT).

Anyway, we proceed in three steps:

Step 1: First we show, using the local limit, that for any \epsilon>0,

\frac{1}{n}\big|L_1(G(n,\lambda/n))\big| \le \zeta_\lambda+\epsilon, with high probability as n\rightarrow\infty.

Step 2: Using a lower bound on the exploration process, for \epsilon>0 small enough

\frac{1}{n}\big|L_1(G(n,\lambda/n))\big| \ge \epsilon, with high probability.

Step 3: Motivated by duality, we count isolated vertices to show

\mathbb{P}(\epsilon n\le |L_1| \le (\zeta_\lambda-\epsilon)n) \rightarrow 0.

We will return to uniqueness at the end.

Step 1

This step is unsurprising. The local limit gives control on how many vertices are in small components of various sizes, and so gives control on how many vertices are in small components of all finite sizes (taking limits in the right order). This gives a bound on how many vertices can be in the giant component. Continue reading

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

Lecture 4 – Hitting time theorem

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.

This lecture consisted of revision of the most relevant theory of Galton-Watson trees, with a focus on the case where the offspring distribution is Poisson, since, as we have seen in previous lectures, this is a strong candidate to approximate the structure of G(n,\lambda/n). It makes sense to cover the theory of the trees before attempting to make rigorous the sense of approximation.

Given a Galton-Watson tree T, it is natural to label the vertices in a breadth-first order as \varnothing=v_1,v_2,\ldots,v_{|T|}. This is easiest if we have constructed the Galton-Watson tree as a subset of the infinite Ulam-Harris tree, where vertices have labels like (3,5,17,4), whose parent is (3,5,17). If this child vertex is part of the tree, then so are (3,5,17,1), (3,5,17,2), and (3,5,17,3). This means our breadth-first order is canonically well-defined, as we have a natural ordering of the children of each parent vertex.

Note: one advantage of using breadth-first order rather than depth-first order (which corresponds to the usual dictionary, or lexicographic ordering of the labels) is that if the tree is infinite, we don’t explore all of it during a depth-first search. (In the sense that there exist vertices which are never given a finite label.) For breadth-first search, a similar problem arises precisely when some vertex has infinitely many children. For a conventional Galton-Watson tree, the latter situation is much less of a problem than the infinite total population problem, which happens with positive probability whenever \mu=\mathbb{E}[X]>1.

Anyway, given the depth-first order, one can consider an exploration process S_0,S_1,S_2,\ldots,S_{|T|} given by

S_0=1,\quad S_i=S_{i-1}+(X_i-1),

where X_i is the number of children of v_i. In this way, we see that

S_i=\big| \Gamma(v_1)\cup\ldots\cup\Gamma(v_i)\backslash \{v_1,\ldots,v_i\}\big|,\quad i\ge 1,

records the number of vertices in some stack containing those which we have ‘seen but not explored’. Some authors prefer to start from 0, in which case one ends up with a similar but slightly different interpretation of the ‘stack’, but that’s fine since we aren’t going to define formally what ‘seen’ and ‘explored’ means in this post.

Essentially, we exhaust the vertices of the tree whenever S_t=0, and so the condition that |T|=n requires

S_n=0,\quad S_m\ge 1,\; m=0,1,\ldots,n-1.

Conveniently, so long as we have avoiding ordering ambiguity, for example by insisting that trees live within the Ulam-Harris tree, we can reconstruct T uniquely from (S_0,S_1,\ldots,S_{|T|}).

Furthermore, if T is a Galton-Watson process, then the numbers of children X_i are IID, and so in fact this exploration process is a random walk, and the size of the tree can be recovered as the hitting time of zero.

Note: making fully rigorous the argument that children in the GW tree are independent of the breadth-first walk fully rigorous is somewhat technical, and not to be dismissed lightly, though not of principle interest at the level of this topics course. See Proposition 1.5 in Section 1.2 of Le Gall’s notes or Section 1.2.2 of my doctoral thesis for further discussion and argument.

The hitting time theorem allows us to study the distribution of the hitting time of a random walk whose increments are bounded below by -1, in terms of the distribution of the value of the random walk.

Theorem: Let (S_n,\, n\ge 0) be a random walk with S_0=0 and IID increments (X_n,n\ge 1) satisfying \mathbb{P}(X_n\ge -1)=1. Let H_{-k}=\inf \left\{n\,:\, S_n=-k\right\} be the hitting time of -k.

Then \mathbb{P}\big( H_{-k}=n\big) = \frac{k}{n}\mathbb{P}\big(S_n=-k).

Commentary: there are local central limit theorem estimates and large deviation estimates that allow good control of the probability on the RHS for a rich class of contexts. So at a meta-level, the hitting time theorem allows us to reduce a complicated (though still classical) problem, to a real classical problem, which is particularly helpful when the LHS is a device for capturing relevant information about our random tree model.

Continue reading

Lecture 3 – Couplings, comparing distributions

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.

In this third lecture, we made our first foray into the scaling regime for G(n,p) which will be the main focus of the course, namely the sparse regime when p=\frac{\lambda}{n}. The goal for today was to give a self-contained proof of the result that in the subcritical setting \lambda<1, there is no giant component, that is, a component supported on a positive proportion of the vertices, with high probability as n\rightarrow\infty.

More formally, we showed that the proportion of vertices contained within the largest component of G(n,\frac{\lambda}{n}) vanishes in probability:

\frac{1}{n} \left| L_1\left(G\left(n,\frac{\lambda}{n}\right)\right) \right| \stackrel{\mathbb{P}}\longrightarrow 0.

The argument for this result involves an exploration process of a component of the graph. This notion will be developed more formally in future lectures, aiming for good approximation rather than bounding arguments.

But for now, the key observation is that when we ‘explore’ the component of a uniformly chosen vertex v\in[n] outwards from v, at all times the number of ‘children’ of v which haven’t already been considered is ‘at most’ \mathrm{Bin}(n-1,\frac{\lambda}{n}). Since, for example, if we already know that eleven vertices, including the current one w are in C(v), then the distribution of the number of new vertices to be added to consideration because they are directly connected to w has conditional distribution \mathrm{Bin}(n-11,\frac{\lambda}{n}).

Firstly, we want to formalise the notion that this is ‘less than’ \mathrm{Bin}(n,\frac{\lambda}{n}), and also that, so long as we don’t replace 11 by a linear function of n, that \mathrm{Bin}(n-11,\frac{\lambda}{n})\stackrel{d}\approx \mathrm{Poisson}(\lambda).

