Existence of Modeling Limits for Sequences of Sparse Structures
A sequence of graphs is FO-convergent if the probability of satisfaction of every first-order formula converges. A graph modeling is a graph, whose domain is a standard probability space, with the property that every definable set is Borel. It was known that FO-convergent sequence of graphs do not always admit a modeling limit, and it was conjectured that this is the case if the graphs in the sequence are sufficiently sparse. Precisely, two conjectures were proposed:
If a FO-convergent sequence of graphs is residual, that is if for every integer the maximum relative size of a ball of radius in the graphs of the sequence tends to zero, then the sequence has a modeling limit.
A monotone class of graphs has the property that every FO-convergent sequence of graphs from has a modeling limit if and only if is nowhere dense, that is if and only if for each integer there is such that no graph in contains the th subdivision of a complete graph on vertices as a subgraph.
In this paper we prove both conjectures. This solves some of the main problems in the area and among others provides an analytic characterization of the nowhere dense–somewhere dense dichotomy.
Combinatorics is at a crossroads of several mathematical fields, including logic, algebra, probability, and analysis. Bridges have been built between these fields (notably at the instigation of Leibniz and Hilbert). From the interactions of algebra and logic is born model theory, which is founded on the duality of semantical and syntactical elements of a language. Several frameworks have been proposed to unify probability and logic, which mainly belong to two kinds: probabilities over models (Carnap, Gaifman, Scott and Kraus, Nilsson, Väänänen, Valiant,…), and models with probabilities (H. Friedman, Keisler and Hoover, Terwijn, Goldbring and Towsner,…). See  for a partial overview.
Recently, new bridges appeared between combinatorics and analysis, which are based on the concept of graph limits (see  for an in-depth exposition). Two main directions were proposed for the study of a “continuous limit” of finite graphs by means of statistics convergence:
the left convergence of a sequence of (dense) graphs, for which the limit object can be either described as an infinite exchangeable random graph (that is a probability measure on the space of graphs over that is invariant under the natural action of ) [2, 16], or as a graphon (that is a measurable function ) [6, 7, 22].
the local convergence of a sequence of bounded degree graphs, for which the limit object can be either described as a unimodular distribution (a probability distribution on the space of rooted connected countable graphs with bounded degrees satisfying some invariance property) , or as a graphing (a Borel graph that satisfies some Intrinsic Mass Transport Principle or, equivalently, a graph on a Borel space that is defined by means of finitely many measure preserving involutions) .
A general (unifying) framework has been introduced by the authors, under the generic name “structural limits” . In this setting, a sequence of structures is convergent if the satisfaction probability of every formula (in a fixed fragment of first-order logic) for a (uniform independent) random assignment of vertices to the free variables converges. The limit object can be described as a probability measure on a Stone space invariant by some group action, thus generalizing approaches of [2, 16] and . This may be viewed as a natural bridge between combinatorics, model theory, probability theory, and functional analysis .
The existence of a graphing-like limit object, called modeling, has been studied in [31, 35], and the authors conjectured that such a limit object exists if and only if the structures in the sequence are sufficiently “structurally sparse”. For instance, the authors conjectured that if a convergent sequence is non-dispersive (meaning that the structures in the sequence have no “accumulation elements”) then a modeling limit exists:
Conjecture 1 ().Every convergent residual sequence of finite structures admits a modeling limit.
Note that this conjecture is known in one direction . To prove the existence of modeling limits for sequences of graphs in a nowhere dense class is the main problem addressed in this paper.
Nowhere dense classes enjoy a number of (non obviously) equivalent characterizations and strong algorithmic and structural properties . For instance, deciding properties of graphs definable in first-order logic is fixed-parameter tractable on nowhere dense graph classes (which is optimal when the considered class is monotone, under a reasonable complexity theoretic assumption) . Modeling limits exist for sequences of graphs with bounded degrees (as graphings are modelings), and this has been so far verified for sequences of graphs with bounded tree-depth , for sequences of trees , for sequences of plane trees and sequences of graphs with bounded pathwidth , and for sequences of mappings  (which is the simplest form of non relational nowhere dense structures). (See also related result on sequences of matroids .)
