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.

Recent Progress and Gromov-Hausdorff Convergence

For the past few weeks I’ve been working on the problem of Cycle-Induced Forest Fires, which I’ve referred to in passing in some recent posts. The aim has been to find a non-contrived process which exhibits self-organised criticality, that is, where the process displays critical characteristics (scaling laws, multiple components at the largest order of magnitude) forever. Note that this is in contrast to the conventional Erdos-Renyi graph process, which is only critical at a single time n/2.

The conjecture is that the largest component in equilibrium typically has size on a scale of n^2/3. An argument based on the equilibrium proportion of isolated vertices gives an upper bound on this exponent. The working argument I have for the lower bound at the moment can comfortably fit on the back of a napkin, with perhaps some context provided verbally. Of course, the current full text is very much larger than that, mainly because the napkin would feature assertions like “event A happens at time $O(n^\beta)$“; whereas the more formal argument has to go like:

“With high probability as $n\rightarrow\infty$, event A happens between times $n^{\beta-\epsilon},n^{\beta+\epsilon},$ for any suitably small $\epsilon>0.$ Furthermore, the probability that A happens after this upper threshold decays exponentially with n for fixed $\epsilon$, and the probability that A happens before the lower threshold is at most $n^{-\epsilon}$. Finally, this is under the implicit assumption that there will be no fragmentations before event A, and this holds with probability $1-o(1)$ etc.”

It’s got to the point where I’ve exhausted the canonical set of symbols for small quantities: $\epsilon,\delta,(\eta ?)$.

This has been a very long way of setting up what was going to be my main point, which is that at many points during undergraduate mathematics, colleagues (and occasionally to be honest, probably myself too) have complained that they “don’t want to have anything to do with analysis. They just want to focus on algebra / number theory / statistics / fluids…” Anyway, the point of this ramble was that I think I’ve realised that it is very hard to think about any sort of open problem without engaging with the sort of ideas that a few years ago I would have thought of (and possibly dismissed) as ‘analysis’.

Much of my working on this problem has been rather from first principles, so haven’t been thinking much about any neat less elementary theory recently.

Ok, so on with the actual post now.

Last month I talked about local limits of graphs, which describe convergence in distribution of (local) neighbourhood structure about a ‘typical’ vertex. This is the correct context in which to make claims like “components of $G(n,\frac{\lambda}{n})$ look like Galton-Watson trees with offspring distribution $\text{Po}(\lambda)$“.

Even from this example, we can see a couple of drawbacks and omissions from this limiting picture. In the sub-critical regime, this G-W tree will be almost surely finite, but the number of vertices in the graph is going to infinity. More concretely, the limit description only tells us about a single component. If we wanted to know about a second component, in this case, it would be roughly independent of the size of the first component, and with the same distribution, but if we wanted to know about all components, it would get much more complicated.

Similarly, this local limit description isn’t particularly satisfactory in the supercritical regime. When the component in question is finite, this description is correct, but with high probability we have a giant component, and so the ‘typical’ vertex is with some positive probability in the giant component. This is reflected by the fact that the G-W tree with supercritical offspring distribution is infinite with some positive probability. However, the giant component does not look like a $\text{Po}(\lambda)$ G-W tree. As we exhaust O(n) vertices, the offspring distribution decreases, in expectation at least. In fact, without the assumption that the giant component is with high probability unique (so $\frac{L_1}{n}=1-\mathbb{P}(|C(v)|<\infty$), we can’t even deduce the expected size of the giant component from the local limit result.

This is all unsurprising. By definition a local limit describes the structure near some vertex. How near? Well, finitely near. It can be arbitrarily large, but still finite, so in particular, the change in the offspring distribution after O(n) vertices as mentioned above will not be covered.

So, if we want to learn more about the global structure of a large discrete object, we need to consider a different type of limit. In particular, the limit will not necessarily be a graph. Rather than try to define a priori a ‘continuum’ version of a graph, it is sensible to generalise from the idea that a graph is a discrete object and instead consider it as a metric space.

In this article, I don’t want to spend much time at all thinking about how to encode a finite graph as a metric space. We have a natural notion of graph distance between vertices, and it is not hard to extend this to points on edges. Alternatively, for sparse graphs, we have an encoding through various functions, which live in some (metric) function space.

However, in general, the graph will be a metric object itself, rather than necessarily a subset of a global metric space. We will be interested in convergence, so we need a suitable style of convergence of different metric spaces.