Couplings to compare distributions

coupling of two random variables (or distributions) X and Y is a realisation (\hat X,\hat Y) on the same probability space with correct marginals, that is

\hat X\stackrel{d}=X,\quad \hat Y\stackrel{d}=Y.

We saw earlier in the course that we could couple G(n,p) and G(n,q) by simulating both from the same family of uniform random variables, and it’s helpful to think of this in general: ‘constructing the distributions from the same source of randomness’.

Couplings are a useful notion to digest at this point, as they embody a general trend in discrete probability theory. Wherever possible, we try to do as we can with the random objects, before starting any calculations. Think about the connectivity property of G(n,p) as discussed in the previous lecture. This can be expressed directly as a function of p in terms of a large sum, but showing it is an increasing function of p is essentially impossible by computation, whereas this is very straightforward using the coupling.

We will now review how to use couplings to compare distributions.

For a real-valued random variable X, with distribution function F_X, we always have the option to couple with a uniform U(0,1) random variable. That is, when U\sim U[0,1], we have (F_X^{-1}(U)\stackrel{d}= X, where the inverse of the distribution function is defined (in the non-obvious case of atoms) as

F_X^{-1}(u)=\inf\left\{ x\in\mathbb{R}\,:\, F(x)\ge u\right\}.

Note that when the value taken by U increases, so does the value taken by F_X^{-1}(U). This coupling can be used simultaneously on two random variables X and Y, as (F_X^{-1}(U),F_Y^{-1}(U)), to generate a coupling of X and Y.

The total variation distance between two probability measures is

d_{\mathrm{TV}}(\mu,\nu):= \sup_{A}|\mu(A)-\nu(A)|,

with supremum taken over all events in the joint support S of \mu,\nu. This is particularly clear in the case of discrete measures, as then

d_{\mathrm{TV}}(\mu,\nu)=\frac12 \sum_{x\in S} \left| \mu\left(\{x\}\right) - \nu\left(\{x\}\right) \right|.

(Think of the difference in heights between the bars, when you plot \mu,\nu simultaneously as a bar graph…)

The total variation distances records how well we can couple two distributions, if we want them to be equal as often as possible. It is therefore a bad measure of distributions with different support. For example, the distributions \delta_0 and \delta_{1/n} are distance 1 apart (the maximum) for all values of n. Similarly, the uniform distribution on [0,1] and the uniform distribution on \{0,1/n,2/n,\ldots, n-1/n, 1\} are also distance 1 apart.

When there is more overlap, the following result is useful.

Proposition: Any coupling (\hat X,\hat Y) of X\sim \mu,\,Y\sim \nu satisfies \mathbb{P}(X=Y)\le 1-d_{\mathrm{TV}}(\mu,\nu), and there exists a coupling such that equality is achieved. Continue reading

Kernels of critical graph components

This post is motivated by G(N,p), the classical Erdos-Renyi random graph, specifically its critical window, when p=p(N)=\frac{1}{N}(1+\lambda N^{-1/3}).

We start with the following observation, which makes no restriction on p. Suppose a component of G(N,p) is a tree. Then, the graph geometry of this component is that of a uniform random tree on the appropriate number of vertices. This is deliberately informal. To be formal, we’d have to say “condition on a particular subset of vertices forming a tree-component” and so on. But the formality is broadly irrelevant, because at the level of metric scaling limits, if we want to describe the structure of a tree component, it doesn’t matter whether it has \log N or \frac{1}{7}N vertices, because in both cases the tree structure is uniform. The only thing that changes is the scaling factor.

In general, when V vertices form a connected component of a graph with E edges, we define the excess to be E-V+1. So the excess is non-negative, and is zero precisely when the component is a tree. I’m reluctant to say that the excess counts the number of cycles in the component, but certainly it quantifies the amount of cyclic structure present. We will sometimes, in a mild abuse of notation, talk about excess edges. But note that for a connected component with positive excess, there is a priori no way to select which edges would be the excess edges. In a graph process, or when there is some underlying exploration of the component, there sometimes might be a canonical way to classify the excess edges, though it’s worth remarking that the risk of size-biasing errors is always extremely high in this sort of situation.

Returning to the random graph process, as so often there are big changes around criticality. In the subcritical regime, the components are small, and most of them, even the largest with high probability, are trees. In the supercritical regime, the giant component has excess \Theta(N), which is qualitatively very different.

It feels like every talk I’ve ever given has begun with an exposition of Aldous’s seminal paper [Al97] giving a distributional scaling limit of the sizes of critical components in the critical window, and a relation between the process on this time-scale and the multiplicative coalescent. And it remains relevant here, because the breadth-first exploration process can also be used to track the number of excess edges.

In a breadth-first exploration, we have a stack of vertices we are waiting to explore. We pick one and look its neighbours restricted to the rest of the graph, that is without the vertices we have already fully explored, and also without the other vertices in the stack. That’s the easiest way to handle the total component size. But we can simultaneously track how many times we would have joined to a neighbour within the stack, which leads to an excess edge, and Aldous derives a joint distributional scaling limit for the sizes of the critical components and their excesses. (Note that in this case, there is a canonical notion of excess edge, but it depends not just on the graph structure, but also on the extra randomness of the ordering within the breadth-first search.)

Roughly speaking, we consider the reflected exploration process, and its scaling limit, which is a reflected parabolically-drifting Brownian motion (though the details of this are not important at this level of exposition, except that it’s a well-behaved non-negative process that hits zero often). The component sizes are given by the widths of the excursions above zero, scaled up in a factor N^{1/3}. Then conditional on the shape of the excursion, the excess is Poisson with parameter the area under the excursion, with no rescaling. That is, a critical component has \Theta(1) excess.

So, with Aldous’s result in the background, when we ask about the metric structure of these critical components, we are really asking: “what does a uniformly-chosen connected component with fixed excess look like when the number of vertices grows?”

I’ll try to keep notation light, but let’s say T(n,k) is a uniform choice from connected graphs on n vertices with excess k.

[Note, the separation of N and n is deliberate, because in the critical window, the connected components have size n = \Theta(N^{2/3}), so I want to distinguish the two problems.]

In this post, we will mainly address the question: “what does the cycle structure of T(n,k) look like for large n?” When k=0, we have a uniform tree, and the convergence of this to the Brownian CRT is now well-known [CRT2, LeGall]. We hope for results with a similar flavour for positive excess k.

2-cores and kernels

First, we have to give a precise statement of what it means to study just the cycle structure of a connected component. From now on I will assume we are always working with a connected graph.

