# The Yule Process

The second problem sheet for classes on the Applied Probability course this term features a long question about the Yule process. This is probably the simplest example of a birth process. It’s named for the British statistician George Udny Yule, though some sources prefer to call it the Yule-Furry process for the American physicist Wendell Furry who used it as a model of a radioactive reaction.

The model is straightforward. At any time there is some number of individuals in the population, and each individual gives birth to an offspring at constant rate $\lambda$, independently from the rest of the population. After a birth has happened, the parent and child evolve independently. In the notation of general birth processes, the birth rate when there are n individuals is $\lambda_n=\lambda n$.

Note that if we start with two or more individuals, the sizes of the two or more families of descendents evolve as a continuous-time Polya’s urn. The arrivals process speeds up with time, but the jump chain is exactly Polya’s urn. Unsurprisingly, the Yule process can be found embedded in preferential attachment models, and other processes which are based around Polya’s urn with extra information.

This is a discrete, random version of exponential growth. Since the geometric distribution is the discrete analogue of the exponential distribution, we probably shouldn’t be surprised to learn that this is indeed the distribution of the process at some fixed time t, when it is started from a single original ancestor. This is all we care about, since the numbers of descendents from each different original ancestors are independent. In general, the distribution of the population size at some fixed time will be negative binomial, that is, a sum of IID geometric distributions.

The standard method here is to proceed using generating functions. Conditioning on the first splitting time gives two independent copies of the original process over a shorter time-scale. One derives an ODE in time for the generating function evaluated at any particular value z. This can be solved uniquely for each z, and patching together gives the generating function of the distribution at any specific time t, which can be seen to coincide with the corresponding generating function of the geometric distribution with parameter $e^{-\lambda t}$.

So we were trying to decide whether there might be a more heuristic argument for this geometric distribution. The method we came up with is not immediate, but does justify the geometric distribution in a couple of steps. First, we say that the birth times are $T_2,T_3,\ldots$, so between times $[T_n,T_{n+1})$ there are n individuals, with $T_1:=0$ for concreteness. Then by construction of the birth process, $T_{n+1}-T_n\stackrel{d}{=}\mathrm{Exp}(\lambda n)$.

We now look at these ‘inter-birth times’ backwards, starting from $T_{n+1}$. Note that $\mathrm{Exp}(\lambda n)$ is the distribution of the time for the first of n IID $\mathrm{Exp}(\lambda)$ clocks to ring. But then, looking backwards, the next inter-birth time is thus the distribution of the time for one of (n-1) IID $\mathrm{Exp}(\lambda)$ clocks to ring. So by memorylessness of the exponential distribution (discussed at great length on the first problem sheet), we can actually take these (n-1) clocks to be exactly those of the original n clocks which did not ring first. Continuing this argument, we can show that the first (in the original time direction) inter-birth time corresponds to the time spent waiting for the final clock to ring. Rewriting this observation formally:

$T_{n+1}\stackrel{d}{=}\max\{X_i : X_1,\ldots,X_n\stackrel{\text{iid}}{\sim}\mathrm{Exp}(\lambda)\}.$ (*)

To return to justifying the geometric form of the distribution, we need to clarify the easiest relationship between the population size at a fixed size and these birth times. As we are aiming for the geometric distribution, the probability of the event $\{X_t>n\}$ will be most useful. Clearly this event is the same as $\{T_{n+1}, and from the description involving maxima of IID exponentials, this is easy to compute as $(1-e^{-\lambda t})^n$, which is exactly what we want.

There are two interesting couplings hidden in these constructions. On closer inspection they turn out to be essentially the same from two different perspectives.

We have specified the distribution of $T_n$ at (*). Look at this distribution on the right hand side. There is a very natural way to couple these distributions for all n, namely to take some infinite sequence $X_1,X_2,\ldots$ of IID $\mathrm{Exp}(\lambda)$ random variables, then use initial sequences of these to generate each of the $T_n$s as described in (*).

Does this coupling correspond to the use of these IID RVs in the birth process? Well, in fact it doesn’t. Examining the argument, we can see that $X_1$ gives a different inter-birth time for each value of t in the correspondence proposed. Even more concretely, in the birth process, almost surely $T_{n+1}>T_n$ for each n. This is not true if we take the canonical coupling of (*). Here, if $X_n<\max\{X_1,\ldots,X_{n-1}\}$, which happens with high probability for large n, we have $T_{n+1}=T_n$ in the process of running maxima.

