Graph limits and exchangeable random graphs

Introduction

DeFinetti’s profound contributions are now woven into many parts of probability, statistics and philosophy. Here we show how developments from deFinetti’s work on partial exchangeability have a direct link to the recent development of a limiting theory for large graphs. This introduction first recalls the theory of exchangeable arrays (Section 1.1). Then, the subject of graph limits is outlined (Section 1.2). Finally, the link between these ideas, which forms the bulk of this paper, is outlined (Section 1.3).

Let $\{X_{i}\}$, $1\leq i<\infty$, be a sequence of binary random variables. They are exchangeable if

for all $n$, permutations $\sigma\in\mathfrak{S}_{n}$ and all $e_{i}\in\{0,1\}$. The celebrated representation theorem says

If $\{X_{i}\}$, $1\leq i<\infty$, is a binary exchangeable sequence, then:

(i) With probability $1$, $X_{\infty}=\lim\frac{1}{n}(X_{1}+\cdots+X_{n})$ exists.

(ii) If $\mu(A)=P\{X_{\infty}\in A\}$, then for all $n$ and $e_{i}$, $1\leq i\leq n$,

It is natural to refine and extend deFinetti’s theorem to allow more general observables ($X_{i}$ with values in a Polish space) and other notions of symmetry (partial exchangeability). A definitive treatment of these developments is given in Kallenberg 2005. Of interest here is the extension of deFinetti’s theorem to two-dimensional arrays.

Let $\{X_{ij}\}$, $1\leq i,j<\infty$, be binary random variables. They are separately exchangeable if

for all $n$, all permutations $\sigma,\tau\in\mathfrak{S}_{n}$ and all $e_{ij}\in\{0,1\}$. They are (jointly) exchangeable if (1.2) holds in the special case $\tau=\sigma$.

The question of two-dimensional versions of deFinetti’s theorem under (separate) exchangeability arose from the statistical problems of two-way analysis of variance. Early workers expected a version of (1.1) with perhaps a two-dimensional integral. The probabilist David Aldous and the logician Douglas Hoover found that the answer is more complicated.

Define a random binary array $\{X_{ij}\}$ as follows: Let $U_{i},V_{j}$, $1\leq i,j<\infty$, be independent and uniform in $$. Let$W(x,y)$be a function from$^{2}$to$$. Let$X_{ij}$be$1$or$0$as a$W(U_{i},V_{j})$-coin comes up heads or tails. Let$P_{W}$be the probability distribution of$\{X_{ij}\}$,$1\leq i,j<\infty$. The family$\{X_{ij}\}$is separately exchangeable because of the symmetry of the construction. The Aldous–Hoover theorem says that any separately exchangeable binary array is a mixture of such$P_{W}$:

Let $X=\{X_{ij}\}$, $1\leq i,j<\infty$, be a separately exchangeable binary array. Then, there is a probability $\mu$ such that

There is a similar result for jointly exchangeable arrays.

The uniqueness of $\mu$ resisted understanding; if $\widehat{W}$ is obtained from $W$ by a measure-preserving change of each variable, clearly the associated process $\{\widehat{X}_{ij}\}$ has the same joint distribution as $\{X_{ij}\}$. Using model theory, Hoover 1979 was able to show that this was the only source of non-uniqueness. A ‘probabilist’s proof’ was finally found by Kallenberg, see [15, Sect. 7.6] for details and references.

These results hold for higher dimensional arrays with $X_{ij}$ taking values in a Polish space with minor change [15, Chap. 7]. The description above has not mentioned several elegant results of the theory. In particular, Kallenberg’s ‘spreadable’ version of the theory replaces invariance under a group by invariance under subsequences. A variety of tail fields may be introduced to allow characterizing when $W$ takes values in $\{0,1\}$ [10, Sect. 4]. Much more general notions of partial exchangeability are studed in .

2. Graph limits

Large graphs, both random and deterministic, abound in applications. They arise from the internet, social networks, gene regulation, ecology and in mathematics. It is natural to seek an approximation theory: What does it mean for a sequence of graphs to converge? When can a large complex graph be approximated by a small graph?

In a sequence of papers , Laszlo Lovász with coauthors (listed here in order of frequency) V. T. Sós, B. Szegedy, C. Borgs, J. Chayes, K. Vesztergombi, A. Schrijver, M. Freedman have developed a beautiful, unifying limit theory. This sheds light on topics such as graph homomorphisms, Szemeredi’s regularity lemma, quasi-random graphs, graph testing and extremal graph theory. Their theory has been developed for dense graphs (number of edges comparable with the square of number of vertices) but parallel theories for sparse graphs are beginning to emerge .

Roughly, a growing sequence of finite graphs $G_{n}$ converges if, for any fixed graph $F$, the proportion of copies of $F$ in $G_{n}$ converges. Section 2 below has precise definitions.

Define a probability distribution on graphs on $n$-vertices as follows. Flip a $\theta$-coin for each vertex (dividing vertices into ‘boys’ and ‘girls’). Connect two boys with probability $p$. Connect two girls with probability $p^{\prime}$. Connect a boy and a girl with probability $p^{\prime\prime}$. Thus, if $p=p^{\prime}=0$, $p^{\prime\prime}=1$, we have a random bipartite graph. If $p=p^{\prime}=1$, $p^{\prime\prime}=0$, we have two disjoint complete graphs. If $p=p^{\prime}=p^{\prime\prime}$, we have the Erdös–Renyi model. As $n$ grows, these models generate a sequence of random graphs which converge almost surely to a limiting object described below.

More substantial examples involving random threshold graphs are in .

If a sequence of graphs converges, what does it converge to? For exchangeable random graphs (defined below), there is a limiting object which may be thought of as a probability measure on infinite random graphs. Suppose $W(x,y)=W(y,x)$ is a function from $^{2}\to$. Choose $\{U_{i}\}$, $1\leq i<\infty$, independent uniformly distributed random variables on $$. Form an infinite random graph by putting an edge from$i$to$j$with probability$W(U_{i},U_{j})$. This measure on graphs (or alternatively$W$) is the limiting object.