There are several equivalent definitions of the 2-core C(G) of a graph G:

  • When the excess is positive, there are some cycles. The 2-core is the union of all edges which form part of some cycle, and any edges which lie on a path between two edges which both form part of some cycle.
  • C(G) is the maximal induced subgraph where all degrees are at least two.
  • If you remove all the leaves from the graph, then all the leaves from the remaining graph, and continue, the 2-core is the state you arrive at where there are no leaves.

It’s very helpful to think of the overall structure of the graph as consisting of the 2-core, with pendant trees ‘hanging off’ the 2-core. That is, we can view every vertex of the 2-core as the root of a (possibly size 1) tree. This is particular clear if we remove all the edges of the 2-core from the graph. What remains is a forest, with one tree for each vertex of the 2-core.

In general, the k-core is the maximal induced subgraph where all degrees are at least k. The core is generally taken to be something rather different. For this post (and any immediate sequels) I will never refer to the k-core for k>2, and certainly not to the traditional core. So I write ‘core’ for ‘2-core’.

As you can see in the diagram, the core consists of lots of paths, and topologically, the lengths of these paths are redundant. So we will often consider instead the kernel, K(G), which is constructed by taking the core and contracting all the paths between vertices of degree greater than 2. The resulting graph has minimal degree at least three. So far we’ve made no comment about the simplicity of the original graphs, but certainly the kernel need not be simple. It will regularly have loops and multiple edges. The kernel of the graph and core in the previous diagram is therefore this:

Kernels of critical components

To recap, we can deconstruct a connected graph as follows. It has a kernel, and each edge of the kernel is a path length of some length in the core. The rest of the graph consists of trees hanging off from the core vertices.

For now, we ask about the distribution of the kernel of a T(n,K). You might notice that the case k=1 is slightly awkward, as when the core consists of a single cycle, it’s somewhat ambiguous how to define the kernel. Everything we do is easily fixable for k=1, but rather than carry separate cases, we handle the case k\ge 2.

We first observe that fixing k doesn’t confirm the number of vertices or edges in the kernel. For example, both of the following pictures could correspond to k=3:

However, with high probability the kernel is 3-regular, which suddenly makes the previous post relevant. As I said earlier, it can introduce size-biasing errors to add the excess edges one-at-a-time, but these should be constant factor errors, not scaling errors. So imagine the core of a large graph with excess k=2. For the sake of argument, assume the kernel has the dumbbell / handcuffs shape. Now add an extra edge somewhere. It’s asymptotically very unlikely that this is incident to one of the two vertices with degree three in the core. Note it would need to be incident to both to generate the right-hand picture above. Instead, the core will gain two new vertices of degree three.

Roughly equivalently, once the size of the core is fixed (and large) we have to make a uniform choice from connected graphs of this size where almost every vertex has degree 2, and \Theta(1) of the rest have degree 3 or higher. But the sum of the degrees is fixed, because the excess is fixed. If there are n vertices in the core, then there are \Theta(n) more graphs where all the vertices have degree 2 or 3, than graphs where a vertex has degree at least 4. Let’s state this formally.

Proposition: The kernel of a uniform graph with n vertices and excess k\ge 2 is, with high probability as n\rightarrow\infty, 3-regular.

This proved rather more formally as part of Theorem 7 of [JKLP], essentially as a corollary after some very comprehensive generating function setup; and in [LPW] with a more direct computation.

In the previous post, we introduced the configuration model as a method for constructing regular graphs (or any graphs with fixed degree sequence). We observe that, conditional on the event that the resulting graph is simple, it is in fact uniformly-distributed among simple graphs. When the graph is allowed to be a multigraph, this is no longer true. However, in many circumstances, as remarked in (1.1) of [JKLP], for most applications the configuration model measure on multigraphs is the most natural.

Given a 3-regular labelled multigraph H with 2(k-1) vertices and 3(k-1) edges, and K a uniform choice from the configuration model with these parameters, we have

\mathbb{P}\left( K \equiv H \right) \propto \left(2^{t(H)} \prod_{e\in E(H)} \mathrm{mult}(e)! \right)^{-1},

where t(H) is the number of loops in H, and mult(e) the multiplicity of an edge e. This might seem initially counter-intuitive, because it looks we are biasing against graphs with multiple edges, when perhaps our intuition is that because there are more ways to form a set of multiple edges we should bias in favour of it.

I think it’s most helpful to look at a diagram of a multigraph as shown, and ask how to assign stubs to edges. At a vertex with degree three, all stub assignments are different, that is 3!=6 possibilities. At the multiple edge, however, we care which stubs match with which stubs, but we don’t care about the order within the multi-edge. Alternatively, there are three choices of how to divide each vertex’s stubs into (2 for the multi-edge, 1 for the rest), and then two choices for how to match up the multi-edge stubs, ie 18 in total = 36/2, and a discount factor of 2.

We mention this because in fact K(T(n,k)) converges in distribution to this uniform configuration model. Once you know that K(T(n,k)) is with high probability 3-regular, then again it’s probably easiest to think about the core, indeed you might as well condition on its total size and number of degree 3 vertices. It’s then not hard to convince yourself that a uniform choice induces a uniform choice of kernel. Again, let’s state that as a proposition.

Proposition: For any H a 3-regular labelled multigraph H with 2(k-1) vertices and 3(k-1) edges as before,

\lim_{n\rightarrow\infty}\mathbb{P}\left( K(T(n,k)) \equiv H \right) \propto \left(2^{t(H)} \prod_{e\in E(H)} \mathrm{mult}(e)! \right)^{-1}.

As we said before, the kernel describes the topology of the core. To reconstruct the graph, we need to know the lengths in the core, and then how to glue pendant trees onto the core. But this final stage depends on k only through the total length of paths in the core. Given that information, it’s a combinatorial problem, and while I’m not claiming it’s easy, it’s essentially the same as for the case with k=1, and is worth treating separately.

It is worth clarifying a couple of things first though. Even the outline of methods above relies on the fact that the size of the core diverges as n grows. Again, the heuristic is that up to size-biasing errors, T(n,k) looks like a uniform tree with some uniformly-chosen extra edges. But distances in T(n,k) scale like n^{1/2} (and thus in critical components of G(N,p) scale like N^{1/3}). And the core will be roughly the set of edges on paths between the uniformly-chosen pairs of vertices, and so will also have length \Theta(n^{1/2}).

