Multitype Branching Processes

Posted on February 25, 2014 by dominicyeo

One of the fundamental objects in classical probability theory is the Galton-Watson branching process. This is defined to be a model for the growth of a population, where each individual in a generation gives birth to some number (possibly zero) of offspring, who form the next generation. Crucially, the numbers of offspring of the individuals are IID, with the same distribution both within generations and between generations.

There are several ways one might generalise this, such as non-IID offspring distributions, or pairs of individuals producing some number of offspring, but here we consider the situation where each individual has some type, and different types have different offspring distributions. Note that if there are K types, say, then the offspring distributions should now be supported on $\mathbb{Z}_{\ge 0}^K$ . Let’s say the offspring distribution from a parent of type i is $\mu^{(i)}$ .

The first question to address is one of survival. Recall that if we want to know whether a standard Galton-Watson process has positive probability of having infinite size, that is never going extinct, we only need to know the expectation of the offspring distribution. If this is less than 1, then the process is subcritical and is almost surely finite. If it is greater than 1, then it is supercritical and survives with positive probability. If the expectation is exactly 1 (and the variance is finite) then the process is critical and although it is still almost surely finite, the overall population size has a power-law tail, and hence (or otherwise) the expected population size is infinite.

We would like a similar result for the multitype process, saying that we do not need to know everything about the distribution to decide what the survival probability should be.

The first thing to address is why we can’t just reduce the multitype change to the monotype setting. It’s easiest to assume that we know the type of the root in the multitype tree. The case where the type of the root is random can be reconstructed later. Anyway, suppose now that we want to know the offspring distribution of a vertex in the m-th generation. To decide this, we need to know the probability that this vertex has a given type, say type j. To calculate this, we need to work out all the type possibilities for the first m generations, and their probabilities, which may well include lots of complicated size-biasing. Certainly it is not easy, and there’s no reason why these offspring distributions should be IID. The best we can say is that they should probably be exchangeable within each generation.

Obviously if the offspring distribution does not depend on the parent’s type, then we have a standard Galton-Watson tree with types assigned in an IID manner to the realisation. If the types are symmetric (for example if M, to be defined, is invariant under permuting the indices) then life gets much easier. In general, however, it will be more complicated than this.

We can however think about how to decide on survival probability. We consider the expected number of offspring, allowing both the type of the parent and the type of the child to vary. So define $m_{ij}$ to be the expected number of type j children born to a type i parent. Then write these in a matrix $M=(m_{ij})$ .

One generalisation is to consider a Galton-Watson forest started from some positive number of roots of various types. Suppose we have a vector $\nu=(\nu_i)$ listing the number of roots of each type. Then the expected number of descendents of each type at generation n is given by the vector $\nu M^n$ .

Let $\lambda$ be the largest eigenvalue of M. As for the transition matrices of Markov chains, the Perron-Frobenius theorem applies here, which confirms that, because the entries of M are positive, the eigenvalue with largest modulus is simple and real, and the associated eigenvector has entirely positive entries. [In fact we need a couple of extra conditions on M, including that it is possible to get from any type to any other type – we say irreducible – but that isn’t worth going into now.]

So in fact the total number of descendents at generation n grows like $\lambda^n$ in expectation, and so we have the same description of subcriticality and supercriticality. We can also make a sensible comment about the left- $\lambda$ -eigenvector of M. This is the limiting proportion of the different types of vertices.

It’s a result (eg. [3]) that the height profile of a depth-first search on a standard Galton-Watson tree converges to Brownian Motion. Another way to phrase this is that a GW tree conditioned to have some size N has the Brownian Continuum Random Tree as a scaling limit as N grows to infinity. Miermont [4] proves that this result holds for the multitype tree as well. In the remainder of this post I want to discuss one idea along the way to the proof, and one application.