For the “boys and girls” example above, $W$ may be pictured as

The theory developed shows that various properties of $G_{n}$ can be well approximated by calculations with the limiting object. There is an elegant characterization of these ‘continuous graph properties’ with applications to algorithms for graph testing (Does this graph contain an Eulerian cycle?) or parameter estimation (What is an approximation to the size of the maximum cut?). There is a practical way to find useful approximations to a large graph by graphs of fixed size . This paper also contains a useful review of the current state of the theory with proofs and references.

We have sketched the theory for unweighted graphs. There are generalizations to graphs with weights on vertices and edges, to bipartite, directed and hypergraphs. The sketch leaves out many nice developments. For example, the useful cut metric between graphs and connections to statistical physics .

3. Overview of the present paper

There is an apparent similarity between the measure $P_{W}$ of the Aldous–Hoover theorem and the limiting object $W$ from graph limits. Roughly, working with symmetric $W$ gives the graph limit theory; working with general $W$ gives directed graphs. The main results of this paper make these connections precise.

Basic definitions are in Section 2 which introduces a probabilist’s version of graph convergence equivalent to the definition using graph homomorphisms. Section 3 uses the well-established theory of weak convergence of a sequence of probability measures on a metric space to get properties of graph convergence. Section 4 carries things over to infinite graphs.

The main results appear in Section 5. This introduces exchangeable random graphs and gives a one-to-one correspondence between infinite exchangeable random graphs and distributions on the space of proper graph limits (Theorem 5.3), which specializes to a one-to-one correspondence between proper graph limits and extreme points in the set of distributions of exchangeable random graphs (Corollary 5.4).

A useful characterization of the extreme points of the set of exchangeable random graphs is in Theorem 5.5. These results are translated to the equivalence between proper graph limits and the Aldous–Hoover theory in Section 6. The non-uniqueness of the representing $W$, for exchangeable random graphs and for graph limits, is discussed in Section 7.

The equivalence involves symmetric $W(x,y)$ and a single permutation $\sigma$ taking $W(U_{i},U_{j})$ to $W(U_{\sigma(i)},U_{\sigma(j)})$. The original Aldous–Hoover theorem, with perhaps non-symmetric $W(x,y)$ and $W(U_{i},V_{j})$ to $W(U_{\sigma(i)},V_{\tau(j)})$ translates to a limit theorem for bipartite graphs. This is developed in Section 8. The third case of the Aldous–Hoover theory for two-dimensional arrays, perhaps non-symmetric $W(x,y)$ and a single permutation $\sigma$, corresponds to directed graphs; this is sketched in Section 9.

The extensions to weighted graphs are covered by allowing $X_{ij}$ to take general values in the Aldous–Hoover theory. The extension to hypergraphs follows from the Aldous–Hoover theory for higher-dimensional arrays. (The details of these extensions are left to the reader.)

Despite these parallels, the theories have much to contribute to each other. The algorithmic, graph testing, Szemeredi partitioning perspective is new to exchangeability theory. Indeed, the “boys and girls” random graph was introduced to study the psychology of vision in Diaconis–Freedman (1981). As far as we know, its graph theoretic properties have not been studied. The various developments around shell-fields in exchangeability, which characterize zero/one $W(x,y)$, have yet to be translated into graph-theoretic terms.

This lecture is an extended version of a talk presented by PD at the 100th anniversary of deFinetti’s birth in Rome, 2006. We thank the organizers. This work was partially funded by the French ANR’s Chaire d’excellence grant to PD.

SJ thanks Christian Borgs and Jennifer Chayes for inspiration from lectures and discussions during the Oberwolfach meeting ‘Combinatorics, Probability and Computing’, held in November, 2006. Parts of the research were completed during a visit by SJ to the Université de Nice - Sophia Antipolis in January 2007.

Definitions and basic properties

All graphs will be simple. Infinite graphs will be important in later sections, but will always be clearly stated to be infinite; otherwise, graphs will be finite. We denote the vertex and edge sets of a graph $G$ by $V(G)$ and $E(G)$, and the numbers of vertices and edges by $v(G):=|V(G)|$ and $e(G):=|E(G)|$. We consider both labelled and unlabelled graphs; the labels will be the integers $1,\dots,n$, where $n$ is the number of vertices in the graph. A labelled graph is thus a graph with vertex set $[n]:=\{1,\dots,n\}$ for some $n\geq 1$; we let ${\mathcal{L}}_{n}$ denote the set of the $2^{\binom{n}{2}}$ labelled graphs on $[n]$ and let ${\mathcal{L}}:=\bigcup_{n=1}^{\infty}{\mathcal{L}}_{n}$. An unlabelled graph can be regarded as a labelled graph where we ignore the labels; formally, we define ${\mathcal{U}}_{n}$, the set of unlabelled graphs of order $n$, as the quotient set ${\mathcal{L}}_{n}/\cong$ of labelled graphs modulo isomorphisms. We let ${\mathcal{U}}:=\bigcup_{n=1}^{\infty}{\mathcal{U}}_{n}={\mathcal{L}}/\cong$, the set of all unlabelled graphs.

Note that we can, and often will, regard a labelled graph as an unlabelled graph.

If $G$ is an (unlabelled) graph and $v_{1},\dots,v_{k}$ is a sequence of vertices in $G$, then $G(v_{1},\dots,v_{k})$ denotes the labelled graph with vertex set $[k]$ where we put an edge between $i$ and $j$ if $v_{i}$ and $v_{j}$ are adjacent in $G$. We allow the possibility that $v_{i}=v_{j}$ for some $i$ and $j$. (In this case, there is no edge $ij$ because there are no loops in $G$.)

We let $G[k]$, for $k\geq 1$, be the random graph $G(v_{1},\dots,v_{k})$ obtained by sampling $v_{1},\dots,v_{k}$ uniformly at random among the vertices of $G$, with replacement. In other words, $v_{1},\dots,v_{k}$ are independent uniformly distributed random vertices of $G$.