Once you have conditioned on the kernel structure, and the (large) number of internal vertices on paths in the core (ie the length of the core), it is natural that the assignment of the degree-2 vertices to core paths / kernel edges is uniform. A consequence of this is that if you record (Y_1,\ldots,Y_m) the lengths of paths in the core, where m=3(k-1), then

\frac{(Y_1,\ldots,Y_m)}{\sum Y_i} \stackrel{d}\rightarrow \mathrm{Dirichlet}(1,1,\ldots,1).

This is stated formally as Corollary 7 b) of [ABG09]. It’s worth noting that this confirms that the lengths of core paths are bounded in probability away from zero after the appropriate rescaling. In seeking a metric scaling limit, this is convenient as it means there’s so danger that two of the degree-3 vertices end up in ‘the same place’ in the scaling limit object.

To recap, the only missing ingredients now to give a complete limiting metric description of T(n,k) are 1) a distributional limit of the total core length; 2) some appropriate description of set of pendant trees conditional on the size of the pendant forest. [ABG09] show the first of these. As remarked before, all the content of the second of these is encoded in the unicyclic k=1 case, which I have written about before, albeit slightly sketchily, here. (Note that in that post we get around size-biasing by counting a slightly different object, namely unicyclic graphs with an identified cyclic edge.)

However, [ABG09] also propose an alternative construction, which you can think of as glueing CRTs directly onto the stubs of the kernel (with the same distribution as before). The proof that this construction works isn’t as painful as one might fear, and allows a lot of the other metric distributional results to be read off as corollaries.

References

[ABG09] – Addario-Berry, Broutin, Goldschmidt – Critical random graphs: limiting constructions and distributional properties

[CRT2] – Aldous – The continuum random tree: II

[Al97] – Aldous – Brownian excursions, critical random graphs and the multiplicative coalescent

[JKLP] – Janson, Knuth, Luczak, Pittel – The birth of the giant component

[LeGall] – Le Gall – Random trees and applications

[LPW] – Luczak, Pittel, Wierman – The structure of a random graph at the point of the phase transition

 

Random walks conditioned to stay positive

In this post, I’m going to discuss some of the literature concerning the question of conditioning a simple random walk to lie above a line with fixed gradient. A special case of this situation is conditioning to stay non-negative. Some notation first. Let (S_n)_{n\ge 0} be a random walk with IID increments, with distribution X. Take \mu to be the expectation of these increments, and we’ll assume that the variance \sigma^2 is finite, though at times we may need to enforce slightly stronger regularity conditions.

(Although simple symmetric random walk is a good example for asymptotic heuristics, in general we also assume that if the increments are discrete they don’t have parity-based support, or any other arithmetic property that prevents local limit theorems holding.)

We will investigate the probability that S_n\ge 0 for n=0,1,…,N, particularly for large N. For ease of notation we write T=\inf\{n\ge 0\,:\, S_n<0\} for the hitting time of the negative half-plane. Thus we are interested in S_n conditioned on T>N, or T=N, mindful that these might not be the same. We will also discuss briefly to what extent we can condition on T=\infty.

In the first paragraph, I said that this is a special case of conditioning SRW to lie above a line with fixed gradient. Fortunately, all the content of the general case is contained in the special case. We can repose the question of S_n conditioned to stay above n\alpha until step N by the question of S_n-n\alpha (which, naturally, has drift \mu-\alpha) conditioned to stay non-negative until step N, by a direct coupling.

Applications

Simple random walk is a perfectly interesting object to study in its own right, and this is a perfectly natural question to ask about it. But lots of probabilistic models can be studied via naturally embedded SRWs, and it’s worth pointing out a couple of applications to other probabilistic settings (one of which is the reason I was investigating this literature).

In many circumstances, we can desribe random trees and random graphs by an embedded random walk, such as an exploration process, as described in several posts during my PhD, such as here and here. The exploration process of a Galton-Watson branching tree is a particularly good example, since the exploration process really is simple random walk, unlike in, for example, the Erdos-Renyi random graph G(N,p), where the increments are only approximately IID. In this setting, the increments are given by the offspring distribution minus one, and the hitting time of -1 is the total population size of the branching process. So if the expectation of the offspring distribution is at most 1, then the event that the size of the tree is large is an atypical event, corresponding to delayed extinction. Whereas if the expectation is greater than one, then it is an event with limiting positive probability. Indeed, with positive probability the exploration process never hits -1, corresponding to survival of the branching tree. There are plenty of interesting questions about the structure of a branching process tree conditional on having atypically large size, including the spine decomposition of Kesten [KS], but the methods described in this post can be used to quantify the probability, or at least the scale of the probability of this atypical event.

In my current research, I’m studying a random walk embedded in a construction of the infinite-volume DGFF pinned at zero, as introduced by Biskup and Louidor [BL]. The random walk controls the gross behaviour of the field on annuli with dyadically-growing radii. Anyway, in this setting the random walk has Gaussian increments. (In fact, there is a complication because the increments aren’t exactly IID, but that’s definitely not a problem at this level of exposition.) The overall field is decomposed as a sum of the random walk, plus independent DGFFs with Dirichlet boundary conditions on each of the annuli, plus asymptotically negligible corrections from a ‘binding field’. Conditioning that this pinned field be non-negative up to the Kth annulus corresponds to conditioning the random walk to stay above the magnitude of the minimum of each successive annular DGFF. (These minima are random, but tightly concentrated around their expectations.)

Conditioning on \{T > N\}

When we condition on \{T>N\}, obviously the resulting distribution (of the process) is a mixture of the distributions we obtain by conditioning on each of \{T=N+1\}, \{T=N+2\},\ldots. Shortly, we’ll condition on \{T=N\} itself, but first it’s worth establishing how to relate the two options. That is, conditional on \{T>N\}, what is the distribution of T?

Firstly, when \mu>0, this event always has positive probability, since \mathbb{P}(T=\infty)>0. So as N\rightarrow\infty, the distribution of the process conditional on \{T>N\} converges to the distribution of the process conditional on survival. So we’ll ignore this for now.

In the case \mu\le 0, everything is encapsulated in the tail of the probabilities \mathbb{P}(T=N), and these tails are qualitatively different in the cases \mu=0 and \mu<0.

When \mu=0, then \mathbb{P}(T=N) decays polynomially in N. In the special case where S_n is simple symmetric random walk (and N has the correct parity), we can check this just by an application of Stirling’s formula to count paths with this property. By contrast, when \mu<0, even demanding S_N=-1 is a large deviations event in the sense of Cramer’s theorem, and so the probability decays exponentially with N. Mogulskii’s theorem gives a large deviation principle for random walks to lie above a line defined on the scale N. The crucial fact here is that the probabilistic cost of staying positive until N has the same exponent as the probabilistic cost of being positive at N. Heuristically, we think of spreading the non-expected behaviour of the increments uniformly through the process, at only polynomial cost once we’ve specified the multiset of values taken by the increments. So, when \mu<0, we have

\mathbb{P}(T\ge(1+\epsilon)N) \ll \mathbb{P}(T= N).