I said initially that there wasn’t a trivial reduction of a multitype process to a monotype process. There is however a non-trivial embedding of a monotype process in a multitype process. Consider all the vertices of type 1, and all the paths between such vertices. Then draw a new tree consisting of just the type 1 vertices. Two of these are joined by an edge if there is no other type 1 vertex on the unique path between them in the original tree. If that definition is confusing, think of the most sensible way to construct a tree on the type 1 vertices from the original, and you’ve probably chosen this definition.

There are two important things about this new tree. 1) It is a Galton-Watson tree, and 2) if the original tree is critical, then this reduced tree is also critical. Proving 1) is heavily dependent on exactly what definitions one takes for both the multitype branching mechanism and the standard G-W mechanism. Essentially, at a type 1 vertex, the number of type 1 descendents is not dependent on anything that happened at previous generations, nor in other branches of the original tree. This gives IID offspring distributions once it is formalised. As for criticality, we note that by the matrix argument given before, under the irreducibility condition discussed, the expectation of the total population size is infinite iff the expected number of type 1 vertices is also infinite. Since the proportion of type 1 vertices is given by the first element of the left eigenvector, which is positive, we can make a further argument that the number of type 1 vertices has a power-law tail iff the total population size also has a power-law tail.

I want to end by explaining why I was thinking about this model at all. In many previous posts I’ve discussed the forest fire model, where occasionally all the edges in some large component are deleted, and the component becomes a set of singletons again. We are interested in the local limit. That is, what do the large components look like from the point of view of a single vertex in the component? If we were able to prove that the large components have BCRT as the scaling limit, this would answer this question.

This holds for the original random graph process. There are two sensible ways to motivate this. Firstly, given that a component is a tree (which it is with high probability if its size is O(1) ), its distribution is that of the uniform tree, and it is known that this has BCRT as a scaling limit [1]. Alternatively, we know that the components have a Poisson Galton-Watson process as a local limit by the same argument used to calculate the increments of the exploration process. So we have an alternative description of the BCRT appearing: the scaling limit of G-W trees conditioned on their size.

Regarding the forest fires, if we stop the process at some time T>1, we know that some vertices have been burned several times and some vertices have never received an edge. What is clear though is that if we specify the age of each vertex, that is, how long has elapsed since it was last burned; conditional on this, we have an inhomogeneous random graph. Note that if we have two vertices of ages s and t, then the probability that there is an edge between them is $1-e^{-\frac{s\wedge t}{n}}$ , ie approximately $\frac{s\wedge t}{n}$ . The function giving the probabilities of edges between different types of vertices is called the kernel, and here it is sufficiently well-behaved (in particular, it is bounded) that we are able to use the results of Bollobas et al in [2], where they discuss general sparse inhomogeneous random graphs. They show, among many other things, that in this setting as well the local limit is a multitype branching process.

So in conclusion, we have almost all the ingredients towards proving the result we want, that forest fire components have BCRT scaling limit. The only outstanding matter is that the Miermont result deals with a finite number of types, whereas obviously in the setting where we parameterise by age, the set of types is continuous. In other words, I’m working hard!

References

[1] Aldous – The Continuum Random Tree III

[2] Bollobas, Janson, Riordan – The phase transition in inhomogeneous random graphs

[3] Le Gall – Random Trees and Applications

[4] Miermont – Invariance principles for spatial multitype Galton-Watson trees

Graph Theory Basics (explanationsfortime.wordpress.com)
The Power Method (lucatrevisan.wordpress.com)
New approach to vertex connectivity could maximize networks’ bandwidth (rdmag.com)
Intuitive Guide to Principal Component Analysis (georgemdallas.wordpress.com)

Branching Random Walk and Amenability

Posted on February 9, 2014 by dominicyeo

This post is about some of the things I learned in an interesting given by Elisabetta Candellero in Oxford last week, based on joint work with Matt Roberts. The paper on which this is based can be found here. The main thing I want to talk about are some properties of graphs which were mentioned near the beginning which I hadn’t heard about before.