For $k\leq v(G)$, we further let $G[k]^{\prime}$ be the random graph $G(v^{\prime}_{1},\dots,v^{\prime}_{k})$ where we sample $v^{\prime}_{1},\dots,v^{\prime}_{k}$ uniformly at random without replacement; the sequence $v^{\prime}_{1},\dots,v^{\prime}_{k}$ is thus a uniformly distributed random sequence of $k$ distinct vertices.

The graph limit theory in and subsequent papers is based on the study of the functional $t(F,G)$ which is defined for two graphs $F$ and $G$ as the proportion of all mappings $V(F)\to V(G)$ that are graph homomorphisms $F\to G$, i.e., map adjacent vertices to adjacent vertices. In probabilistic terms, $t(F,G)$ is the probability that a uniform random mapping $V(F)\to V(G)$ is a graph homomorphism. Using the notation introduced above, we can, equivalently, write this as, assuming that $F$ is labelled and $k=v(F)$,

Note that both $F$ and $G[k]$ are graphs on $[k]$, so the relation $F\subseteq G[k]$ is well-defined as containment of labelled graphs on the same vertex set, i.e. as $E(F)\subseteq E(G[k])$. Although the relation $F\subseteq G[k]$ may depend on the labelling of $F$, the probability in (2.1) does not, by symmetry, so $t(F,G)$ is really well defined by (2.1) for unlabelled $F$ and $G$.

With $F$, $G$ and $k$ as in (2.1), we further define, again following (and the notation of ) but stating the definitions in different but equivalent forms,

Since the probability that a random sample $v_{1},\dots,v_{k}$ of vertices in $G$ contains some repeated vertex is $\leq k^{2}/(2v(G))$, it follows that

The basic definition of Lovász and Szegedy and Borgs, Chayes, Lovász, Sós and Vesztergombi is that a sequence $(G_{n})$ of graphs converges if $t(F,G_{n})$ converges for every graph $F$. We can express this by considering the map $\tau:{\mathcal{U}}\to^{{\mathcal{U}}}$ defined by

Then $(G_{n})$ converges if and only if $\tau(G_{n})$ converges in $^{{\mathcal{U}}}$, equipped with the usual product topology. Note that $^{{\mathcal{U}}}$ is a compact metric space; as is well known, a metric can be defined by, for example,

where $F_{1},F_{2},\dots$ is some enumeration of all unlabelled graphs.

We define ${\mathcal{U}}^{*}:=\tau({\mathcal{U}})\subseteq^{{\mathcal{U}}}$ to be the image of ${\mathcal{U}}$ under this mapping $\tau$, and let $\overline{{\mathcal{U}}^{*}}$ be the closure of ${\mathcal{U}}^{*}$ in $^{{\mathcal{U}}}$. Thus $\overline{{\mathcal{U}}^{*}}$ is a compact metric space. (For explicit descriptions of the subset $\overline{{\mathcal{U}}^{*}}$ of $^{{\mathcal{U}}}$ as a set of graph functionals, see Lovász and Szegedy .)

As pointed out in and (in equivalent terminology), $\tau$ is not injective; for example, $\tau(K_{n,n})$ is the same for all complete bipartite graphs $K_{n,n}$. Nevertheless, as in and , we can consider a graph $G$ as an element of ${\mathcal{U}}^{*}$ by identifying $G$ and $\tau(G)$ (thus identifying graphs with the same $\tau(G)$), and then convergence of $(G_{n})$ as defined above is equivalent to convergence in $\overline{{\mathcal{U}}^{*}}$. The limit is thus an element of $\overline{{\mathcal{U}}^{*}}$, but typically not a graph in ${\mathcal{U}}^{*}$. The main result of Lovász and Szegedy is a representation of the elements in $\overline{{\mathcal{U}}^{*}}$ to which we will return in Section 6.

As said above, $\overline{{\mathcal{U}}^{*}}$ is a compact metric space, and it can be given several equivalent metrics. One metric is the metric (2.8) inherited from $^{{\mathcal{U}}}$, which for graphs becomes $d(G,G^{\prime})=\sum_{i}2^{-i}|t(F_{i},G)-t(F_{i},G^{\prime})|$. Another metric, shown by Borgs, Chayes, Lovász, Sós and Vesztergombi to be equivalent, is the cut-distance $\delta_{\square}$, see for definitions. Further characterizations of convergence of sequences of graphs in $\overline{{\mathcal{U}}}$ are given in Borgs, Chayes, Lovász, Sós and Vesztergombi 2007+, Borgs, Chayes, Lovász, Sós and Vesztergombi 2007+.

The identification of graphs with the same image in ${\mathcal{U}}^{*}$ (i.e., with the same $t(F,\cdot)$ for all $F$) is sometimes elegant but at other times inconvenient. It can be avoided if we instead let ${\mathcal{U}}^{+}$ be the union of ${\mathcal{U}}$ and some one-point set $\{*\}$ and consider the mapping $\tau^{+}:{\mathcal{U}}\to^{{\mathcal{U}}^{+}}=^{{\mathcal{U}}}\times$ defined by

Consequently, we can identify ${\mathcal{U}}$ with its image $\tau^{+}({\mathcal{U}})\subseteq^{{\mathcal{U}}^{+}}$ and define $\overline{{\mathcal{U}}}\subseteq^{{\mathcal{U}}^{+}}$ as its closure. It is easily seen that a sequence $(G_{n})$ of graphs converges in $\overline{{\mathcal{U}}}$ if and only if either $v(G_{n})\to\infty$ and $(G_{n})$ converges in $\overline{{\mathcal{U}}^{*}}$, or the sequence $(G_{n})$ is constant from some $n_{0}$ on. Hence, convergence in $\overline{{\mathcal{U}}}$ is essentially the same as the convergence considered by by Lovász and Szegedy , but without any identification of non-isomorphic graphs of different orders.

We will in the sequel prefer to use $\overline{{\mathcal{U}}}$ rather than $\overline{{\mathcal{U}}^{*}}$, thus not identifying some graphs of different orders, nor identifying finite graphs with some limit objects in ${\mathcal{U}}_{\infty}$.