Therefore, conditioning on \{T\ge N\} in fact concentrates T on N+o(N). Whereas by contrast, when \mu=0, conditioning on \{T\ge N\} gives a nontrivial limit in distribution for T/N, supported on [1,\infty).

A related problem is the value taken by S_N, conditional on {T>N}. It’s a related problem because the event {T>N} depends only on the process up to time N, and so given the value of S_N, even with the conditioning, after time N, the process is just an unconditioned RW. This is a classic application of the Markov property, beloved in several guises by undergraduate probability exam designers.

Anyway, Iglehart [Ig2] shows an invariance principle for S_N | T>N when \mu<0, without scaling. That is S_N=\Theta(1), though the limiting distribution depends on the increment distribution in a sense that is best described through Laplace transforms. If we start a RW with negative drift from height O(1), then it hits zero in time O(1), so in fact this shows that conditonal on \{T\ge N\}, we have T= N +O(1) with high probability. When \mu=0, we have fluctuations on a scale \sqrt{N}, as shown earlier by Iglehart [Ig1]. Again, thinking about the central limit theorem, this fits the asymptotic description of T conditioned on T>N.

Conditioning on T=N

In the case \mu=0, conditioning on T=N gives

\left[\frac{1}{\sqrt{N}}S(\lfloor Nt\rfloor ) ,t\in[0,1] \right] \Rightarrow W^+(t), (*)

where W^+ is a standard Brownian excursion on [0,1]. This is shown roughly simultaneously in [Ka] and [DIM]. This is similar to Donsker’s theorem for the unconditioned random walk, which converges after rescaling to Brownian motion in this sense, or Brownian bridge if you condition on S_N=0. Skorohod’s proof for Brownian bridge [Sk] approximates the event \{S_N=0\} by \{S_N\in[-\epsilon \sqrt{N},+\epsilon \sqrt{N}]\}, since the probability of this event is bounded away from zero. Similarly, but with more technicalities, a proof of convergence conditional on T=N can approximate by \{S_m\ge 0, m\in[\delta N,(1-\delta)N], S_N\in [-\epsilon \sqrt{N},+\epsilon\sqrt{N}]\}. The technicalities here emerge since T, the first return time to zero, is not continuous as a function of continuous functions. (Imagine a sequence of processes f^N for which f^N(x)\ge 0 on [0,1] and f^N(\frac12)=\frac{1}{N}.)

Once you condition on T=N, the mean \mu doesn’t really matter for this scaling limit. That is, so long as variance is finite, for any \mu\in\mathbb{R}, the same result (*) holds, although a different proof is in general necessary. See [BD] and references for details. However, this is particularly clear in the case where the increments are Gaussian. In this setting, we don’t actually need to take a scaling limit. The distribution of Gaussian *random walk bridge* doesn’t depend on the mean of the increments. This is related to the fact that a linear transformation of a Gaussian is Gaussian, and can be seen by examining the joint density function directly.

Conditioning on T=\infty

When \mu>0, the event \{T=\infty\} occurs with positive probability, so it is well-defined to condition on it. When \mu\le 0, this is not the case, and so we have to be more careful.

First, an observation. Just for clarity, let’s take \mu<0, and condition on \{T>N\}, and look at the distribution of S_{\epsilon N}, where \epsilon>0 is small. This is approximately given by

\frac{S_{\epsilon N}}{\sqrt{N}}\stackrel{d}{\approx}W^+(\epsilon).

Now take \epsilon\rightarrow\infty and consider the RHS. If instead of the Brownian excursion W^+, we instead had Brownian motion, we could specify the distribution exactly. But in fact, we can construct Brownian excursion as the solution to an SDE:

\mathrm{d}W^+(t) = \left[\frac{1}{W^+(t)} - \frac{W^+(t)}{1-t}\right] \mathrm{d}t + \mathrm{d}B(t),\quad t\in(0,1) (**)

for B a standard Brownian motion. I might return in the next post to why this is valid. For now, note that the first drift term pushes the excursion away from zero, while the second term brings it back to zero as t\rightarrow 1.

From this, the second drift term is essentially negligible if we care about scaling W^+(\epsilon) as \epsilon\rightarrow 0, and we can say that W^+(\epsilon)=\Theta(\sqrt{\epsilon}).

So, returning to the random walk, we have

\frac{S_{\epsilon N}}{\sqrt{\epsilon N}}\stackrel{d}{\approx} \frac{W^+(\epsilon)}{\sqrt{\epsilon}} = \Theta(1).

At a heuristic level, it’s tempting to try ‘taking N\rightarrow\infty while fixing \epsilon N‘, to conclude that there is a well-defined scaling limit for the RW conditioned to stay positive forever. But we came up with this estimate by taking N\rightarrow\infty and then \epsilon\rightarrow 0 in that order. So while the heuristic might be convincing, this is not the outline of a valid argument in any way. However, the SDE representation of W^+ in the \epsilon\rightarrow 0 regime is useful. If we drop the second drift term in (**), we define the three-dimensional Bessel process, which (again, possibly the subject of a new post) is the correct scaling limit we should be aiming for.

Finally, it’s worth observing that the limit \{T=\infty\}=\lim_{N\rightarrow\infty} \{T>N\} is a monotone limit, and so further tools are available. In particular, if we know that the trajectories of the random walk satisfy the FKG property, then we can define this limit directly. It feels intuitively clear that random walks should satisfy the FKG inequality (in the sense that if a RW is large somewhere, it’s more likely to be large somewhere else). You can do a covariance calculation easily, but a standard way to show the FKG inequality applies is by verifying the FKG lattice condition, and unless I’m missing something, this is clear (though a bit annoying to check) when the increments are Gaussian, but not in general. Even so, defining this monotone limit does not tell you that it is non-degenerate (ie almost-surely finite), for which some separate estimates would be required.

A final remark: in a recent post, I talked about the Skorohod embedding, as a way to construct any centered random walk where the increments have finite variance as a stopped Brownian motion. One approach to conditioning a random walk to lie above some discrete function is to condition the corresponding Brownian motion to lie above some continuous extension of that function. This is a slightly stronger conditioning, and so any approach of this kind must quantify how much stronger. In Section 4 of [BL], the authors do this for the random walk associated with the DGFF conditioned to lie above a polylogarithmic curve.