Perhaps more interestingly, we might observe that this birth process gives a coupling of the geometric distributions. If we want to recover the standard parameterisation of the geometric distribution, we should reparameterise time. [And thus generate an essentially inevitable temptation to make some joke about now having a Yule Log process.]

Let’s consider what the standard coupling might be. For a binomial random variable, either on [n] or some more exotic set, as in percolation, we can couple across all values of the parameter by constructing a family independent uniform random variables, and returning a 1 if $U_i>1-p$ and so on, where p is the parameter of a specific binomial realisation.

We can do exactly the same here. A geometric distribution can be justified as the first success in a sequence of Bernoulli trials, so again we can replace the relevant Bernoulli distribution with a uniform distribution. Take $U_1,U_2,\ldots$ to be IID U[0,1] random variables. Then, we have:

$X_t=\stackrel{d}{=}\bar X_t:= \max\{n: U_1,\ldots,U_{n-1}\ge e^{-\lambda t}\}.$

The equality in distribution holds for any particular value of t by constructing. But it certainly doesn’t hold uniformly in t. Note that if we define $\bar X_t$ as a process, then typically the jumps of this process will be greater than 1, which is forbidden in the Yule process.

So, we have seen that this Yule process, even though its distribution at a fixed time has a standard form, provides a coupling of such distributions that is perhaps slightly surprising.

# The Coupon Collector Problem

This post talks about a problem which I’ve mentioned in passing a few times previously. A research problem I’ve been thinking about in the past week involves some careful calculations for a related setup, so I thought it would be worth writing some short notes about the classical problem. As an child who has tried to collect football cards or the sets of toys you find inside cereal packets will know, it always takes ages to pick up the last few cards, since you will mainly be picking up types that you already have.

The classical coupon collector problem is exactly a mathematical statement of this situation. We have a finite set, that we might as well take to be [n]. Then we have a collection of IID random variables which are uniformly distributed on [n]. Let’s call these $X_1,X_2,\ldots$. We are interested in how many samples we need before we have seen each element of [n]. We could define this finishing time as

$T=\inf_{k\in\mathbb{N}}\{\{X_1,\ldots,X_k\}=[n]\}.$

We are interested in the distribution of this random variable T. The key initial observation is that we can specify the distribution of the number of samples needed between seeing new objects as geometric. For example, given that we have seen k possible values, the number of Xs until we see a new value is distributed as Geom(n-k/n). This of course has expectation n/(n-k), and so the expectation of T is

$\mathbb{E}T=\frac{n}{n}+\frac{n}{n-1}+\ldots+\frac{n}{1}.$

We can immediately do a crude approximation of this as

$n\int_1^n \frac{dx}{x}=\log n.$

We should ask just how approximate this is, and so shortly we will move to a discussion of the harmonic numbers

$H_n:=1+\frac12+\ldots+\frac{1}{n}.$

First though, we should settle the concentration question. Note that this is concretely different to the question of approximating the harmonic numbers. Concentration asks how tightly the distribution is packed around its mean. Later, we will ask exactly how close the mean is to n log n.

The most crude measure is variance. Note that in the above calculation we used linearity of expectation, which holds generally. We also have a result concerning linearity of variance, which Wikipedia asserts is commonly called Bienayme’s formula, provided the summand random variables are independent. And that is the case here. Recall that the variance of a Geom(p) random variable is $\frac{q}{p^2}$, where p+q=1. So we obtain

$\mathrm{Var}(T)= \sum_{k=1}^n \frac{n(n-k)}{k^2}$

$\le \sum_{k=1}^n \frac{n^2}{k^2}\le n^2\frac{\pi^2}{6}.$

So the standard deviation is an order of magnitude less than the expectation, hence by Chebyshev or similar, we can get some concentration bounds.

Harmonic Numbers

We return now to the question of how good an estimate log n is for

$H_n:=1+\frac12+\ldots+\frac{1}{n}.$

First, we note that log n must be an underestimate. In the following picture

log n is the area under the black curve between 1 and n, whereas H_n also includes the block between n and n+1, and the red area above the curve. So, in the limit, the estimate is off by at least the total red area (as the first error term decays to zero). The value of this area is defined to be the Euler-Mascheroni constant, a name attributed to the two men who first made a serious attempt to approximate it.