We summarize the results above on convergence.

A sequence $(G_{n})$ of graphs converges in the sense of Lovász and Szegedy if and only if it converges in the compact metric space $\overline{{\mathcal{U}}^{*}}$. Moreover, if $v(G_{n})\to\infty$, the sequence $(G_{n})$ converges in this sense if and only if it converges in $\overline{{\mathcal{U}}}$.

The projection $\pi:^{{\mathcal{U}}^{+}}=^{{\mathcal{U}}}\times\to^{{\mathcal{U}}}$ maps $\tau^{+}(G)$ to $\tau(G)$ for every graph $G$, so by continuity it maps $\overline{{\mathcal{U}}}$ into $\overline{{\mathcal{U}}^{*}}$. For graph $G\in{\mathcal{U}}$, $\pi(G)=\tau(G)$ is the object in $\overline{{\mathcal{U}}^{*}}$ corresponding to $G$ considered above, and we will in the sequel denote this object by $\pi(G)$; recall that this projection ${\mathcal{U}}\to\overline{{\mathcal{U}}^{*}}$ is not injective. (We thus distinguish between a graph $G$ and its “ghost” $\pi(G)$ in $\overline{{\mathcal{U}}^{*}}$. Recall that when graphs are considered as elements of $\overline{{\mathcal{U}}^{*}}$ as in and , certain graphs are identified with each other; we avoid this.) On the other hand, an element $G$ of $\overline{{\mathcal{U}}}$ is by definition determined by $\tau(G)$ and $v(G)^{-1}$, cf. (2.9), so the restriction $\pi:{\mathcal{U}}_{n}\to\overline{{\mathcal{U}}^{*}}$ is injective for each $n\leq\infty$. In particular, $\pi:{\mathcal{U}}_{\infty}\to\overline{{\mathcal{U}}^{*}}$ is injective. Moreover, this map is surjective because every element $G\in\overline{{\mathcal{U}}^{*}}$ is the limit of some sequence $(G_{n})$ of graphs in ${\mathcal{U}}$ with $v(G_{n})\to\infty$; by Theorem 2.1, this sequence converges in $\overline{{\mathcal{U}}}$ to some element $G^{\prime}$, and then $\pi(G^{\prime})=G$. Since ${\mathcal{U}}_{\infty}$ is compact, the restriction of $\pi$ to ${\mathcal{U}}_{\infty}$ is thus a homeomorphism, and we have the following theorem, saying that we can identify the set ${\mathcal{U}}_{\infty}$ of proper graph limits with $\overline{{\mathcal{U}}^{*}}$.

The projection $\pi$ maps the set ${\mathcal{U}}_{\infty}:=\overline{{\mathcal{U}}}\setminus{\mathcal{U}}$ of proper graph limits homeomorphically onto $\overline{{\mathcal{U}}^{*}}$.

Convergence of random graphs

A random unlabelled graph is a random element of ${\mathcal{U}}$ (with any distribution; we do not imply any particular model). We consider convergence of a sequence $(G_{n})$ of random unlabelled graphs in the larger space ${\mathcal{U}}$; recall that this is a compact metric space so we may use the general theory set forth in, for example, Billingsley .

We begin with convergence in distribution.

For every finite family $F_{1},\dots,F_{m}$ of (non-random) graphs, the random variables $t(F_{1},G_{n}),\dots,t(F_{m},G_{n})$ converge jointly in distribution.

For every (non-random) $F\in{\mathcal{U}}$, the random variables $t(F,G_{n})$ converge in distribution.

(iii)$\implies$(iv). Immediate, since $t$ is bounded (by 1).

Specializing to the case of a non-random limit $G\in{\mathcal{U}}_{\infty}$, we obtain the corresponding result for convergence in probability.

We observe another corollary to Theorem 3.1 (and its proof).

If $\Gamma$ is a random element of ${\mathcal{U}}_{\infty}=\overline{{\mathcal{U}}}\setminus{\mathcal{U}}\cong\overline{{\mathcal{U}}^{*}}$, then, for every sequence $F_{1},\dots,F_{m}$ of graphs, possibly with repetitions,

Convergence to infinite graphs

It is sometimes convenient to regard ${\mathcal{L}}_{n}$ for a finite $n$ as a subset of ${\mathcal{L}}_{\infty}$: we can identify graphs in ${\mathcal{L}}_{n}$ and ${\mathcal{L}}_{\infty}$ with the same edge set. In other words, if $G\in{\mathcal{L}}_{n}$ is a graph with vertex set $[n]$, we add an infinite number of isolated vertices $n+1,n+2,\dots$ to obtain a graph in ${\mathcal{L}}_{\infty}$.

Conversely, if $H\in{\mathcal{L}}_{\infty}$ is an infinite graph, we let $H|_{[n]}\in{\mathcal{L}}_{n}$ be the induced subgraph of $H$ with vertex set $[n]$.

If $G$ is a (finite) graph, let $\widehat{G}$ be the random labelled graph obtained by a random labelling of the vertices of $G$ by the numbers $1,\dots,v(G)$. (If $G$ is labelled, we thus ignore the labels and randomly relabel.) Thus $\widehat{G}$ is a random finite graph with the same number of vertices as $G$, but as just said, we can (and will) also regard $\widehat{G}$ as a random graph in ${\mathcal{L}}_{\infty}$.

We use the same notation $\widehat{G}$ also for a random (finite) graph $G$ given a random labelling.

Let $G$ be a labelled graph and consider the graph $\widehat{G}|_{[k]}$, assuming $k\leq v(G)$. This random graph equals $G[k]^{\prime}=G(v^{\prime}_{1},\dots,v^{\prime}_{k})$, where $v^{\prime}_{1},\dots,v^{\prime}_{k}$ are $k$ vertices sampled at random without replacement as in Section 2. Hence, by (2.3), for every $F\in{\mathcal{L}}_{k}$,

Applied to the random graph $G_{n}$, this yields

for every $k\geq 1$ and every $F\in{\mathcal{L}}_{k}$.