Our paper is organized as follows: In Section 2 we recall all necessary notions, definitions, and notations. In Section 3 we will deal with limits with respect to the fragment of all first-order formulas with at most one free variable. In Section 5 we deduce a proof of Conjecture 1 and, using a characterization of nowhere denses from , we prove that Conjecture 2 holds. Finally, we discuss some possible developments in Section 6. The general proof strategy is depicted bellow:
2. Preliminaries, Definitions, and Notations
2.1. Structures and Formulas
A signature is a set of function or relation symbols, each with a finite arity. In this paper we consider finite or countable signatures. A -structure is defined by its domain , and by the interpretation of the symbols in , either as a relation (for a relation symbol ) or as a function (for a function symbol ). A signature also defines the (countable) set of all first-order formulas built using the relation and function symbols in , equality, the standard logical conjunctives, and quantification over elements of the domain. The quotient of by logical equivalence has a natural structure of countable Boolean algebra, the Lindenbaum-Tarski algebra of .
For a formula with free variables and a structure we denote by the set of all satisfying assignments of in , that is
If is a finite structure (or a structure whose domain is a probability space), we define the Stone pairing of and as the probability of satisfaction of in for a random assignments of the free variables. Hence if is finite (and no specific probability measure is specified on the domain of ) it holds
Generally, if the domain of is a probability space (with probability measure ) and is measurable then
where denotes the product measure on .
For a -structure we denote by the graph with vertex set , such that two (distinct) vertices and are adjacent in if both belong to some relation in (that is if ).
2.2. Stone Space and Representation by Probability Measures
The term of Stone pairing comes from a functional analysis point of view: Let be the Stone dual of the Boolean algebra . Points of are equivalently described as the ultrafilters on , the homomorphisms from to the two-element Boolean algebra, or the maximal consistent sets of formulas from (point of view we shall make use of here). The space is a compact totally disconnected Polish space, whose topology is generated by its clopen sets
Let be a finite -structure (or a -structure on a probability space such that every first-order definable set is measurable). Identifying with the indicator function of the clopen set , the map uniquely extends to a continuous linear form on the space . By Riesz representation theorem there exists a unique probability measure such that for every it holds
Note that the permutation group defines a (subgroup of the) group of automorphisms of (by permuting free variables) and acts naturally on . The probability measure associated to the structure is obviously invariant under the -action.
For more details on this representation theorem we refer the reader to .
2.3. Structural Limits
Let be a signature, and let be a fragment of . A sequence of -structures is -convergent if converges as grows to infinity or, equivalently, if the associated probability measures on converge weakly .
In our setting, the strongest notion of convergence is -convergence (corresponding to the full fragment of all first-order formulas). Convergence with respect to the fragment (of all quantifier-free formulas without equality) is equivalent to the left convergence introduced by Lovász et al [4, 6, 22]. (It is also equivalent to convergence with respect to the fragment of all quantifier-free formulas, provided that the sizes of the structures in the sequence tend to infinity.) For bounded degree graphs, convergence with respect to the fragment of local formulas with a single free variable is equivalent to the local convergence introduced by Benjamini and Schramm . (Recall that a formula is local if its satisfaction only depends on a fixed neighborhood of its free variables.) Also, in this case, local convergence is equivalent to convergence with respect to the fragment of all local formulas, provided that the sizes of the structures in the sequence tend to infinity. For a discussion on the different notions of convergence arising from different choices of the considered fragment of first-order logic, we refer the interested reader to [29, 31, 35].
Note that the equivalence of -convergence with the weak convergence of the probability measures on associated to the finite structures in the sequence is stated in  as a representation theorem, which generalizes both the representation of the left limit of a sequence of graphs by an infinite random exchangeable graph  and the representation of the local limit of a sequence of graphs with bounded degree by an unimodular distribution on the space of rooted connected countable graphs .
2.4. Non-standard Limit Structures
A construction of a non-standard limit object for FO-convergent sequences has been proposed in , which closely follows Elek and Szegedy construction for left limits of hypergraphs . One proceeds as follows:
Let be a sequence of finite -structures and let be a non-principal ultrafilter. Let and let be the equivalence relation on defined by if . Then the ultraproduct of the structures is the structure , whose domain is the quotient of by , and such that for each relational symbol it holds is defined by
As proved by Łoś , for each formula and each we have
In  a probability measure is constructed from the normalised counting measures of via the Loeb measure construction, and it is proved that every first-order definable set of the ultraproduct is measurable. The ultraproduct is then a limit object for the sequence . In particular, for every first-order formula with free variables it holds:
Moreover, the above integral is invariant by any permutation on the order of the integrations.