Why should this be finite? Well note that if we translated each red bit so they lay one on top of the other, they would not overlap (apart from boundaries) and be contained within the block with width [1,2] and height 1. So the constant is bounded above by 1. Since the function 1/x is strictly convex, approximating each red sector by a triangle with the same boundary points would be an underestimate, so the constant $\gamma>1/2$. The value turns out to be about 0.577.

So thus far we have the approximation

$H_n\approx \log n + \gamma.$

In fact a strong asymptotic statement is

$H_n=\log n +\gamma+\frac{1}{2n}+o\left(\frac{1}{n}\right).$

Rather than address that, I will show that, asymptotically:

$\log n+\gamma< H_n<\log n +\gamma+\frac{1}{n}.$

We treat the upper bound first. Note that $\log(n+1)=\log n + \log (1+\frac{1}{n})=\log n +\frac{1}{n}+O(\frac{1}{n^2})$. So, just by bounding with areas, noting that we need to take the upper limit of the integral to be (n+1), we get:

$H_n < \int_1^{n+1} \frac{dx}{x} + \gamma=\log(n+1)+\gamma\leq \log n + \frac{1}{n}+\gamma.$

For the lower bound, it is required to show that

$\int_n^{n+1}\frac{dx}{x}$

is strictly greater than the red area contributing to $\gamma$ that lies to the right of x=n+1 But for this we can use the same argument as before, by translating all the red areas as far to the left as possible so they all lie on top of each other, and in particular within a box of dimensions 1 x 1/n+1, which is less than the integral.

We conclude this article by relating these harmonic numbers to the Stirling numbers of the first kind, which are a particular example of Bell polynomials. The easiest way to state the result is

$H_n=\frac{1}{n!}\#\{\text{permutations of }[n+1]\text{ with 2 cycles}\}.$

To see why this might be, consider the number of permutations of [n+1] with two cycles of length k and n+1-k, where for now we assume these are not equal. Then the number of such permutations is

$\binom{n+1}{k}(k-1)! (n-k)!=\frac{(n+1)!}{k\cdot(n-k+1)}=n!\left[\frac{n+1}{k\cdot(n-k+1)}\right]= n!\left[\frac{1}{k}+\frac{1}{n-k+1}\right].$

After checking the case k=n+1/2 if applicable, and making sure all the bounds match up correctly, the result follows.

# 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.

# Diameters of Trees and Cycle Deletion

In the past two posts, we introduced two models of random trees. The Uniform Spanning Tree chooses uniformly at random from the set of spanning trees for a given underlying graph. The Minimum Spanning Tree assigns IID weights to each edge in the underlying graph, then chooses the spanning tree with minimum induced total weight. We are interested to know whether these are in fact the same distribution, and if they are not, what properties can be used to distinguish them asymptotically.

While investigating my current research problem, I was interested in the diameter of large random trees under various models. Specifically, I am considering what happens if you take a standard Erdos-Renyi process on n vertices, where edges appear at constant rate between pairs of vertices chosen uniformly at random, and add an extra mechanism to prevent the components becoming too large. For this particular model, our mechanism consists of removing any cycles as they are formed. Thus all the components remain trees as time advances, so it is not unreasonable to think that there might be some sort of equilibrium distribution.

Now, by definition, any tree formed by the Erdos-Renyi process is a uniform tree. Why? Well, the probability of a configuration is determined entirely by the number of edges present, so once we condition that a particular set of vertices are the support of a tree, all possible tree structures are equally likely. Note that this relies on sampling at a single fixed time. If we know the full history of the process, then it is no longer uniform. For example, define a k-star to be a tree on k vertices where one ‘centre’ vertex has degree k-1. The probability that a uniform tree on k vertices is a k-star is $\frac{k}{k^{k-2}}=k^{-(k-3)}$. But a star can only be formed by successively adding single vertices to an existing star. That is, we cannot join a 3-tree and a 4-tree with a edge to get a 7-star. So it is certainly not immediately clear that once we’ve incorporated the cycle deletion mechanism, the resulting trees will be uniform once we condition on their size.

In fact, the process of component sizes is not itself Markovian. For a concrete example, observe first that there is, up to isomorphism, only one tree on any of {0,1,2,3} vertices, so the first possible counterexample will be splitting up a tree on four vertices. Note that cycle deletion always removes at least three edges (ie a triangle), so the two possibilities for breaking a 4-tree are:

(4) -> (2,1,1) and (4) -> (1,1,1,1)

I claim that the probabilities of each of these are different in the two cases: a) (4) is formed from (2,2) and b) (4) is formed from (3,1). This is precisely a counterexample to the Markov property.