Exchangeable random graphs

Equivalently, if $X_{ij}:=\boldsymbol{1}[ij\in H]$ is the indicator of there being an edge $ij$ in $H$, then the array $\{X_{ij}\}$, $1\leq i,j\leq\infty$, is (jointly) exchangeable as defined in Section 1.

Let $H$ be a random infinite graph in ${\mathcal{L}}_{\infty}$. Then the following are equivalent.

$H|_{[k]}$ has a distribution invariant under all permutations of $[k]$, for every $k\geq 1$.

The limit $H$ is Theorem 4.1 is exchangeable.

Moreover, Theorem 4.1 implies the following connection with random elements of ${\mathcal{U}}_{\infty}$.

There is a one-to-one correspondence between distributions of random elements $\Gamma\in{\mathcal{U}}_{\infty}$ (or $\overline{{\mathcal{U}}^{*}}$) and distributions of exchangeable random infinite graphs $H\in{\mathcal{L}}_{\infty}$ given by

There is a one-to-one correspondence between elements $\Gamma$ of ${\mathcal{U}}_{\infty}\cong\overline{{\mathcal{U}}^{*}}$ and extreme points of the set of distributions of exchangeable random infinite graphs $H\in{\mathcal{L}}_{\infty}$. This correspondence is given by

The extreme points of the set of distributions on ${\mathcal{U}}_{\infty}$ are the point masses, which are in one-to-one correspondence with the elements of ${\mathcal{U}}_{\infty}$. ∎

We can characterize these extreme point distributions of exchangeable random infinite graphs as follows.

Let $H$ be an exchangeable random infinite graph. Then the following are equivalent.

The distribution of $H$ is an extreme point in the set of exchangeable distributions in ${\mathcal{L}}_{\infty}$.

The restrictions $H|_{[k]}$ and $H|_{[k+1,\infty)}$ are independent for every $k$.

Let $\mathcal{F}_{n}$ be the $\sigma$-field generated by $H|_{[n,\infty)}$. Then the tail $\sigma$-field $\bigcap_{n=1}^{\infty}\mathcal{F}_{n}$ is trivial, i.e., contains only events with probability $0$ or $1$.

(i)$\implies$(ii). By Corollary 5.4, $H$ corresponds to some (non-random) $\Gamma\in{\mathcal{U}}_{\infty}$ such that

Furthermore, since $\Gamma$ is non-random, Corollary 3.3 yields $t(F_{1}\cup F_{2},\Gamma)=t(F_{1},\Gamma)t(F_{2},\Gamma)$. Hence,

(ii)$\implies$(iii). By inclusion–exclusion, as for (2.3), (ii) implies that if $1\leq k<l<\infty$, then for any graphs $F_{1}$ and $F_{2}$ with $V(F_{1})=\{1,\dots,k\}$ and $V(F_{2})=\{k+1,\dots,k+l\}$, the events $H|_{[k]}=F_{1}$ and $H|_{\{k+1,\dots,l\}}=F_{2}$ are independent. Hence $H|_{[k]}$ and $H|_{\{k,\dots,l\}}$ are independent for every $l>k$, and the result follows.

(iv)$\implies$(i). Let $F\in{\mathcal{L}}_{k}$ for some $k$ and let $F_{n}$ be $F$ with all vertices shifted by $n$. Consider the two indicators $I=\boldsymbol{1}[H\supseteq F]$ and $I_{n}=\boldsymbol{1}[H\supseteq F_{n}]$. Since $I_{n}$ is $\mathcal{F}_{n}$-measurable,

Let $\Gamma$ be a random element of ${\mathcal{U}}_{\infty}$ corresponding to $H$ as in Theorem 5.3. By (5.2) and (3.2), (5.6) can be written

Hence the random variable $t(F,\Gamma)$ has variance 0 so it is a.s. constant. Since this holds for every $F\in{\mathcal{L}}$, it follows that $\Gamma$ is a.s. constant, i.e., we can take $\Gamma$ non-random, and (i) follows by Corollary 5.4. ∎

Representations of graph limits and exchangeable graphs

As said in the introduction, the exchangeable infinite random graphs were characterized by Aldous and Hoover , see also Kallenberg , and the graph limits in ${\mathcal{U}}_{\infty}\cong\overline{{\mathcal{U}}^{*}}$ were characterized in a very similar way by Lovász and Szegedy . We can now make the connection between these two characterizations explicit.

Let $\mathcal{W}$ be the set of all measurable functions $W:^{2}\to$ and let $\mathcal{W}_{\mathsf{s}}$ be the subset of symmetric functions. For every $W\in\mathcal{W}_{\mathsf{s}}$, we define an infinite random graph $G(\infty,W)\in{\mathcal{L}}_{\infty}$ as follows: we first choose a sequence $X_{1},X_{2},\dots$ of i.i.d. random variables uniformly distributed on $$, and then, given this sequence, for each pair$(i,j)$with$i$we draw an edge$ij$with probability$W(X_{i},X_{j})$, independently for all pairs$(i,j)$with$i$(conditionally given$\{X_{i}\}$). Further, let$G(n,W)$be the restriction$G(\infty,W)|_{[n]}$, which is obtained by the same construction with a finite sequence$X_{1},\dots,X_{n}$.

It is evident that $G(\infty,W)$ is an exchangeable infinite random graph, and the result by Aldous and Hoover is that every exchangeable infinite random graph is obtained as a mixture of such $G(\infty,W)$; in other words as $G(\infty,W)$ with a random $W$.

Considering again a deterministic $W\in\mathcal{W}_{\mathsf{s}}$, it is evident that Theorem 5.5(ii) holds, and thus Theorem 5.5 and Corollary 5.4 show that $G(\infty,W)$ corresponds to an element $\Gamma_{W}\in{\mathcal{U}}_{\infty}$. Moreover, by Theorem 5.3 and Remark 5.1, $G(n,W)\to\Gamma_{W}$ a.s. as ${n\to\infty}$, and (5.3) shows that if $F\in{\mathcal{L}}_{k}$, then

The main result of Lovász and Szegedy is that every element of ${\mathcal{U}}_{\infty}\cong\overline{{\mathcal{U}}^{*}}$ can be obtained as $\Gamma_{W}$ satisfying (6.1) for some $W\in\mathcal{W}_{\mathsf{s}}$.