References

[BD] – Bertoin, Doney – 1994 – On conditioning a random walk to stay nonnegative

[BL] – Biskup, Louidor – 2016 – Full extremal process, cluster law and freezing for two-dimensional discrete Gaussian free field

[DIM] – Durrett, Iglehart, Miller – 1977 – Weak convergence to Brownian meander and Brownian excursion

[Ig1] – Iglehart – 1974 – Functional central limit theorems for random walks conditioned to stay positive

[Ig2] – Iglehart – 1974 – Random walks with negative drift conditioned to stay positive

[Ka] – Kaigh – 1976 – An invariance principle for random walk conditioned by a late return to zero

[KS] – Kesten, Stigum – 1966 – A limit theorem for multidimensional Galton-Watson processes

[Sk] – Skorohod – 1955 – Limit theorems for stochastic processes with independent increments

Parking on a ring, linear hashing

I’ve spent most of my doctorate trying to analyse how adding destructive dynamics affects the behaviour of a particular random growth process, the classical random graph. In this post I’m going to talk about another random growth process, which is slightly less natural, but for which one can show some similar qualitative properties.

The model, and the additive coalescent

Consider m places arranged in a circle, and for consistency of analogy we think of these as parking spaces. Some number n of cars will arrive one at a time. Each car will arrive at a space chosen uniformly at random. If it is empty they will park in it, otherwise they will look clockwise until they find an empty space, and park there. For now we are only interested in growth, so we assume cars never leave. We are interested in the sizes of blocks of consecutively parked cars.

The reason to consider this slightly unnatural statement is its equivalence to the problem of hashing with linear probing, apparently a key topic in computer science, which I won’t pretend that I know anything about. In any case, it’s a nice model, and it seems reasonable that it would have a basis in more realistic search algorithms.

So, how does the sequence of sizes of blocks of consecutively parked cars grow? Well, given the sequence of block sizes, it is reasonably easy to convince yourself that the order of the blocks around the circle is uniformly random, and the number of empty spaces between adjacent blocks is also uniformly random.

Assume for now that there are at least three blocks. A block of size x can merge with a block of size y with the arrival of the next car only if the blocks are adjacent, with exactly one empty space between them. The chance of this is uniform among all pairs of blocks. Now suppose this is the case, and that the block of size y lies clockwise from the block of size x. Then they will merge precisely if the next car arrives at any of the x occupied spaces in that block, or at the empty space between the pair of blocks. This has probability \frac{x+1}{m}. There’s also the opposite ordering to consider, where the block of size x lies clockwise from the other. The total probability of this merge \{x,y\}\mapsto \{x+y+1\} is therefore proportional to (x+y+2).

So the process of block sizes looks a bit like the additive coalescent, at least for large blocks. This is in contrast to the random graph process, where the sequence of component sizes behaves exactly like a multiplicative coalescent, where blocks merge at a rate proportional to the product of their sizes.

Asymptotics

As in the random graph process, it’s interesting to ask roughly how large the largest block will be in such a configuration. Pittel [3] considers the case where the number of empty places \ell = m-n \approx \beta m, for some \beta\in (0,1).

A less interesting model would be to choose the positions of the n cars uniformly at random. But then the size of a block is roughly geometric with parameter \beta, and there are \Theta(m) blocks with high probability. Relatively straightforward calculations in extreme value theory suggest that the largest block is likely to have size on the order of \log m in this setting.

Of course, the actual model is slightly more complicated, because the size of a block is self-reinforcing, since larger blocks are more likely to grow than smaller blocks. However, we can still get somewhere with naïve estimates. Let’s label the places clockwise. Then in order for there to be a block starting at 0 and stretching beyond \alpha \log m, a necessary condition is that at least \alpha \log m cars arrive at those places. The number of cars which arrive at those places is binomial, since there are n cars, and each arrives at a place chosen uniformly, and independently of the other cars. So this event corresponds to

\mathrm{Bin}(n,\frac{\alpha \log m}{m}) \ge \alpha \log m.

Then, since n\approx (1-\beta)n, this event corresponds approximately to

\mathrm{Po}((1-\beta)\alpha \log m) \ge \alpha \log m.

The probability that a Poisson RV is at least a constant multiple larger than its mean decays exponentially with the mean, hence in this case the probability is asymptotically some negative power of m, depending on the value of \alpha. But there are O(m) possible places for such a block to start, so whether we can apply a union bound usefully or not depends on whether the power of m is strictly less than -1.

Since all of this depends on \alpha, it is reasonable that everything is fine, and the largest block does have size at least \alpha \log m when \alpha is small, and very unlikely when \alpha is large. This heuristic argument fits with Pittel’s theorem. Indeed, his result shows much stronger concentration: that the fluctuations of the size of the largest block are O(1).

Critical regime and empirical processes

The following is a paraphrase of the introduction and some methods from [2].

Obviously, once m=m cars have arrived, there’s no room for manoeuvre and definitely all the places are taken in one giant block. But it’s not obvious in general what scaling for the number of gaps will give rise to giant blocks of \Theta(m) cars.

As for the random graph, we can find a process similar to the exploration process of a (random) graph which encodes much of the information we care about. Let Y_k be the number of cars which arrive at place k. So the sum of the Y_ks will be n, the total number of cars. Now consider the process

C_0=0, \ldots, C_{k+1}=C_k + Y_{k+1}-1.

A block has the property that the number of arrivals within that set of places is equal to the number of places. So every time this *empirical process* C drops below its previous running minimum, this indicates the end of a block. To make this equivalence precise, we need to be a bit careful about where we start counting. It works exactly if we start at the beginning of a block. If not, it might introduce some unwanted divisions within the first block.

What we have is a process that looks roughly like a random walk that is constrained to pass through the point (m,n-m), which is equal to (m,-l). Even if we aren’t totally precise about how this is like a random walk, we would expect to see Brownian fluctuations after rescaling. Indeed, we might expect to see a Brownian bridge added to a deterministic linear function with negative gradient. But this is only meaningful if the random part is at least as large as the deterministic part, and since the fluctuations have order \sqrt{m}, if l is much larger than this, the rescaled empirical process is essentially deterministic, so we won’t see any macroscopic excursions above the minimum.

If l is substantially smaller than \sqrt{m}, then there is no real difference between (m,-l) and (m,0), and what we see is just a Brownian bridge. At this point, where we choose to start the process is actually important. If we were to start it at the minimum of the Brownian bridge instead, we would have seen a Brownian excursion, which corresponds to one block occupying (almost) all of the places.