Branching Random Walk (hereafter BRW) is a model to which much attention has been paid, because of its natural applications in a range of physical and genetic settings. As with many of the best models, the definition is pretty much in the title. We take the ingredients for a random walk on a graph, which is a graph, and a transition matrix P on that graph. For most of the time we will consider simple random walk, so the graph G exactly specifies P. This requires the additional condition that the graph G is locally finite. We will introduce a branching mechanism, so at discrete times {0,1,2,…} we will track both the number of particles, and their current locations. We start at time 0 with a single particle at some vertex. Then at each time-step, all the vertices present die, and each gives birth independently to some number of offspring according to a fixed probability distribution $\mu$ . These offspring then perform one move according to transition matrix P. Note that if you want the system to carry the appearance of having no death, then taking the support of the offspring distribution to be {1,2,3,…} achieves precisely this. The properties we consider will not be very interesting unless G is infinite, so assume that from now on.

There are almost limitless ways we could think of to generalise these dynamics. The offspring distribution could be allowed to depend on the vertex the particle is occupying. The joint transition probabilities of the offspring at a vertex could be biased in favour or against the offspring moving to the same site next. The environment could be chosen in advance before the process starts, but random.

The classical question about BRW is that of recurrence and transience. The definition extends naturally from that of a Markov chain (which any non-branching random walk on a graph is). As in that setting, we say a BRW is recurrent if every vertex is almost surely visited infinitely often by particles of the graph.

Heuristically, we should observe that in some sense, it is quite difficult for simple random walk on an infinite graph to be recurrent. We have examples in $\mathbb{Z},\mathbb{Z}^2$ , but these are about as ‘small’ as an infinite graph can be. An idea might be that if the number of sites some distance away from where we start grows rapidly as the distance grows, then there isn’t enough ‘pull’ back to visit the sites near where we start infinitely often. Extending this argument, it is easier for a BRW to be recurrent, as we have the option to make the branching rate large, which means that there are lots of particles at large times, hence more possibility for visiting everywhere. Note that if the offspring distribution is subcritical, we don’t stand a chance of having interesting properties. If we ignore the random walk part, we just have a subcritical Galton-Watson process, which dies out almost surely.

We need a measure of the concept discussed in the heuristic for how fast the number of vertices in the graph grows as we consider bands of vertices further and further away from the starting vertex. The standard measure for this is the spectral radius, which is defined not in terms of number of vertices, but through the limiting probability of returning to a fixed vertex at large time n. Precisely

$\rho:= \limsup \mathbb{P}_i(X_n=i)^{1/n},$

so in some approximation sense

$\mathbb{P}_i(X_n=i)\sim \rho^{n},$

which explains why $\rho\le 1$ . Note that by considering the sum of such terms, if simple random walk on G is recurrent, then $\rho=1$ , but the converse does not hold. (Consider SRW on $\mathbb{Z}^3$ for example.)

It’s also worth remarking that $\rho$ is a class property. In particular, for a connected graph, the value of $\rho$ is independent of i. This is not surprising, as if d is the graph distance between vertices i and j, then

$p_{ii}^{(n)}\ge p_{ij}^{(d)}p_{jj}^{(n-2d)}p_{ji}^{(d)},$

and vice versa, which enables us to sandwich usefully for the limits.

Really, $\rho$ is a function of the transition matrix P. In fact, we can be more specific, by considering diagonalising P. The only case we care about is when P is infinite, so this is not especially nice, but it makes it clear why $p_{ii}^{(n)}$ decays like $|\rho|^n$ where $\rho$ is the largest eigenvalue of P. Indeed this is an alternative definition of the spectral radius. Note that Perron-Frobenius theory (which seems to keep coming up on the blog this week…) says that since $|\rho|\le 1$ , then if $|\rho|=1$ , we must have $\rho=1$ . So the spectral radius being 1 is precisely equivalent to having an invariant measure. We don’t know whether we can normalise it, but P-F guarantees the relevant left-eigenvector is non-negative, and hence a measure.

Next we give this situation a name. Say that a random walk is amenable if $\rho(P)=1$ . We can extend this property to say that a graph is amenable if SRW on it is amenable.