It is now clear that the representation theorems of Aldous–Hoover and Lovász and Szegedy are connected by Theorem 5.3 and Corollary 5.4 above, and that one characterization easily follows from the other.

The representations by $W$ are far from unique, see Section 7. Borgs, Chayes, Lovász, Sós and Vesztergombi call an element $W\in\mathcal{W}_{\mathsf{s}}$ a graphon. They further define a pseudometric (called the cut-distance) on $\mathcal{W}_{\mathsf{s}}$ and show that if we consider the quotient space $\widehat{\mathcal{W}}_{\mathsf{s}}$ obtained by identifying elements with cut-distance 0, we obtain a compact metric space, and the mapping $W\mapsto\Gamma_{W}$ yields a bijection $\widehat{\mathcal{W}}_{\mathsf{s}}\to\overline{{\mathcal{U}}^{*}}\cong{\mathcal{U}}_{\infty}$, which furthermore is a homeomorphism.

As remarked in Lovász and Szegedy , we can more generally consider a symmetric measurable function $W:{\mathcal{S}}^{2}\to$ for any probability space $({\mathcal{S}},\mu)$, and define $G(\infty,W)$ as above with $X_{i}$ i.i.d. random variables in ${\mathcal{S}}$ with distribution $\mu$. This does not give any new limit objects $G(\infty,W)$ or $\Gamma_{W}$, since we just said that every limit object is obtained from some $W\in\mathcal{W}_{\mathsf{s}}$, but they can sometimes give useful representations.

An interesting case is when $W$ is the adjacency matrix of a (finite) graph $G$, with ${\mathcal{S}}=V(G)$ and $\mu$ the uniform measure on ${\mathcal{S}}$; we thus let $X_{i}$ be i.i.d. random vertices of $G$ and $G(n,W)$ equals the random graph $G[n]$ defined in Section 2. It follows from (6.1) and (2.1) that $t(F,\Gamma_{W})=t(F,G)$ for every $F\in{\mathcal{U}}$, and thus $\Gamma_{W}=G$ as elements of $\overline{{\mathcal{U}}^{*}}$. In other words, $\Gamma_{W}\in{\mathcal{U}}_{\infty}=\pi(G)$, the “ghost” of $G$ in ${\mathcal{U}}_{\infty}\cong\overline{{\mathcal{U}}^{*}}$.

For the asymptotic behavior of $G(n,W)$ in another, sparse, case, with $W$ depending on $n$, see .

Non-uniqueness

The functions $W$ on $^{2}$ used to represent graph limits or exchangeable arrays are far from unique. (For a special case when there is a natural canonical choice, which much simplifies and helps applications, see .) For example, it is obvious that if $\varphi:\to$ is any measure preserving map, then $W$ and $W\circ\varphi$, defined by $W\circ\varphi(x,y):=W\bigl(\varphi(x),\varphi(y)\bigr)$, define the same graph limit and the same (in distribution) exchangeable array.

Although in principle, this is the only source on non-uniqueness, the details are more complicated, mainly because the measure preserving map $\varphi$ does not have to be a bijection, and thus the relation $W^{\prime}=W\circ\varphi$ is not symmetric: it can hold without there being a measure preserving map $\varphi^{\prime}$ such that $W=W^{\prime}\circ\varphi^{\prime}$. (For a 1-dimensional example, consider $f(x)=x$ and $f^{\prime}(x)=\varphi(x)=2x\bmod 1$; for a 2-dimensional example, let $W(x,y)=f(x)f(y)$ and $W^{\prime}(x,y)=f^{\prime}(x)f^{\prime}(y)$.)

For exchangeable arrays, the equivalence problem was solved by Hoover 1979, who gave a criterion which in our case reduces to (vi) below; this criterion involves an auxiliary variable, and can be interpreted as saying $W=W^{\prime}\circ\varphi^{\prime}$ for a random $\varphi^{\prime}$. This work was continued by Kallenberg, see , who gave a probabilistic proof and added criterion (v). For graph limits, Borgs, Chayes, Lovász, Sós and Vesztergombi 2007+ gave the criterion (vii) in terms of the cut-distance, and Bollobás and Riordan 2007+ found the criterion (v) in this context. Further, Borgs, Chayes, Lovász, Sós and Vesztergombi 2007+ announced the related criterion that there exists a measurable function $U:^{2}\to$ and two measure preserving maps $\varphi,\varphi^{\prime}:\to$ such that $W=U\circ\varphi$ and $W^{\prime}=U\circ\varphi^{\prime}$ a.e.; the proof of this will appear in .

As in Section 6, these two lines of work are connected by the results in Section 5, and we can combine the previous results as follows.

Let $W,W^{\prime}\in\mathcal{W}_{\mathsf{s}}$. Then the following are equivalent.

$\Gamma_{W}=\Gamma_{W^{\prime}}$ for the graph limits $\Gamma_{W},\Gamma_{W^{\prime}}\in{\mathcal{U}}_{\infty}$.

$t(F,\Gamma_{W})=t(F,\Gamma_{W^{\prime}})$ for every graph $F$.

The exchangeable random infinite graphs $G(\infty,W)$ and $G(\infty,W^{\prime})$ have the same distribution.

The random graphs $G(n,W)$ and $G(n,W^{\prime})$ have the same distribution for every finite $n$.

There exist measure preserving maps $\varphi,\varphi^{\prime}:\to$ such that $W\circ\varphi=W\circ\varphi^{\prime}$ a.e. on $^{2}$, i.e., $W\bigl(\varphi(x),\varphi(y)\bigr)=W^{\prime}\bigl(\varphi^{\prime}(x),\varphi^{\prime}(y)\bigr)$ a.e.

There exists a measure preserving map $\psi:^{2}\to$ such that $W(x_{1},x_{2})=W^{\prime}\bigl(\psi(x_{1},y_{1}),\psi(x_{2},y_{2})\bigr)$ a.e. on $^{4}$.

