Lecture 10 – the configuration model

Posted on January 15, 2019 by dominicyeo

I am aiming to write a short post about each lecture in my ongoing course on Random Graphs. Details and logistics for the course can be found here.

As we enter the final stages of the semester, I want to discuss some extensions to the standard Erdos-Renyi random graph which has been the focus of most of the course so far. Although we will not get far into the details during this course, the overall goal is to develop models which are close to Erdos-Renyi in terms of ease of analysis, while also allowing more of the features characteristic of networks observed in the real world.

One of the more obvious deficiencies of the sparse regime of Erdos-Renyi random graphs for modelling ‘real-world phenomena’ concerns the degree sequence. Indeed, the empirical degree distribution of G(n,c/n) converges to Poisson(c). By contrast, in real-world networks, a much wider range of degrees is typically observed, and in many cases it is felt that these should follow a power law, with a small number of a very highly connected agents.

One way around this problem to construct random graphs where we insist that the graph has a given sequence of degrees. The configuration model, which is the subject of this lecture and this post (and about which I’ve written before), offers one way to achieve this.

Definition and notes

Let $n\ge 1$ and let $d=(d_1,d_2,\ldots,d_n)$ be a sequence of non-negative integers such that $\sum_{i=1}^n d_i$ is even. Then the configuration model with degree sequence d is a random multigraph with vertex set [n], constructed as follows:

To each vertex $i\in[n]$ , assign $d_i$ half-edges;
Then, take a uniform matching of these half-edges;
Finally, for each pair of half-edges in the matching, replace the two half-edges with a genuine edge, to obtain the multigraph $CM_n(d)$ , in which, by construction, vertex i has degree $d_i$ .

One should note immediately that although the matching is uniform, the multigraph is not uniform amongst multigraphs with that degree sequence. Note also that the condition on the sums of the degrees is necessary for any graph, and in this context means that the number of half-edges is even, without which it would not be possible to construct a matching.

This effect is manifest in the simplest possible example, when n=2 and d=(3,3). There are two possible graphs, up to isomorphism, which are shown below:

For obvious reasons, we might refer to these as the handcuffs and the theta , respectively. It’s helpful if we, temporarily, assume the half-edges are distinguishable at the moment we join them up in the configuration model construction. Because then there are 3×3=9 ways to join them up to form the handcuffs (think of which half-edge ends up forming the edge between the two vertices) while there are 3!=6 ways to pair up the half-edges in the theta.

In general, for multigraphs H with the correct degree sequence, we have

$\mathbb{P}( CM_n(d)\simeq H) \propto \left( 2^{\# \text{loops}(H)} \prod_{e\in E(H)} \text{mult}(e)! \right),$

where $\text{mult}(e)$ is the multiplicity with which a given edge e appears in H.

Note: it might seem counterintuitive that this procedure is biased against multiple edges and self-loops, but it is really just saying that there are more ways to form two distinct edges than to form two equal edges (ie a multiedge pair) when we view the half-edges as distinguishable. (See this post for further discussion of this aspect in the 3-regular setting.)

However, a consequence of this result is that if we condition on the event that $CM_n(d)$ is simple, then the resulting random graph is uniform on the set of simple graphs satisfying the degree property. Note that the same example as above shows that there’s no guarantee that there exists a simple graph whose degrees are some given sequence.

d-regular configuration model

In general, from a modelling point of view, we are particularly interested in simple, connected graphs, and so it is valuable to study whether the large examples of the configuration model are likely to have these properties. In this lecture, I will mainly focus on the case where the multigraphs are d-regular, meaning that all the vertices have degree equal to d. For the purposes of this lecture, we denote by $G^d(n)$ , the d-regular configuration model $CM_n(d,\ldots,d)$ .

d=1: to satisfy the parity condition on the sums of degrees, we must have n even. But then $G^1(n)$ will consist of n/2 disjoint edges.
d=2: $G^2(n)$ will consist of some number of disjoint cycles, and it is a straightforward calculation to check that when n is large, with high probability the graph will be disconnected.

In particular, I will focus on the case when d=3, which is the first interesting case. Most of the results we prove here can be generalised (under various conditions) to more general examples of the configuration model. The main goal of the lecture is revision of some techniques of the course, plus one new one, in a fresh setting, and the strongest possible versions of many of these results can be found amongst the references listed at the end.

Connectedness

In the lecture, we showed that $G^3(2n)$ is connected with high probability. This is, in fact, a very weak result, since in fact $G^d(n)$ is d-connected with high probability for $d\ge 3$ [Bol81, Wor81]. Here, d-connected means that one must remove at least d vertices in order to disconnect the graph, or, equivalently, that there are d disjoint paths between any pair of vertices. Furthermore, Bollobas shows that for $d\ge 3$ , $G^d(n)$ is a (random) expander family [Bol88].