This is not the standard definition of amenability. This property is originally defined (by von Neumann) in the context of groups. A group G is said to be amenable if there exists a left-invariant probability measure on G, ie $\mu$ such that

$\forall A\subset G, \forall g\in G, \mu(gA)=A.$

The uniform distribution shows that any finite group is amenable.

It turns out that in general there are several conditions for a group which are equivalent to amenability. One is that, given G finitely generated by B, the Cayley graph for G with edges given by elements of B does not satisfy a strong isoperimetric inequality. Such an inequality is an alternative way of saying that the graph grows rapidly. It says that the size of the boundary of a subset of the vertices is uniformly large relative to the size of the set. Precisely, there exists a constant c>0 such that whenever U is a finite subset of the vertices, we have $|\partial U|\ge c|U|$ . (Note that finiteness of U is important – we would not expect results like this to hold for very large subsets.)

Kesten proved that it is further equivalent to the statement that simple random walk on $Cay(G,B)$ is amenable in our original sense. This technical and important result links the two definitions.

We finish by declaring the main classical result in BRW, which is a precise condition for transience. As motivated earlier, the rate of branching and the spectral radius have opposing effects on whether the system is recurrent or transient. Note that at some large time, the expected number of particles which have returned to the starting vertex is given by the expected number of particles in the system multiplied by the probability that any one of them is back at its origin, ie $\sim \mu^n\rho^n$ . So the probability that there is a particle back at the origin at this time is (crudely transferring from expectation to probability) $1\wedge (\mu \rho)^n$ . We can conclude that the chain is recurrent if $\mu > \rho^{-1}$ and transient if $\mu<\rho^{-1}$ . This result is due to Benjamini and Peres.

The remaining case, when $\mu=\rho^{-1}$ is called, unsurprisingly, critical BRW. It was proved in ’06 by Gantert and Muller that, in fact, all critical BRWs are transient too. This must exclude the amenable case, as we could think of SRW on $\mathbb{Z}$ as a critical BRW by taking the branching distribution to be identically one, as the spectral radius is also 1.

In the end, the material in this post is rather preliminary to the work presented in EC’s talk, which concerned the trace of BRW, and whether there are infinitely many essentially different paths to infinity taken by the particles of the BRW. They show that this holds in a broad class of graphs with symmetric properties.

Rado graph and 0-1 law for finite graphs (chiasme.wordpress.com)
How much is the stacks project graph like a random graph? (quomodocumque.wordpress.com)
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Critical Components in Erdos-Renyi

Posted on November 7, 2013 by dominicyeo

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

Erdox-Renyi no isolated nodes (cs.stackexchange.com)
The Erdős-Rényi Random Graph (jeremykun.com)
FEATURE: The man who turned coffee into theorems (sciencealert.com.au)
Welcome to Random Graphs and Complex Networks, T-79.7003 (randomgraphsaalto.wordpress.com)

Recent Research Activity

Posted on October 25, 2013 by dominicyeo

I’ve spent this week in Luminy, near Marseille, attending a summer school run by ALEA, the organisation of French probabilists. We’ve been staying in CIRM, a dedicated maths research conference centre at the edges of the calanques, the area of mountains and jagged coastal inlets between Marseille and Cassis. The walking possibilities have been excellent, as have the courses and lectures, on a range of topics in probability theory.

Anyway, the time here has been an excellent moment to reflect on my research progress, and try to come up with the sort of fresh ideas that are perhaps slightly inhibited by sitting at a desk with an endless supply of paper on which to try calculations. When I get back, I have to submit a first-year report, so at least for a little while I will have to suppress the desire to make further progress and instead diligently assemble the progress I have made.