$\delta_{\square}(W,W^{\prime})=0$, where $\delta_{\square}$ is the cut-distance defined in .

(i)$\iff$(ii). By our definition of ${\mathcal{U}}_{\infty}\subset\overline{{\mathcal{U}}}$.

(iii)$\implies$(v). The general form of the representation theorem as stated in [15, Theorem 7.15, see also p. 304] is (in our two-dimensional case) $X_{ij}=f(\xi_{\emptyset},\xi_{i},\xi_{j},\xi_{ij})$ for a function $f:^{4}\to$, symmetric in the two middle variables, and independent random variables $\xi_{\emptyset}$, $\xi_{i}$ ($1\leq i$) and $\xi_{ij}$ ($1\leq i<j$), all uniformly distributed on , and where we further let $\xi_{ji}=\xi_{ij}$ for $j>i$. We can write the construction of $G(\infty,W)$ in this form with

Note that this $f$ does not depend on $\xi_{\emptyset}$. (In general, $\xi_{\emptyset}$ is needed for the case of a random $W$, which can be written as a deterministic function of $\xi_{\emptyset}$, but this is not needed in the present theorem.)

are a.s. equal. Conditioned on $\xi_{\emptyset},\xi_{1}$ and $\xi_{2}$, the random variable $g_{k,2}(\xi_{\emptyset},\xi_{1},\xi_{2},\xi_{12})$ is uniformly distributed on $$, and it follows (e.g., by taking the conditional expectation) that a.s.

For a.e. value $x_{0}$ of $\xi_{\emptyset}$, this thus holds for a.e. values of $\xi_{1}$ and $\xi_{2}$, and we may choose $\varphi(x)=g_{1,1}(x_{0},x)$ and $\varphi^{\prime}(x):=g_{2,1}(x_{0},x)$ for some such $x_{0}$.

(iii)$\iff$(vi). Similar, using [15, Theorem 7.28(iii)].

Bipartite graphs

The definitions and results above have analogues for bipartite graphs, which we give in this section, leaving some details to the reader. The proofs are straightforward analogues of the ones given above and are omitted. Applications of the results of this section to random difference graphs are in .

A bipartite graph will be a graph with an explicit bipartition; in other words, a bipartite graph $G$ consists of two vertex sets $V_{1}(G)$ and $V_{2}(G)$ and an edge set $E(G)\subseteq V_{1}(G)\times V_{2}(G)$; we let $v_{1}(G):=|V_{1}(G)|$ and $v_{2}(G):=|V_{2}(G)|$ be the numbers of vertices in the two sets. Again we consider both the labelled and unlabelled cases; in the labelled case we assume the labels of the vertices in $V_{j}(G)$ are $1,\dots,v_{j}(G)$ for $j=1,2$. Let $\mathcal{B}^{L}_{n_{1}n_{2}}$ be the set of the $2^{n_{1}n_{2}}$ labelled bipartite graphs with vertex sets $[n_{1}]$ and $[n_{2}]$, and let $\mathcal{B}_{n_{1}n_{2}}$ be the quotient set $\mathcal{B}^{L}_{n_{1}n_{2}}/\cong$ of unlabelled bipartite graphs with $n_{1}$ and $n_{2}$ vertices in the two parts; further, let $\mathcal{B}^{L}:=\bigcup_{n_{1},n_{2}\geq 1}\mathcal{B}^{L}_{n_{1}n_{2}}$ and $\mathcal{B}:=\bigcup_{n_{1},n_{2}\geq 1}\mathcal{B}_{n_{1}n_{2}}$.

In analogy with (2.7), we now define $\tau:\mathcal{B}\to^{\mathcal{B}}$ by

We define $\mathcal{B}^{*}:=\tau(\mathcal{B})\subseteq^{\mathcal{B}}$ to be the image of $\mathcal{B}$ under this mapping $\tau$, and let $\overline{\mathcal{B}^{*}}$ be the closure of $\mathcal{B}^{*}$ in $^{\mathcal{B}}$; this is a compact metric space.

Again, $\tau$ is not injective; we may consider a graph $G$ as an element of $\mathcal{B}^{*}$ by identifying $G$ and $\tau(G)$, but this implies identification of some graphs of different orders and we prefer to avoid it. We let $\mathcal{B}^{+}$ be the union of $\mathcal{B}$ and some two-point set $\{*_{1},*_{2}\}$ and consider the mapping $\tau^{+}:\mathcal{B}\to^{\mathcal{B}^{+}}=^{\mathcal{B}}\times\times$ defined by

Then $\tau^{+}$ is injective and we can identify $\mathcal{B}$ with its image $\tau^{+}(\mathcal{B})\subseteq^{\mathcal{B}^{+}}$ and define $\overline{\mathcal{B}}\subseteq^{\mathcal{B}^{+}}$ as its closure; this is a compact metric space.

Note that in the bipartite case there are other limit objects too in $\overline{\mathcal{B}}$; in fact, $\overline{\mathcal{B}}$ can be partitioned into $\mathcal{B}$, $\mathcal{B}_{\infty\infty}$, and the sets $\mathcal{B}_{n\infty}$, $\mathcal{B}_{\infty n}$, for $n=1,2,\dots$, where, for example, $\mathcal{B}_{n_{1}\infty}$ is the set of limits of sequences $(G_{n})$ of bipartite graphs such that $v_{2}(G_{n})\to\infty$ but $v_{1}(G_{n})=n_{1}$ is constant. We will not consider such degenerate limits further here, but we remark that in the simplest case $n_{1}=1$, a bipartite graph in $\mathcal{B}^{L}_{1n_{2}}$ can be identified with a subset of $[n_{2}]$, and an unlabelled graph in $\mathcal{B}_{1n_{2}}$ thus with a number in $m\in\{0,\dots,n_{2}\}$, the number of edges in the graph, and it is easily seen that a sequence of such unlabelled graphs with $n_{2}\to\infty$ converges in $\overline{\mathcal{B}}$ if and only if the proportion $m/n_{2}$ converges; hence we can identify $\mathcal{B}_{1\infty}$ with the interval .

