1 Introduction
Here, we show a construction of an infinitely exchangeable family of random graphs which is based on an associated Poisson point process on the power set of the natural numbers . We provide a necessary and sufficient condition for the induced random graph to be infinitely exchangeable and discuss a potential use for this model in the area of cluster analysis and stochastic classification, which has been previously studied in a statistical and machine learning context in previous work by Jordan [2] (with Blei and Ng) and [7] (with Broderick and Pitman) McCullagh [10, 11], but outside of the realm of network analysis.
We now introduce preliminary material and notation which is critical to our treatment.
1.1 Projective systems
A projective system associates with each finite set a set and with each onetoone injective map , , a projection which maps into such that

if is the identity then is the identity and

if , , and is its associated projection, then the composition satisfies .
If is the set of subsets of , i.e. the space of directed graphs with vertices, one can define the projection either by restriction or deleteandrepair. Each can be represented as an matrix with entries in such that if and otherwise. For each , let be the operation on which restricts to the complement of . In matrix form, is the matrix obtained from by removing the last row and last column of and keeping the rest of the entries unchanged. It is clear that the compositions for are welldefined as the restriction of to by removing the last rows and columns of .
For , we write to denote the symmetric group of permutations of , i.e. onetoone maps . Each acts on each element componentwise in the usual way. That is, if and only if . The restriction maps together with permutation maps and their compositions make a projective system.
Another way to specify a projective system on is by deleteandrepair. For , let act on by removing the th row and column of and directing an edge from each in to each in . In other words, is obtained by deleting the vertex labeled from and connecting two vertices and by a directed edge from to if both and are elements of , i.e. there is a directed path in .
For , define . Plainly, is welldefined since for each , and . The deleteandrepair maps together with permutation maps and compositions also make a projective system, which differs from the above projective system based on restriction maps.
Throughout the rest of this paper, for a system we write to represent the restriction maps for collection and to represent the corresponding deleteandrepair maps. We write , resp. , to denote the projective system on described by the restriction, resp. deleteandrepair, maps together with permutation maps.
1.2 Power set of
Let denote the power set of all subsets of , the power set of . For each , we define the partial order on by
Therefore, but and have no relation under .
We define restriction maps as follows. For , we define . The maps preserve on since
and so
Permutations act on elements of componentwise, i.e. for , and , we have . We write to denote the projective system together with restriction maps and permutation maps and their compositions.
1.3 Monotone sets
A subset is monotone if implies . For example, the set is a monotone set with maximal elements and , which constitute the generating class of , written An element of the generating class of a monotone set is a maximal element of in the sense that no other element contains as a subset. The generating class of a monotone subset consists of all maximal elements of . We write as the set of monotone sets taking elements in . Note that a monotone set is uniquely determined by its generating class, and so we will write and to describe the same object, i.e. the monotone set , whenever it is convenient to do so. Every subset of has a least monotone cover in , which we denote by and is given by .
For each , is a partially ordered set induced by the partial order inclusion on , i.e. for , we say if and only if each is a subset of some . And for any pair with , the intervals and are welldefined subsets of . Note that we intend the symbols and to have strictly different meanings in this paper. In particular, we write to mean is any subset of , while we write to mean is a proper subset of , i.e. but . This distinction becomes important in the next section.
We define the operation restriction as the operation which maps . That is, for with , . For , we define in the usual way by composition, and the collection of restriction maps together with permutation maps makes into a projective system. written . Here, a permutation acts on a monotone set by acting componentwise on its generating class, i.e. if and only if .
Also note that the inverse mapping associates with each an interval, and also maps intervals in to intervals in . In particular, for and , we have
1.4 Undirected graphs
For , an undirected graph is a pair of vertices and edges whereby the number of vertices of is , and without loss of generality we assume , and the edges are a subset of such that implies .
For each , the elements of correspond to the symmetric subsets of , e.g. for , if and only if , and hence is a projective system under both restriction and deleteandrepair, as described in section 1.1. For each , is a poset under where for
For clarity of notation, write and to denote the operations of restriction and deleteandrepair respectively on the projective system and respectively.
1.5 Some category theory
The relationship between the collections , and is described in a straightforward way by elementary concepts in category theory [1].
A category C consists of objects and arrows between objects so that for each arrow C there are objects and in C, the domain and codomain respectively of , and we write . Given C such that , the composite is an arrow in C. Also, for each object C there is an identity arrow , and all arrows of C must satisfy associativity and preservation under composition with the identity functions.
In each of the categories we define below, there is at most one arrow between any two objects. Therefore, if an arrow corresponds to the composition of arrows , these must represent the same arrow. Under this assumption, for a pair of objects we need not make explicit all of the various compositions of arrows which result in an arrow between these objects, as they are implicitly assumed to be there, and are all the same arrow.
We define three categories as follows. We write for the category with objects given by the elements of and arrows given by the restriction and permutation maps, and compositions of these maps. That is, for and , there is an arrow . We denote the arrow between and by . Likewise, we define and to be the category with objects given by the elements of and respectively, and arrows defined by the restriction maps, and resp., and permutation maps. For example, in , we have
A functor between categories C and D is a map which takes objects in C to objects in D and arrows in C to arrows in D. There are natural functors and , which we define as follows.