Anyway, for the purposes of this course, the main tool is direct enumeration. The matching number $M_{2k}$ satisfies

$M_{2k}=(2k-1)\times (2k-3)\times\ldots\times 3\times 1 = \frac{(2k)!}{2^k \cdot k!},$

and so Stirling’s approximation gives the asymptotics

$M_{2k} = (\sqrt{2}+o(1)) \left(\frac{2}{e}\right)^k k^k,$

although it will be useful to use the true bounds

$c \left(\frac{2}{e}\right)^k k^k \le M_{2k}\le C\left(\frac{2}{e}\right)^k k^k,\quad \forall k,$

instead in some places. Anyway, in $G^3(2n)$ , there are 6n half-edges in total, and so the probability that the graph may be split into two parts consisting of $2\ell,2m$ vertices, with $2\ell+2m=2n$ , and with no edges between the classes is $\frac{\binom{2n}{2\ell} M_{6\ell}M_{6m}}{M_{6n}}.$ Continue reading →

Random 3-regular graphs

Posted on November 1, 2017 by dominicyeo

A graph is d-regular if every vertex has degree d. Probably the easiest examples of d-regular graphs are the complete graph on (d+1) vertices, and the infinite d-ary tree. A less trivial example is the Petersen graph, which is 3-regular. 3-regular graphs will be the main focus for some of this post, but initially we lose nothing by considering general d.

Throughout, a necessary condition for the existence of a d-regular graph with N vertices is that at least one of d and N is even, as the sum of the degrees of a graph must be even. We will always assume that this holds, so that when d=3, we are always taking N to be even.

A natural pair of questions for a probabilist is ‘can we sample a d-regular graph with N vertices uniformly at random?’ and ‘what does a typical large d-regular graph look like?’

In a rather old post, I addressed some aspects of the first question, but revisit it briefly here. A good idea, due to Bollobas [B80] is to assign to all the vertices d stubs (or half-edges), and choose a matching of the Nd stubs uniformly at random. This works as a method to generate a random graph with any fixed degree sequence.

If you want your graphs to be simple, this can go wrong, because there’s a chance you get loops (that is, an edge from a vertex v to itself) and multiple edges between the same pair of vertices. It would be nice the graph formed in this fashion was simple with high probability when $N\rightarrow\infty$ . Unfortunately that’s not the case, however the probability that the graph is simple remains asymptotically bounded away from 0 and 1. Indeed, because the presence of a loop / multiple edge is asymptotically independent of the presence of a loop / multiple edge elsewhere, it’s unsurprising we have a Poisson limit for the number of such occurences. So from a sampling point of view, it’s reasonable to sample a graph in this way until you find a simple one. This takes O(1) steps, and it’s O(N) steps to check whether a given multigraph is simple.

It’s clear that conditional on the graph generated in this fashion being simple, its distribution is uniform on the set of simple graphs with the correct degree distribution. If you are happy for your graphs to have loops, then it’s a little bit more complicated, because if an edge has multiplicity k, these can appear in k! ways in the configuration construction.

Other asymptotic properties

Loops and multiple edges can be thought of as cycles of length 1 and 2 respectively if you want. We might ask about other small cycles. A calculation in expectation is relatively straightforward. Given three vertices, the probability they form a triangle (in at least one way) is $\Theta(N^{-3})$ , and there are $\Theta(N^3)$ ways to choose three vertices. Thus the expected number of triangles is $\Theta(1)$ . Finally, the edge structure induced on disjoint triples is asymptotically independent, and hence a Poisson limit. (See [J06] for details, including more detail on the general configuration construction.) The same result holds for the same reasons for cycles of any fixed finite length.

We might also ask about connectivity. At a heuristic level, there are two ways for the graph to be disconnected: it could have some small components; or it could have two components of size $\Theta(N)$ . The smallest possible component is $K_4$ , and an argument like for the cycles above shows that the number of copies of $K_4$ vanishes in expectation. Now, consider having two components of size roughly N/2. There are $\binom{N}{N/2} \sim 2^{2N}$ ways to make this choice. However, given such a choice, we can handle the probability that all the stubs from one class match within that class by going through the class one stub at a time:

$\frac{\frac{3N}{2}-1}{3N-1} \times \frac{\frac{3N}{2}-3}{3N-3} \times \cdots \times \frac{1}{\frac{3N}{2}+1}.$

We approximate this as

$\frac{\sqrt{(3N/2)!}}{\sqrt{ (3N)!}} \sim e^{3N/2} 2^{-3N/2} \left(3N\right)^{-3N/2},$

and this dominates the number of choices powerfully enough that we might believe it remains valid for a broader range of class sizes. In fact we have a much stronger statement, namely that G(N,3) is 3-connected with high probability. This means that the graph cannot be disconnected by removing two vertices, or equivalently that there are three vertex-disjoint paths between any pair of vertices in the graph, essentially one emerging from each stub. See this note by David Ellis for a quick proof. We might return to this later.

You might ask about planarity. It’s clear from degree consideration that there are no induced copies of $K_5$ in any random 3-regular graph, and since $K_{3,3}$ contains a cycle of length 4, and with high probability G(N,3) doesn’t, that takes care of that possibility too. However, there might be minors of this form. This seemed a good example of the Kuratowski criterion not actually being that useful, since I certainly don’t find the minors of the 3-regular graph an obvious structure to handle.

