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 . 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 . 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 .

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

Note that this drift is positive for and negative for , so we would expect 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 scaling by the time this occurs. As a result, we expect the length of the x to y path to have magnitude , 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 ? 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 . 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!

Related articles

• Minimum Spanning Trees (eventuallyalmosteverywhere.wordpress.com)
• Uniform Spanning Trees (eventuallyalmosteverywhere.wordpress.com)

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