For each and , is the least monotone cover of .

For each and , where if and only if for some .

For , and so that .

For every , and for all .
Functors between posets preserve partial ordering. For , we have defined the partial order in previous sections. For and such that , it holds that . Hence, for each fixed , the restriction maps define a functor , where we take the elements of and as objects and the partial order defines the arrows, i.e. there is an arrow in if and only if .
We can also regard the collections as partially ordered sets with partial order , , defined as follows. For , , ,
The functors and above preserve the partial orders and .
Below, we show a construction of infinitely exchangeable random elements of . In particular, we construct an infinitely exchangeable random monotone set by projecting from a Poisson point process on the power set to a random subset of the power set to the least monotone cover , which corresponds to a random graph . This procedure looks like this
Projective systems in statistics
The relevance of category theory and projective systems in statistical modeling is discussed in detail by McCullagh [9]. Here we have introduced projective systems for the collection of subsets of , their associated monotone subsets of and their associated undirected graphs in .
The choice of projection, i.e. either restriction or deleteandrepair, on is intended to reflect the notion of subsampling in statistics, and each admits its own statistical interpretation in terms of subsampling which may be appropriate depending on the application, and the actual way in which observations are made. In the study of directed graphs, in particular permutations, deleteandrepair is often used, see e.g. the Chinese restaurant process (CRP) on permutations [12]. However, for our purposes, restriction maps are a natural choice.
To be specific, if we think of a graph as modeling a social network, the edges of the graph represent social links among the agents (nodes) in the population. When observing a social network, there are several sampling methods which are reasonable. The most intuitive of these are perhaps node sampling and snowball sampling. In node sampling, we first sample a set of nodes and subsequently observe any edges between these nodes. In this setting, restriction accurately reflects our sampling method since removal of a node from our sample, e.g. , removes any edges which involves that node from our sampled graph. In snowball sampling, we start with a set of nodes, without loss of generality suppose we start with one node , and proceed as follows. Given , sample , the set of nodes adjacent to some node in in the graph we are observing. Subsequently, put . In snowball sampling, we stop at some arbitrary level . This way of sampling results in a sample of all nodes at radius or less from , and depends on our choice of and the network structure. In this case, removal of a node results in removal of all other nodes such that lies along every path between and . The result of this type of subsampling is not described simply via restriction. Below, we discuss only those projective systems characterized by the restriction maps, and the implications of this method of sampling on inference.
1.6 Infinite exchangeability
A family
of probability measures on a projective system
is infinitely exchangeable if it is invariant under both the action of permutations, called finite exchangeability, and selection according to the projection maps associated with the system, called consistency. For example, a family of measures on is infinitely exchangeable if
for each and , for every and

for every , for every .
An infinitely exchangeable collection of measures uniquely characterizes a measure on the infinite space associated with through its finitedimensional distributions and invariance under projection and permutation maps.
Above we have defined functors and . One property of functors is that . Hence, we have the following elementary lemma which is useful for our purposes below.
Lemma 1.1.
Let be a functor between categories and where the objects of and of form a projective system under mappings and respectively, and the arrows are defined by the partial ordering induced by the projections and . Let be a probability measure on and let be the distribution induced on the objects of by through the functor . If is invariant under , i.e. , then is invariant under .
Proof.
Since is a functor, we have for any arrows such that . Let be an object in and , so that we have an arrow which we denote by , and the action of the functor is such that
and we have and invariance of the induced measure follows. Indeed, for and ,
∎
This will simplify our proofs below for infinite exchangeability of the random graph induced by a random subset of .
2 Construction of an infinitely exchangeable random graph
Let be a Poisson point process on with mean measure so that
} is a collection of independent Poisson random variables with each
having mean . By ignoring multiplicities, each realization of this process defines a random subset which consists of those points for which . The distribution of is given byIn general, will not be monotone, but, as discussed above, it will have a least monotone cover given by .
It is straightforward to compute the induced distribution of on under the partial ordering . That is, for , let denote the complement of in , then
For each , let and be the minimal element of and respectively, i.e. , and define to be the probability measure on and the probability measure on induced by through , shown above, for some nonnegative mean measure on .
For a subset , let denote the cardinality of , i.e. the number of elements in . We now show that a necessary and sufficient condition for , and hence , to be infinitely exchangeable under action of is that for every , for some collection of nonnegative real numbers which satisfy
(1) 
Theorem 2.1.
The collection of probability measures on and on are infinitely exchangeable if and only if the collection of mean measures satisfy

for every , for all for some collection of measures on the nonnegative real numbers and