The Model

I’ve defined some of these processes in past posts, but I see no harm in doing so again. We take the standard Erdos-Renyi random graph process, where edges are added one-at-a-time uniformly at random between n vertices, and amend it by adding a deletion mechanism. The aim is to arrive at a process which looks in equilibrium more like the critical random graph than either the subcritical or supercritical regimes, where the components are very small, and dominated by one giant component respectively. Rath, Toth and others have studied the process where each vertex is hit by lightning at uniform rate. When this happens, we delete all the edges in the component containing that vertex. Naturally, big components will be hit by lightning more often than small components, and so this acts as a mechanism to prevent the formation of giant components, if scaled correctly.

We take a different approach. We observe that criticality in the original random graph process is denoted by the first appearance of a giant component, but also by the first appearance of a) lots of cycles, and b) large cycles. In particular, it is very unlikely that a giant component could form without containing any cycles. We will therefore use the appearance of a cycle to trigger some form of deletion mechanism.

Our final goal is to treat the so-called ‘Cycle Deletion’ model. Here, whenever a cycle appears, we delete all the edges in that cycle immediately. There are several challenges in treating this model, because the rate at which cycles emerge in a tree is a function of the tree structure. The trees in this model will not be Uniform Spanning Trees (though it is very possible that they will be ‘almost USTs’ in some sense – we need to investigate this further) so it will be hard to make nice statements about the rates. For the standard random graph process, if we are only interested in the sizes of the components, we are actually allowed to ignore the graph structure entirely. The component sizes evolve as a discrete, stochastic version of the multiplicative coalescent (sometimes called a Marcus-Lushnikov process). We would like a deletion mechanism that has a nice interpretation as a fragmentation operation in the same sense. The rate at which a component fragments will be quadratic in the size of the component, since there are $O(k^2)$ possible edges between k vertices forming a component, and adding any of precisely these will create a cycle.

I’ve talked previously about how to overcome the problems with the tree structure in Cycle Deletion with the so-called Uniform Cycle Deleting model. In any case, as a starting point we might consider the Cycle-Induced Forest Fire model. Here, whenever a cycle appears, we delete all the edges, including the new one, in the whole component which contains the cycle.

We suspect this model may resemble the critical random graph at all times. The main characteristic of G(n,1/n) is that the largest component is of size O(n^2/3), and indeed there are arbitrarily many components of this size, with high probability in the limit. Since CIFF is recurrent for any fixed n, meaning that it will visit any state infinitely often (rather than tending to infinity or similar), we should ask what the largest component is typically in the equilibrium distribution. Our aim is to prove that it is O(n^2/3). We might suspect that the typical size of the largest component will be greater in the Cycle Deletion model, since each fragmentation event is less severe there, removing fewer edges.

An Upper Bound

The nice thing about Markov chains is that they have an ergodic property, which means that if you run them for long enough, the proportion of time spent in any state is given by the stationary probability of being in that state. It doesn’t matter whether or not you start in equilibrium, since it will converge anyway. Thus it is meaningful to talk about properties like the average number of isolated vertices as a time-average as well as an average with respect to some distribution.

This quantity is the key to an upper bound. We can equally talk about the average change in the number of isolated vertices in a time-step. This will increase when a component fragments, and will decrease when an isolated vertex coalesces with another component. In particular, the largest possible decrease in the number of isolated vertices in a single time-step is 2, corresponding to an edge appearing between two isolated vertices.

Suppose that with probability $\Theta(1)$ there is a component of size $n^\alpha$ for some $\alpha>2/3$ . Then such a component makes a contribution to the expected change in the number of isolated vertices of

$\Theta(1) n^\alpha \left(\frac{n^\alpha}{n}\right)^2.$ (*)

Where does this come from? Well, we are tracking the contributions from the event that the largest component is of this size and that it fragments, giving $n^\alpha$ new isolated vertices. So the $\Theta(1)$ accounts for the probability that there is such a component to begin with. Then, conditional on that, the probability that it gets fragmented in the next time-step is the probability that both ends of the next edge added lie in that component. Since the edge is chosen uniformly at random, the probability of this is $n^\alpha/n$ . Note that this is under a slightly odd definition of an edge, that allows loops. Basically, I don’t want to have lots of correction terms involving $\binom{n}{2}$ floating around. However, it would make no difference to the orders of magnitude if we to do it with these.