In the case where (4) is formed from (2,2), the 4-tree is certainly a path of length 4. Therefore, with probability 1/3, the next edge added creates a 4-cycle, which is deleted to leave components (1,1,1,1). In the case where (4) is formed from (3,1), then with probability 2/3 it is a path of length 4 and with probability 1/3 it is a 4-star (a ‘T’ shape). In this second case, no edge can be added to make a 4-cycle, so after cycle deletion the only possibility is (2,1,1). Thus the probability of getting (1,1,1,1) is 2/9 in this case, confirming that the process is non-Markovian. However, we might remark that we are unlikely to have O(n) vertices involved in fragmentations until at least the formation of the giant component in the underlying E-R process, so it is possible that the cycle deletion process is ‘almost Markov’ for any property we might actually be interested in.

When we delete a cycle, how many vertices do we lose? Well, for a large tree on n vertices, the edge added which creates the cycle is chosen uniformly at random from the pairs of vertices which are not currently joined by an edge. Assuming that n is w(1), that is we are thinking about a limit of fairly large trees, then the number of edges present is much smaller than the number of possible edges. So we might as well assume we are choosing uniformly from the possible edges, rather than just the possible edges which aren’t already present.

If we choose to add an edge between vertices x and y in the tree, then a cycle is formed and immediately deleted. So the number of edges lost is precisely the length of the path between x and y in the original tree. We are interested to know the asymptotics for this length when x and y are chosen at random. The largest path in a graph is called the diameter, and in practice if we are just interested in orders of magnitude, we might as well assume diameter and expected path length are the same.

So we want to know the asymptotic diameter of a UST on n vertices for large n. This is generally taken to be $n^{1/2}$. Here’s a quick but very informal argument that did genuinely originate on the back of a napkin. I’m using the LERW definition. Let’s start at vertex x and perform LERW, and record how long the resultant path is as time t advances. This is a Markov chain: call the path length at time t $X_t$.

Then if $X_t=k$, with probability $1-\frac k n$ we get $X_{t+1}=k+1$, and for each j in {0,…,k-1}, with probability 1/n we have $X_{t+1}=j$, as this corresponds to hitting a vertex we have already visited. So

$\mathbb{E}\Big[X_{t+1}|X_t=k\Big]=\frac{nk-k^2/2}{n}.$

Note that this drift is positive for $k<< \sqrt n$ and negative for $k>>\sqrt n$, so we would expect $n^{-1/2}$ to be the correct scaling if we wanted to find an equilibrium distribution. And the expected hitting time of vertex y is n, by a geometric distribution argument, so in fact we would expect this Markov chain to be well into the equilibrium window with the $n^{-1/2}$ scaling by the time this occurs. As a result, we expect the length of the x to y path to have magnitude $n^{1/2}$, and assume that the diameter is similar.

So this will be helpful for calculations in the cycle deletion model, provided that the trees look like uniform trees. But does that even matter? Do all sensible models of random trees have diameter going like $n^{1/2}$? Well, a recent paper of Addario-Berry, Broutin and Reed shows that this is not the case for the minimum spanning tree. They demonstrate that the diameter in this case is $n^{1/3}$. I found this initially surprising, so tried a small example to see if that shed any light on the situation.

The underlying claim is that MSTs are more likely to be ‘star-like’ than USTs, a term I am not going to define. Let’s consider n=4. Specifically, consider the 4-star with centre labelled as 1. There are six possible edges in K_4 and we want to see how many of the 6! weight allocations lead to this star. If the three edges into vertex 1 have weights 1, 2 and 3 then we certainly get the star, but we can also get this star if the edges have weights 1, 2 and 4, and the edge with weight 3 lies between the edges with weights 1 and 2. So the total number of possibilities for this is 3! x 3! + 3! x 2! = 48. Whereas to get a 4-path, you can assign weights 1, 2 and 3 to the edges of the path, or weights 1, 2 and 4 provided the 4 is not in the middle, and then you have the 3 joining up the triangle formed by 1 and 2. So the number of possibilities for this is 3! x 3! + 4 x 2! = 44.

To summarise in a highly informal way, in a star-like tree, you can ‘hide’ some fairly low-scoring weights on edges that aren’t in the tree, so long as they join up very low-scoring edges that are in the tree. Obviously, this is a long way from getting any formal results on asymptotics, but it does at least show that we need to be careful about diameters if we don’t know exactly what mechanism is generating the tree!