However, the constructed object is difficult to handle. In particular, the sigma-algebra constructed on is not separable. For a discussion we refer the reader to [8, 10]. The ultraproduct construction is used in the proof of Lemma 2 to prove consistency of some theories in Friedman’s logic (see Section 2.6).
By similarity with graphings, which are limit objects for local convergent sequences of graphs with bounded degrees , the authors proposed the term of modeling for a structure built on a standard Borel space , endowed with a probability measure , and such that every first-order definable set is Borel . Such structures naturally avoid pathological behaviours (for instance, every definable set is either finite, countable, or has the cardinality of continuum). The definition of Stone pairing obviously extends to modeling by setting
An -convergent sequence has modeling -limit (or simply modeling limit when ) if is a modeling such that for every it holds
Let be a class of structures. We say that admits modeling limits if every -convergent sequence of structures with has a modeling limit.
Note that not every -convergent sequence has a modeling limit: Consider a sequence of graphs, where is a graph of order , with edges drawn randomly (independently) with edge probability . Then with probability the sequence is FO-convergent. However, this sequence has no modeling limit, and even no modeling -limit: Assume for contradiction that has a modeling -limit . Because the probability measure is atomless thus is uncountable. As is a standard Borel space, there exists zero-measure sets and , and a bijective measure preserving map . By the equivalence of -convergence and left-convergence the modeling defines a -valued graphon , which is a left limit of by:
But a left limit of is the constant graphon , which is not weakly equivalent to (as it should, according to ) thus we are led to a contradiction.
This example is prototypal, and this allows us to prove that if a monotone class of graphs admits modeling limits then this class has to be nowhere dense . The proof involves the characterization of nowhere dense classes by the model theoretical notions of stability and independence property , their relation to VC-dimension , and the characterization of sequences of graphs admitting a random-free (i.e. almost everywhere -valued) left limit graphon . Conjecture 2 asserts that the converse is true as well.
2.6. H. Friedman’s -logic
Friedman [11, 12] studied a logical system where the language is enriched by the quantifier “there exists x in a non zero-measure set …”, for which he studied axiomatizations, completeness, decidability, etc. A survey including all these results was written by Steinhorn [37, 38]. In particular, H. Friedman considered specific type of models, which he calls totally Borel, which are (almost) equivalent to our notion of modeling: A totally Borel structure is a structure whose domain is a standard Borel space (endowed with implicit Borel measure) with the property that every first-order definable set (with parameters) is measurable.
In this context, Friedman introduced a new quantifier , which is to be understood as expressing “there exists non-measure many”, and initiated the study of the extension of first order logic, whose axioms are all the usual axiom schema for first-order logic together with the following ones :
, where is an -formula in which does not occur and is the result of replacing each free occurrence of by ;
The rules of inference for are the same as for first-order logic: modus ponens and generalization. Let the proof system just described be denoted by .
The standard semantic for is as follows: for a structure on a probability space such that every first-order definable (with parameters) is measurable (for probability measure ) it holds
Note that the set of -sentences satisfied by (for this semantic) is obviously consistent in .
A set of sentences in has a totally Borel model if and only if is consistent in .
It has been noted that one can require the domain of the totally Borel model to be a Borel subset of with Lebesgue measure .
3. Modeling -limits
Let be an FO-convergent sequence of finite structures, and let be the union of a complete theory of an elementary limit of together with, for each first order formula with free variables ,
The ultraproduct construction provides a model for :
For every FO-convergent sequence of finite structures, the theory is consistent in .
Using the standard semantic for it is immediate that any ultraproduct is a model for hence is consistent in . ∎
For every integer , there exists an integer and formulas (with a single free variable) defining the local -types up to quantifier rank in the sense that all of these formulas are local and have quantifier rank , they induce a partition (formalized as if and ), and for every local formula with quantifier rank and for every either it holds , or .
Define . Define the probability measure on as follows: for every Borel subset of define
Obviously weakly converges to some probability measure . Let be the modeling obtained by endowing with the probability measure . Note that is absolutely continuous with respect to by construction. ∎
Theorem 3 immediately implies
Corollary 1.Every -convergent sequence has a modeling -limit.
4. Modeling Limits of Residual Sequences
We know that in general an -convergent sequence does not have a modeling limit (hence Corollary 1 does not extend to full ). This nicely relates to sparse–dense dichotomy.