However, we can use Euler’s formula V – E + F = 2 for planar graphs. Here V = N, E = 3N/2. Faces are described by (a subset of the) cycles, and we there are asymptotically O(1) small cycles, so most faces include a large number of edges. But each edge corresponds to at most two faces. So we have $F \ll E$ , and so with high probability Euler’s formula can’t hold in G(N,3) for large N.

We can also ask about the local limit of G(N,3). Since the vertices are exchangeable, we don’t need to worry about whether we choose the root uniformly at random (often referred to as the Benjamini-Schramm sense) or by some other method.

The root has up to three neighbours, and with high probability it has exactly three neighbours. These neighbours have at most two other neighbours themselves. However, we’ve already seen that there are asymptotically O(1) cycles, and so with high probability there are no small cycles near a fixed root vertex. So the six neighbours-of-neighbours are with high probability different to the root and the root’s neighbours and to each other. We can make this argument at arbitrary finite radius from the root, to conclude that the local limit of G(N,3) is the infinite 3-ary tree.

Spectral expansion

[Caveat – this is something I read about and wanted to mention, but I really don’t know much at all about any of this theory, and it’s definitely not certain that what follows wouldn’t be better replaced by a set of links.]

This straightforward local limit offers good heuristics on some of the more global properties. Almost by definition, the d-ary tree expands as rapidly as is possible away from the root among infinite d-regular graphs. There are a number of ways to measure the expansion of a graph, and some methods transfer better to the infinite setting than others. The adjacency matrix of an infinite graph can be defined similarly to that of a finite graph, and it remains possible to talk about eigenfunctions and spectrum. As for the finite setting, d is an eigenvalue because the tree is d-regular, and -d is an eigenvalue because it is also bipartite.

The next largest eigenvalue $\lambda_2$ governs the spectral gap $d-\lambda_2$ which is a measure of the expansion of a graph. A graph is a good (spectral) expander if all the non-trivial eigenvalues are close to zero. A priori, all we know is that $|\lambda_2|\le d$ . For the infinite d-ary tree, we have $\lambda_2 = 2\sqrt{d-1}$ . This blog post by Luca Trevisan gives a very readable proof.

A key result is that finite graphs can have $\lambda_2 \le 2\sqrt{d-1}$ , but not asymptotically. That is, taking N to be the number of vertices:

$\lambda_2 \ge 2\sqrt{d-1} - o_N(1).$

This is the content of the Alon-Boppana theorem [Al86]. In fact the error can be quantified as $O(\frac{1}{\log N})$ – the diamater of the graph is relevant here. A finite d-regular graph for which $\lambda_2\le 2\sqrt{d-1}$ is called a Ramanujan graph. The existence of Ramanujan graphs has been much studied, and various constructions often rely on number theoretic properties of N, and lie at the interface of disparate branches of mathematics where my understanding is zero rather than epsilon.

Now return to our view of the d-ary tree as the local limit of a d-regular graph on N vertices for large N. We might expect from everything above that the uniform d-regular graph is a good expander. Bollobas shows that in the sense of edge-expansion, asymptotically almost all d-regular graphs have edge-expansion bounded away from zero. (See Section 2 of [Ell], including history of the d=3 case.) Friedman [Fri08] proves the conjecture of Alon that for every $\epsilon>0$ , a.a.s. $\lambda_2$ for G(N,d) is at most $2\sqrt{d-1}+\epsilon$ . In this sense, G(N,d) is asymptotically ‘almost Ramanujan’. (See also [Bor17] for another proof and an introduction including history, context and references.)

Some other links: The Wikipedia page on expanders, which includes a discussion of the different descriptions of expansion, and the Cheeger inequalities and other relations between them; slides for a talk by Spielman on spectra and Ramanujan graphs; a survey by Murty on Ramanujan graphs;.

What next?

This post took a slightly different direction from what I had intended, and rather than make a halting U-turn back to my planned finale, I’ll postpone this. However, a short overture is that I’m interested in the structure of critical components of random graphs during the critical window. This is the window during which the largest components first have cycles with probability $\Theta(1)$ . Indeed, the critical components have size $\Theta(N^{2/3})$ and $\Theta(1)$ surplus edges. Conditional on their size, and number of surplus edges, the choice of the graph structure on the component is uniform among such (connected) graphs.

Addario-Berry, Broutin and Goldschmidt [ABG09] study scaling limits of such components. Central to this analysis is the 2-core of such components, which can be described in terms of 3-regular (multi)graphs. Various processes we are now interested in running on the critical components of critical RGs can then be studied in terms of related processes on random 3-regular graphs.

References

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

[Al86] – Alon – Eigenvalues and expanders

[B80] – Bollobas – A probabilistic proof of an asymptotic formula for the number of labelled regular graphs

[B88] – Bollobas – The isoperimetric number of random regular graphs

[Bor17] – Bordenave – A new proof of Friedman’s second eigenvalue theorem and its extension to random lifts. Arxiv.

[Ell] – Ellis – The expansion of random regular graphs

[Fri08] – Friedman – A proof of Alon’s second eigenvalue conjecture and related problems

[J06] – Janson – The probability that a random multigraph is simple by

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Lecture 10 – the configuration model

Random 3-regular graphs