So, this is only one contribution to the typical rate of gain of isolated vertices. Now note that if $\alpha>2/3$ , then this expression is >> 1. This is bad since the negative contributions to this expected flux in the number of isolated vertices is O(1). So this suggests that over time, the number of isolated vertices will keep growing. This is obviously ridiculous since a) we are in equilibrium, so the expected flux should be 0 and b) the number of isolated vertices cannot exceed n, for clear reasons.

This gives us an upper bound of n^2/3 as the typical scale of the largest component. We can come up with a similar argument for the cycle deleting model. The most helpful thing to track there is the number of edges in the graph. Note that since the graph is at all times a forest on n vertices, the number of edges is equal to n minus the number of (tree) components. We use the fact that the typical fragmentation of a component of size k creates $O(\sqrt{k})$ new components. It is possible to argue via isolated vertices here too, but the estimates are harder, or at least less present in the literature.

Lower Bounds?

The problem with lower bounds is that it is entirely possible that the flux in the number of isolated vertices is not driven by typical behaviour. Suppose for example we had a different rule. We begin a random graph process, and the first time we see a cycle in a component with size larger than n^2/3, we delete all the edges in the whole graph. Then we will see a sequence of random graph processes starting with the empty graph and stopped at some point close to criticality (in fact, with high probability in the *critical window*), and these will all be glued together. So then, most of the time the process will look subcritical, but the gains in isolated vertices will occur only during the critical periods, which are only an asymptotically small proportion of the time.

At the moment, my approach to the lower bound is instead to prove that the upper bound is tight. I mean this in the following sense. Suppose we wanted to be sure that (*) was in fact equal to the average rate of gain of isolated vertices. We would have to check the following:

That the total contributions from all other components were similar or smaller than from the component(s) of size roughly $n^{\alpha}$ .
That there were only a few components of size $n^{\alpha}$ . In particular, the estimate would be wrong if there were $n^\epsilon$ such components for any $\epsilon>0$ .
That it cannot be the case that for example, some small proportion of the time there is a component of size roughly $n^{\alpha+\epsilon}$ , and over a large enough time these make a greater contribution to the average gain in isolated vertices.

A nice way to re-interpret this is to consider some special vertex and track the size of its component in time. It will be involved in repeated fragmentations over the course of time, so it is meaningful to talk about the distribution of the size of the component containing the vertex when it is fragmented. Our aim is to show that this distribution is concentrated on the scaling $O(n^\alpha)$ .

So this has turned out to be fairly hard. Rather than try to explain some of the ideas I’ve employed in attempting to overcome this, I will finish by giving one reason why it is hard.

We have seen that the component sizes in random graphs evolve as the multiplicative coalescent, but at a fixed moment in time, we can derive good estimates from an analogy with branching processes. We might like to do that here. If we know what the system looks like most of the time, we might try to ‘grow’ a multiplicative coalescent, viewing it like a branching process, with distribution given by the typical distribution. The problem is that when I do this, I find that the expectation of the offspring distribution is $\Theta(1)$ . This looks fine, since 1 is the threshold for extinction with probability 1. However, throughout the analysis, I have only been paying attention to the exponent of n in all the time and size estimates. For example, I view $n^\alpha$ and $n^\alpha \log n$ as the same. This is a problem, as when I say the expectation is $\Theta(1)$ , I am really saying it is $\sim n^0$ . This means it could be $\frac{1}{\log n}$ or $\log n$ . Of course, there is a massive difference between these, since a branching process grows expectationally!

So, this approach appears doomed in its current form. I have some other ideas, but a bit more background may be required before going into those. I’m going to be rather busy with teaching on my return to the office, so unfortunately it is possible that there may be many posts about second year probability and third year applied probability before anything more about CIFF.

Eventually Almost Everywhere

A blog about probability and olympiads by Dominic Yeo

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