Recall that a class of (finite) graphs is nowhere dense if, for every integer , there exists an integer such that the -th subdivision of the complete graph on vertices is the subgraph of no graph in [27, 30]. (Note a subgraph needs not to be induced.) Based on a characterization by Lovász and Szegedy  or random-free graphon and a characterization of nowhere-dense classes in terms of VC-dimension (Adler and Adler  and Laskowski ) the authors derived in  the following necessary condition for a monotone class to have modeling limits.
Let be a monotone class of graphs. If every FO-convergent of graphs from has a modeling limit then the class is nowhere dense.
However, there is a particular case where a modeling limit for an FO-convergent sequence will easily follow from Theorem 3. That will be done next.
A sequence is residual if, for every integer it holds
where denotes the set of elements of at distance at most from (in the Gaifman graph of ). Equivalently, is residual if, for every integer , it holds
The notion of residual sequence is linked to the one of residual modeling: A residual modeling is a modeling, all components of which have zero measure (that is if and only if for every integer , every ball of radius has zero measure).
By an interplay of these notions we now can prove Conjecture 1.
Theorem 6.Every -convergent residual sequence has a modeling limit.
The main characteristic of residual sequences is that a residual sequence is -convergent if and only if it is -convergent . Consider the modeling limit obtained in Theorem 3 for a -convergent residual sequence. Then for every integer it holds
It follows that is residual, and thus the convergence of to for first-order formulas with (at most) one free variable (i.e. -convergence) implies convergence for all first-order formulas (i.e. -convergence). ∎
5. Modeling Limits of Quasi-Residual Sequences
Here we prove our main result in the form of a generalization of Section 4 for quasi-residual sequences. The motivation for the introduction of the definition of quasi-residual sequences is the following:
Known constructions of modeling limits for some nowhere dense classes with unbounded degrees [14, 31, 35] are based on the construction of a countable “skeleton” on which residual parts are grafted. We shall use the same idea here for the general case. The countable skeleton will be built thanks to the following characterization of nowhere dense classes proved in :
Let be a class of graphs. Then is nowhere dense if and only if for every integer and every there is an integer with the following property: for every graph , and every subset of vertices of , there is with such that no ball of radius in has order greater than .
This theorem justifies the introduction of the following relaxation of the notion of residual sequence:
A sequence (with ) is quasi-residual if, for every integer and every there exists an integer such that it holds
In other words, is quasi-residual if, for every distance and every there exists an integer so that (for sufficiently large ) one can remove at most vertices in the Gaifman graph of so that no ball of radius will contain at least proportion of .
The next result directly follows from Theorem 7.
Let be a nowhere dense class of graphs and let be a sequences of graphs from such that . Then is quasi-residual.
5.1. -residual Sequences
We now consider a relaxation of the notion of residual sequence and show how this allows to partially reduce the problem of finding modeling -limits to finding modeling -limits.
Let be an integer and let be a positive real. A sequence is -residual if it holds
Similarly, a modeling is -residual if it holds
Let and let be a positive real. Assume is a -convergent -residual sequence of graphs and assume is a -residual modeling -limit of .
Then for every -local formula with free variables it holds
By restricting the signature to the symbols in if necessary, we can assume that is finite. Let be the quantifier rank of . Then there exists finitely many local formula with quantifier rank at most (expressing the rank -local type) such that:
every element of every model satisfies exactly one of the (formally, and if );
two elements and satisfies the same local first-order formulas of quantifier rank at most if and only if they satisfy the same .
Let be the formula . By -locality of there exists a subset such that
Let . For every structure it holds
As is a modeling -limit of it holds , hence
On the other hand, as , for every structure holds
Note that is nothing but the expected measure of a ball of radius in . In particular, if is -residual, then it holds . Thus,
5.2. Marked Quasi-residual sequences
To allow an effective use of the properties of quasi-residual sequences, we use a (lifted) variant of the notion of quasi-residual sequence.
Let be a countable signature and let be the signature obtained by adding to countably many unary symbols and .
For integers we define the formulas and as
In other words, holds if belongs to the ball of radius centered at the element marked , and holds if belongs to the -neighborhood of elements marked by .
A sequence (with ) of -structures is a marked quasi-residual sequence if the following condition holds:
For every integers it holds (i.e. at most one element in is marked by );
For every distinct integers and every integer , no element of is marked both and ;
For every integer there is a non-decreasing unbounded function with the property that for every integer it holds
For every integer and every positive real there is such that
(In other words, every ball of radius in contains less than proportion of all the vertices, as soon as is sufficiently large.)