We have the following basic result, cf. Theorem 2.1.

Let $(G_{n})$ be a sequence of bipartite graphs with $v_{1}(G_{n})$, $v_{2}(G_{n})\to\infty$. Then the following are equivalent.

$t(F,G_{n})$ converges for every $F\in\mathcal{B}$.

$G_{n}$ converges in $\overline{\mathcal{B}}$.

For convergence of random unlabelled bipartite graphs, the results in Section 3 hold with trivial changes.

For every finite family $F_{1},\dots,F_{m}$ of (non-random) bipartite graphs, the random variables $t(F_{1},G_{n}),\dots,t(F_{m},G_{n})$ converge jointly in distribution.

For every (non-random) $F\in\mathcal{B}$, the random variables $t(F,G_{n})$ converge in distribution.

If $G$ is a bipartite graph, let $\widehat{G}$ be the random labelled bipartite graph obtained by random labellings of the vertices in $V_{j}(G)$ by the numbers $1,\dots,v_{j}(G)$, for $j=1,2$. This is a random finite bipartite graph, but we can also regard it as a random element of $\mathcal{B}^{L}_{\infty\infty}$ by adding isolated vertices.

A random infinite bipartite graph $H\in\mathcal{B}^{L}_{\infty\infty}$ is exchangeable if its distribution is invariant under every pair of finite permutations of $V_{1}(H)$ and $V_{2}(H)$.

There is a one-to-one correspondence between distributions of random elements $\Gamma\in\mathcal{B}_{\infty\infty}$ (or $\overline{\mathcal{B}^{*}}$) and distributions of exchangeable random infinite graphs $H\in\mathcal{B}^{L}_{\infty\infty}$ given by

There is a one-to-one correspondence between elements $\Gamma$ of $\mathcal{B}_{\infty\infty}\cong\overline{\mathcal{B}^{*}}$ and extreme points of the set of distributions of exchangeable random infinite graphs $H\in\mathcal{B}^{L}_{\infty\infty}$. This correspondence is given by

Let $H$ be an exchangeable random infinite bipartite graph. Then the following are equivalent.

The distribution of $H$ is an extreme point in the set of exchangeable distributions in $\mathcal{B}^{L}_{\infty\infty}$.

The result by Aldous in the non-symmetric case is that every exchangeable infinite random bipartite graph is obtained as a mixture of such $G(\infty,\infty,W)$; in other words as $G(\infty,\infty,W)$ with a random $W$.

By Theorem 8.5 and Corollary 8.6 above, this implies (and is implied by) the fact that every element of $\overline{\mathcal{B}}$ equals $\Gamma^{\prime\prime}_{W}$ for some (non-unique) $W\in\mathcal{W}$; the bipartite version of the characterization by Lovász and Szegedy .

Directed graphs

A directed graph $G$ consists of a vertex set $V(G)$ and an edge set $E(G)\subseteq V(G)\times V(G)$; the edge indicators thus form an arbitrary zero–one matrix $\{X_{ij}\}$, $i,j\in V(G)$. Note that we allow loops, corresponding to the diagonal indicators $X_{ii}$. The definitions and results above have analogues for directed graphs too, with mainly notational differences. We sketch these in this section, leaving the details to the reader.

Let $\mathcal{D}^{L}_{n}$ be the set of the $2^{n^{2}}$ labelled directed graphs with vertex set $[n]$ and let $\mathcal{D}_{n}$ be the quotient set $\mathcal{D}^{L}_{n}/\cong$ of unlabelled directed graphs with $n$ vertices; further, let $\mathcal{D}^{L}:=\bigcup_{n\geq 1}\mathcal{D}^{L}_{n}$ and $\mathcal{D}:=\bigcup_{n\geq 1}\mathcal{D}_{n}$.

We define $\mathcal{D}^{*}:=\tau(\mathcal{D})\subseteq^{\mathcal{D}}$ to be the image of $\mathcal{D}$ under this mapping $\tau$, and let $\overline{\mathcal{D}^{*}}$ be the closure of $\mathcal{D}^{*}$ in $^{\mathcal{D}}$; this is a compact metric space.

All results in Sections 2–5 are valid for directed graphs too, with at most notational differences.

The main difference for the directed case concerns the representations discussed in Section 6. Since two vertices may be connected by up to two directed edges (in opposite directions), and the events that the two possible edges occur typically are dependent, a single function $W$ is no longer enough. Instead, we have a representation using several functions as follows.

Let $\mathcal{W}_{5}$ be the set of quintuples $\mathbf{W}=(W_{00},W_{01},W_{10},W_{11},w)$ where $W_{\alpha\beta}:^{2}\to$ and $w:\to\{0,1\}$ are measurable functions such that $\sum_{\alpha,\beta=0}^{1}W_{\alpha\beta}=1$ and $W_{\alpha\beta}(x,y)=W_{\beta\alpha}(y,x)$ for $\alpha,\beta\in\{0,1\}$ and $x,y\in$. For $\mathbf{W}\in\mathcal{W}_{5}$, we define a random infinite directed graph $G(\infty,\mathbf{W})$ by specifying its edge indicators $X_{ij}$ as follows: we first choose a sequence $Y_{1},Y_{2},\dots$ of i.i.d. random variables uniformly distributed on $$, and then, given this sequence, let$X_{ii}=w(Y_{i})$and for each pair$(i,j)$with$i$choose$X_{ij}$and$X_{ji}$ at random such that

this is done independently for all pairs $(i,j)$ with $i<j$ (conditionally given $\{Y_{i}\}$). In other words, for every $i$ we draw a loop at $i$ if $w(Y_{i})=1$ and for each pair $(i,j)$ with $i<j$ we draw edges $ij$ and $ji$ at random such that (9.2) holds. Further, let $G(n,\mathbf{W})$ be the restriction $G(\infty,\mathbf{W})|_{[n]}$, which is obtained by the same construction with a finite sequence $Y_{1},\dots,Y_{n}$.