for every , for .
Moreover, if (a) and (b) hold, the measure induced on by through , i.e. , is infinitely exchangeable.
Proof.
Suppose (a) and (b) hold. Let and suppose . Then
Clearly, the are finitely exchangeable for each as the measure depends only on the cardinality of the elements of , which directly depends on the cardinality of the elements of , which are invariant under permutations.
As noted in section 1.1, the pullback under the restriction maps takes intervals to intervals. Hence, for , the interval maps to the interval , where is the unique maximal element of in . To simplify notation, write in what follows.
Hence, for consistency under sampling by restriction maps we must have
(2) 
Consistency follows from this since we now have
which reduces (2) to
(3) 
which is a sum over subsets of the power set of and respectively. Under restriction, each corresponds to a two element subset of , namely . Hence, for , we have and (3) is just
Infinite exchangeability for is endowed by through , as discussed in section 1.6.
For the reverse implication, note that we start with the condition on the mean measures
and exchangeability requires for all with so we can reduce this to the collection of measures taking values in the nonnegative real numbers, as we have done above. And consistency requires of either or requires (b) to hold.
The infinite exchangeability of the measures on is a direct corollary of the infinite exchangeability of and the action of the functor . ∎
Corollary 2.2.
Suppose is a doubly indexed sequence of nonnegative real numbers satisfying (1), then there exists a measure on , the space of graphs with vertex set such that
where denotes the restriction of to the vertex set and are the induced measures on given above.
We make note of a correspondence between the solutions to (1
) and the classical Hausdorff moment problem. Connection between the Hausdorff moment problem and de Finetti’s theorem have been shown by Diaconis and Freedman
[8] and are well known throughout the literature. Some usable choices for are
for and

.
3 Cluster analysis
Given a measure on which is based on a collection which satisfy (1), we can easily calculate the marginal distribution that a triple of vertices, e.g. , is transitive, i.e. , given two are adjacent, e.g. and , by a standard exchangeability argument. In particular, for any and we have
which, unlike the ErdösRényi process, does not correspond to the clustering coefficient. This calculation only accounts for the subgraph comprised of the vertices and any edges between them. In general, if we observe a graph and we are told that and but not told if there is an edge between and or not, but we also know the rest of the graph structure, this information is relevant for determining whether or not in , and has implications in the inference of missing links in network data sets.
For example, consider the two graphs and below where
and
and the presence or absence of an edge between and is unknown, but the rest of this network is known. Given what we do know about , there are three possible monotone sets which correspond to ,
On the other hand, there are five monotone sets corresponding to ,
So the information given by the rest of the network, and any edges, or lack thereof, which involve any of the of interest, affects the conditional probability of , e.g. in this case.
In general, the clustering of this process is expected to be larger than the marginal probability expression above as the presence of a cluster of three vertices increases the probability of other clusters which involve these vertices. The nature of the construction, e.g. description of the functor , leads to overlapping of the various subsets of which is “forgotten” in the projection onto and provides various ways by which clustering can occur.
The construction also allows us to move between the space of graphs and that of monotone sets, which is helpful in calculations.
3.1 Detecting clusters
The nature of this construction naturally lends itself to methods in cluster analysis, which has been studied in certain applications in statistics and machine learning [2, 10, 11]. The setting is as follows. Let be the size of a sample for which we label statistical units, e.g. individuals, arbitrarily in and observe a network for this sample, i.e. an undirected graph . Along with , let be a collection of different equivalence relations on . A collection of labels is said to form a cluster, or community, in our network if, for some , for every . Inferring clusters in networks has implications, for example, in the problem of data deduplication and parsing for semistructured text data sets as well as inferring communities and missing links in social networks.
In a statistical setting for social networks could represent different ‘types’ of relationships among individuals. That is, individuals in a social network are associated by certain relationships which underlie the network, e.g. a 4node clique in a network could arise from 4 nodes belonging to the same cluster, or due to the overlap of two 3node clusters and a 2node cluster, e.g. and , which both result in the presence of the clique in the projected network.
3.1.1 A statistical model
In the setting of section 2 consider an infinite population of units from which we sample a finite number which we label in , i.e. our sample is , and we observe for this sample a network, or graph, which we assume to have been generated according to the Poisson point process (PPP) construction on which we laid out above. In particular, let be a family of mean measures on such that for every and the satisfy (1). Let and be the measures on and defined in above sections. Then we have the following model:
We now imagine reversing this process to infer whether a given complete subgraph of units represents a cluster of . This amounts to computing the conditional probability that given that .
Suppose we observe a network with complete subgraph . Under the inverse image of the functor , we have that is a collection of monotone sets which correspond to . Furthermore, the inverse image of the least monotone cover is a collection of possible subsets of which have least monotone covers corresponding to . As we have shown in theorem 2.1, the consistency condition in (1) guarantees infinite exchangeability of as well as and on and respectively.
Hence is the collection of random subsets in which correspond to . Define to be the elements of which contain the set , and the complement of in , i.e. . Then the conditional probability that is a cluster in the subsample of given is
(4) 
The conditional probability calculation in (4) is valid for inference of a cluster if we are interested in the presence specifically of a cluster with exactly the elements of and not a cluster