For every integer the following limit equality holds:
The main purpose of this admittedly technical definition is to allow to make use of the sets arising in the definition of quasi-residual sequences by first-order formula, by means of the marks . The role of the marks is to allow a kind of of limit exchange. (Note that is nothing but the ball of radius of centered at the element marked by .)
For every quasi-residual sequence of -structures there exists an -convergent marked quasi-residual sequence of -structures such that is a subsequence of .
Let be the signature obtained by adding to countably many unary symbols . For we define the -structure has the -structure obtained from by defining marks are assigned in such a way that for every and there is such that letting it holds
This is obviously possible, thanks to the definition of a quasi-residual sequence.
Considering an FO-convergent subsequence we may assume that is -convergent.
For we define the constant
(Note that the values exist as is -convergent and that they form, for increasing , a non-decreasing sequence bounded by .)
Then for each there exists a non-decreasing function such that and
Then we define to be the sequence obtained from by marking by all the elements in . Now we let to be a converging subsequence of . ∎
Let be the formula asserting that the ball of radius centered at contains but no element marked , that is
Let be a marked quasi-residual sequence. Then
Assume for contradiction that is strictly positive.
According to the definition of a marked quasi-residual sequence, there exists an integer such that no ball of radius in contains more than elements. Let be such that , and let be such that holds for every .
Then there exists such that the ball of radius centered at contains no element marked (hence no element marked ) and contains more than elements, what contradicts the fact that this ball is a ball of radius in . ∎
In general, a modeling -limit of a -residual sequence does not need to be -residual. However, if we consider a sequence that is also marked quasi-residual, and if we assume that the modeling -limit satisfies the additional properties asserted by Theorem 3 then we can conclude that the modeling is -residual, as proved in the next lemma.
If the sequence is -residual and is a modeling with the properties asserted by Theorem 3 then is -residual.
We first prove that the set of vertices such that the ball of radius centered at has measure greater than has zero measure. According to Lemma 13, it holds hence . This implies that the set of such that the ball of radius centered at contains no element marked and has measure at least has zero measure. Hence we only have to consider vertices in the -neighborhood of . Let
Let . There exists such that
which means that at least proportion of is at distance at most from elements marked .
However, according to (6), and as is a modeling -limit of it holds
which means that a proportion of is at distance at most from elements marked (which include elements marked ). Thus the set of vertices in the -neighborhood of but not in the -neighborhood of has measure at most .
Let be in the -neighborhood of . Then the ball of radius centered at is included in the ball of radius centered at a vertex marked , for some . But this ball has measure . As the sequence is -residual, it holds for sufficiently large . Hence the ball of of radius centered at (which is included in the ball of radius centered at the vertex marked ) has measure less than .
It follows that the set of such that the ball of radius centered at has measure at least is included in hence has zero measure.
Now assume for contradiction that there exists a vertex such that the ball of radius centered at has measure at least . Then for every the ball of radius centered at has measure at least , what contradicts the fact that the measure of is positive. ∎
5.3. Color Coding and Mark Elimination
We now consider how to turn a marked quasi-residual into a -residual marked quasi-residual sequence.
The idea here, is to encode each relation with arity with relations plus a sentence. The sentence expresses the behaviour of when restricted to elements marked . The relations expresses which tuples of non-marked elements can be extended (and how) with elements marked to form a -tuple of .
As above, let be a countable signature with unary relations and . Let .
We define the signature as the signature obtained from by adding, for each symbol with arity the relation symbols of arity , where and .
Let be a -structure.
We define the structure as the -structure , which has same domain as , same unary relations, and such that for every symbol with arity , for every and , denoting the elements of and the elements of , it holds
Note that the Gaifman graph of can be obtained from the Gaifman graph of by removing all edges incident to a vertex marked .
We now explicit how the relation in can be retrieved from .
For , with arity , and let be defined as follows:
and let be the following sentence, which expresses that encodes the set of all the tuples of elements marked in .
The following lemma sums up the main properties of our construction.
Let be a -structure, and let .
Let be a relation symbol with arity . Then
there exists a unique subset of such that
for this and for every it holds
This lemma straightforwardly follows from the above definitions. ∎
Let be fixed.
An elimination theory is a set containing, for each with arity , exactly one sentence (for some ). For a -structure , the elimination theory of is the set of all sentences