Unsurprisingly, the story is completed by considering \ell=\Theta(\sqrt{m}), where the rescaled empirical process looks like a slanted Brownian bridge, that is Brownian motion conditioned to pass through $(1,-\frac{\ell}{\sqrt{m})$. There isn’t an obvious fix to the question of where to start the process, but it turns out that the correct way is now adding a Brownian excursion onto the deterministic linear function with gradient - \frac{\ell}{\sqrt{m}}. It’s now reasonable that the excursions above the minimum should macroscopic.

This scaling limit works dynamically as well, where the same Brownian excursion is used for different gradients of the deterministic line, corresponding to \ell moving through the critical window m-\Theta(\sqrt{m}). Finally, a direction to Bertoin’s recent paper [1] for the model with an additional destructive property. Analogous to the forest fire, blocks of cars are removed at a rate proportional to their size (as a result, naturally, of ‘Molotov cocktails’…). Similar effects of self-organised criticality are seen when the rate of bombs is scaled appropriately.

References

[1] – Bertoin – Burning cars in a parking lot (paper / slides)

[2] – Chassaing + Louchard – Phase transition for parking blocks, Brownian excursion and coalescence (arXiv)

[3] – Pittel – Linear probing: the probable largest search time grows logarithmically with the number of records

Critical Components in Erdos-Renyi

In various previous posts, I’ve talked about the phase transition in the Erdos-Renyi random graph process. Recall the definition of the process. Here we will use the Gilbert model G(n,p), where we have n vertices, and between any pair of vertices we add an edge, independently of other pairs with probability p. We are interested in the sparse scaling, where the typical vertex has degree O(1) in n, and so p=c/n for constant c>0, and we assume throughout that n is large. We could alternatively have considered the alternative Erdos-Renyi model where we choose uniformly at random from the set of graphs with n vertices and some fixed number of edges. Almost all the results present work equally well in this setting.

As proved by Erdos and Renyi, the typical component structure of such a graph changes noticeably around the threshold c=1. Below this, in the subcritical regime, all the components are small, meaning of size at most order O(log n). Above this, in the supercritical regime, there is a single giant component on some non-zero proportion of the vertices. The rest of the graph looks subcritical. The case c=1 exhibits a phase transition between these qualitatively different behaviours. They proved that here, the largest component is with high probability O(n^2/3). It seems that they thought this result held whenever c=1-o(1), but it turns out that this is not the case. In this post, I will discuss some aspects of behaviour around criticality, and the tools needed to treat them.

The first question to address is this: how many components of size n^{2/3} are there? It might be plausible that there is a single such component, like for the subsequent giant component. It might also be plausible that there are n^1/3 such components, so O(n) vertices are on such critical components. As then it is clear how we transition out of criticality into supercriticality – all the vertices on critical components coalesce to form the new giant component.

In fact neither of these are correct. The answer is that for all integers k>0, with high probability the k-th largest component is on a size scale of n^2/3. This is potentially a confusing statement. It looks like there are infinitely many such components, but of course for any particular value of n, this cannot be the case. We should think of there being w(1) components, but o(n^b) for any b>0.

The easiest way to see this is by a duality argument, as we have discussed previously for the supercritical phase. If we remove a component of size O(n^2/3), then what remains is a random graph with n-O(n^2/3) vertices, and edge probability the same as originally. It might make sense to rewrite this probability 1/n as

\frac{1}{n-O(n^{2/3})}\cdot \frac{n-O(n^{2/3})}{n}=\frac{1-O(n^{-1/3})}{n-O(n^{2/3})}.

The approximation in the final numerator is basically the same as

1-o\left(n-O(n^{2/3})\right).

Although we have no concrete reasoning, it seems at least plausible that this should look similar in structure to G(n,1/n). In particular, there should be another component of size

O\left([n-O(n^{2/3})]^{2/3}\right)=O(n^{2/3}).

In fact, the formal proof of this proceeds by an identical argument, only using the exploration process. Because I’ve described this several times before, I’ll be brief. We track how far we have gone through each component in a depth-first walk. In both the supercritical and subcritical cases, when we scale correctly we get a random path which is basically deterministic in the limit (in n). For exactly the same reasons as visible CLT fluctuations for partial sums of RVs with expectation zero, we start seeing interesting effects at criticality.

The important question is the order of rescaling to choose. At each stage of the exploration process, the number of vertices added to the stack is binomial. We want to distinguish between components of size O(n^{2/3}) so we should look at the exploration process at time sn^{2/3}. The drift of the exploration process is given by the expectation of a binomial random variable minus one (since we remove the current vertex from the stack as we finish exploring it). This is given by

\mathbb{E}=\left[n-sn^{2/3}\right]\cdot \frac{1}{n}-1=-sn^{-1/3}.

Note that this is the drift in one time-step. The drift in n^{2/3} time-steps will accordingly by sn^{1/3}. So, if we rescale time by n^{2/3} and space by n^{1/3}, we should get a nice stochastic process. Specifically, if Z is the exploration process, then we obtain:

\frac{1}{n^{1/3}}Z^{(n)}_{sn^{2/3}} \rightarrow_d W_s,

where W is a Brownian motion with inhomogeneous drift -s at time s. The net effect of such a drift at a fixed positive time is given by integrating up to that time, and hence we might say the process has quadratic drift, or is parabolic.

We should remark that our binomial expectation is not entirely correct. We have discounted those sn^{2/3} vertices that have already been explored, but we have not accounted for the vertices currently in the stack. We should also be avoiding considering these. However, we now have a heuristic for the approximate number of these. The number of vertices in the stack should be O(n^{1/3}) at all times, and so in particular will always be an order of magnitude smaller than the number of vertices already considered. Therefore, they won’t affect this drift term, though this must be accounted for in any formal proof of convergence. On the subject of which, the mode of convergence is, unsurprisingly, weak convergence uniformly on compact sets. That is, for any fixed S, the convergence holds weakly on the random functions up to time sn^{2/3}.

Note that this process will tend to minus infinity almost surely. Component sizes are given by excursions above the running minimum. The process given by the height of the original process above the running minimum is called reflected. Essentially, we construct the reflected process by having the same generator when the current value is positive, and forcing the process up when it is at zero. There are various ways to construct this more formally, including as the scaling limit of some simple random walks conditioned never to stay non-negative.

The cute part of the result is that it holds equally well in a so-called critical window either side of the critical probability 1/n. When the probability is \frac{1+tn^{-1/3}}{n}, for any t\in \mathbb{R}, the same argument holds. Now the drift at time s is t-s, though everything else still holds.