The natural candidate for this is the Gromov-Hausdorff metric, and the corresponding Gromov-Hausdorff convergence.

The Hausdorff distance between two subsets X, Y of a metric space is defined as follows. Informally, we say that $d_H(X,Y)<\epsilon$ if any point of X is within distance $\epsilon$ from some point of Y, in the sense of the original metric. Formally

$d_H(X,Y):=\max \{\sup_{x\in X}\inf_{y\in Y}d(x,y), \sup_{y\in Y}\inf_{x\in X}d(x,y)\}.$

It is not particularly illuminating to prove that this is in fact a metric. In fact, it isn’t a metric as the definition stands, but rather a pseudo-metric, which is exactly the same, only allowing d(X,Y)=0 when X and Y are not equal. Note that

$d(X^\circ,\bar X)=0,$

for any set X, so this gives an example, provided X is not both open and closed. Furthermore, if the underlying metric space is unbounded, then the Hausdorff distance between two sets might be infinite. For example in $\mathbb{R}$,

$d_H(\mathbb{R}_{<0},\mathbb{R}_{>0})=\infty.$

We can overcome this pair of objections by restricting attention to closed, bounded sets. In practice, many spaces under consideration will be real in flavour, so it makes sense to define this for compact sets when appropriate.

But this still leaves the underlying problem, which is how to define a distance function on metric spaces. If two metric spaces X and Y were both subspaces of some larger metric space then it would be easy, as we now have the Hausdorff distance. So this is in fact how we proceed in general. We don’t need any knowledge of this covering space a priori, we can just choose the one which minimises the resulting Hausdorff distance. That is

$d_{GH}(X,Y)=\inf\{d_H(\phi(X),\psi(Y))\},$

where the infimum is taken over all metric spaces (E,d), and isometric embeddings $\phi: X\rightarrow E, \psi: Y\rightarrow E$.

The first observation is that this will again be a pseudometric unless we demand that X, Y be closed and bounded. The second is that this index set is not a set. Fortunately, this is quickly rectified. Instead consider all metrics on the disjoint union of sets X and Y, which is set, and contains the subset of those metrics which restrict to the correct metric on each of X and Y. It can be checked that this forms a true metric on the set of compact metric spaces up to isometry.

We have an alternative characterisation. Given compact sets X and Y, a correspondence between X and Y is a set of pairs in $X\times Y$, such that both projection maps are surjective. Ie for any x in X, there is some pair (x,y) in the correspondence. Let $\mathcal{C}(X,Y)$ be the set of such correspondences. We then define the distortion of correspondence $\mathcal{R}$ by:

$\text{dis}(\mathcal{R}):=\sup\{|d_X(x_1,x_2)-d_2(y_1,y_2)|: (x_i,y_i)\in\mathcal{R}\}.$

Then

$d_{GH}(X,Y)=\frac{1}{2}\inf_{\mathcal{R}\in\mathcal{C}(X,Y)}\text{dis}(\mathcal{R}).$

In particular, this gives another reason why we don’t have to worry about taking an infimum over a proper class.

Gromov-Hausdorff convergence then has the natural definition. Note that this does not respect topological equivalence, ie homeomorphism. For example,

$\bar{B(0,\frac{1}{n})}\stackrel{GH}{\rightarrow} \{0\},$

where the latter has the trivial metric. In particular, although all the closed balls are homeomorphic, the G-H limit is not.

A final remark is that the trees we might be looking at are not necessarily compact, so it is useful to have a notion of how this might be extended to non-compact spaces. The answer is to borrow the idea from local limits of considering large finite balls around a fixed central point. In the case of trees, this is particularly well-motivated, as it is often quite natural to have a canonical choice for the ‘root’.

Then with identified points $p_n\in X_n$, say $(X_n,p_n)\rightarrow (X,p)$ if for any R>0 the R-ball around p_n in X_n converges to the R-ball around p in X. We adjust the definition of distortion to include the condition that the infimum be over correspondences for which $(p_X,p_Y)$ is an element.

REFERENCES

This article was based on some lecture notes by Jean-Francois Le Gall from the Clay Institute Summer School which can be found on the author’s website here (about halfway down the page). This material is in chapter 3. I also used Nicolas Curien’s tutorials on this chapter to inform some of the examples. The resolution of the proper class problem was mentioned by several sources I examined. These notes by Jan Christina were among the best.