This result was established by Aldous in [1], and gives a mechanism for calculating distributions of component sizes and so on through this critical window.

In particular, we are now in a position to answer the original question regarding how many such components there were. The key idea is that because whenever we exhaust a component in the exploration process, we choose a new vertex uniformly at random, we are effectively choosing a component according to the size-biased distribution. Roughly speaking, the largest components will show up near the beginning. Note that a critical O(n^{2/3}) component will not necessarily be exactly the first component in the exploration process, but the components that are explored before this will take up sufficiently few vertices that they won’t show up in the scaling of the limit.

In any case, the reflected Brownian motion ‘goes on forever’, and the drift is eventually very negative, so there cannot be infinitely wide excursions, hence there are infinitely many such critical components.

If we care about the number of cycles, we can treat this also via the exploration process. Note that in any depth-first search we are necessarily only interested in a spanning tree of the host graph. Anyway, when we are exploring a vertex, there could be extra edges to other vertices in the stack, but not to vertices we’ve already finished exploring (otherwise the edge would have been exposed then). So the expected number of excess edges into a vertex is proportional to the height of the exploration process at that vertex. So the overall expected number of excess edges, conditional on the exploration process is the area under the curve. This carries over perfectly well into the stochastic process limit. It is then a calculation to verify that the area under the curve is almost surely infinite, and thus that we expect there to be infinitely many cycles in a critical random graph.

REFERENCES

[1] Aldous D. – Brownian excursions, critical random graphs and the multiplicative coalescent

Long Paths and Expanders

I’m in Birmingham this week for the LMS-EPSRC summer school on Random Graphs, Geometry and Asymptotic Structure. The event consists of three five-hour mini-courses, a plenary lecture, leaving plenty of time for problem sheet and discussion. I thought it would be worth trying to say a couple of interesting things each day – I do not know whether this will succeed, but I might as well try.

Today, a few thoughts on the first two lectures of Michael Krivelevich’s course on Long Paths and Hamiltonicity in Random Graphs. The aim is to develop tools to investigate the threshold for the presence of a Hamiltonian cycle in G(n,p). In this first part of the course, we were mainly thinking about long paths.

One tool we used a lot was the Depth-First Search algorithm. This is very similar to the exploration process I’ve talked about before. Essentially, here we consider trying to explore the graph in a depth-first way, but instead of viewing all the edges incident to a vertex we have just arrived at, we only look to see whether there is an edge out of the new vertex. If there is, we explore it, then come back eventually to look for more. It really comes down to a difference in the information we are storing. In this DFS, we store the vertices which we haven’t finished exploring, which is the set of vertices on the explored path between the root and the current vertex. So the size of this set evolves like the contour process. In particular, we can read off the sizes of paths from this description. These dynamics are useful in particular because we know there are no edges between the set of vertices we have finished exploring, and the ones we have yet to explore. The stack of ‘processing’ vertices must glue everything else together.

We can translate one of the arguments back into the language for the old exploration process. Recall the increments of the exploration process are \mathrm{Bin}(\alpha n,\frac{c}{n}) -1 once we have explored \alpha n vertices. We don’t need to worry about the -1 bit for now. Observe that because we are exploring in a depth-first way, if a subsequence of the Binomial variables of length k are all positive, this corresponds to a path of length (k-1).

So to prove, for example, that the longest path in a subcritical random graph is O(log n), it suffices to prove that there are O(log n) consecutive positive entries in the sequence of n binomial entries. Since the distribution changes continuously, it is convenient to prove that there are O(log n) consecutive positive entries in the first \epsilon n binomial entries. The probability that any of these entries is positive is bounded below by some p, so it suffices to consider instead a sequence of Bernoulli RVs with parameter p. So if we never have clog n consecutive, this gives control of the sequence of geometric random variables corresponding to the gaps between 0s in the sequence. Precisely, these are Geom(q), and we must have \frac{\epsilon n}{c\log n} of them independently being less than clog n. We have to chase a few constants, and use the fact that if f(n)\rightarrow\infty, \frac{g(n)}{f(n)}\rightarrow\infty, then

(1-\frac{1}{f(n)})^{g(n)}\rightarrow 0,

by comparison with the standard asymptotic result for $e^{-x}$. In any case, we get that this probability tends to 0 if we choose c small enough, and so with high probability there is a path of length clog n.

This is interesting, because we knew already that the largest component in a subcritical random graph had size O(log n). But we also knew that all the components were trees, or ‘almost trees’, and were uniformly chosen from the set of trees (or trees + an edge or two) with appropriate size. And the largest path in a UST on n vertices is O(n^{1/2}) with high probability. So we learn that there are enough components of size \geq c\log n that it is actually very probable that one of them will have the unlikely property of being much more path-like than a typical tree.

Krivelevich also showed a pleasant elementary proof of the result that a supercritical random graph has a path of length O(n), using a similar idea.

The other definition of major interest was an expander graph. Often when doing calculations about neighbourhoods of sets of vertices, we run into the problem that the neighbourhoods may overlap, and so we cannot get the total outer neighbourhood (or outer boundary) just by summing over the individual neighbourhood sizes. In an expander graph, we demand that all small sets of vertices have neighbourhood at least as large as some constant multiple of the set size, essentially giving us a bound on the above problem. Concretely, G is a (k,\alpha)-expander is for any set of vertices |U|\leq k, |N(U)|\geq \alpha |U|.

There’s a very nice argument using Posa’s lemma, where we consider all the possible ways to rearrange the vertices in some longest path into a different longest path, and then focus on the endpoints of all these paths. With this so-called rotation-extension technique, we can show that a (k,2)-expander has a path of length at least 3k-1.

There are structural similarities between expander graphs and regular graphs, so it seems natural that there will be some interesting spectral properties. I don’t know much about this, but perhaps it will come up later in the week. But, returning to the random graph long path problem, it now suffices to show subcritical G(n,p) is a (clog n,2)-expander for some c. Expander properties are in some sense the opposite of clustering properties, and independence of a RG inhibit most clustering properties (as discussed in much greater detail in some of the posts about network models). Unfortunately, this doesn’t actually work, as in a subcritical graph, the typical expansion coefficient, even of a small set will be c, for G(n,c/n), which is not large enough. However, if you chose the constants carefully, such an argument should work for c>2, so long as you chose k=an, with a small enough that the probability of a vertex elsewhere in the graph being joined to (at least) two of the k vertices in the set, was small compared with (c-2).

REFERENCES

The course notes are not available, though chapter 3 from these 2010 notes by the same lecturer are related and interesting.