Multigraph limits, unbounded kernels, and Banach space decorated graphs
Introduction
The motivation for this paper was to provide a framework for a theory of convergence and limits of graphs with unbounded edge multiplicities, along the lines of the limit theory for dense simple graphs developed by Borgs, Chayes, Lovász, Sós and Vesztergombi [2], [3] and Lovász and Szegedy [10]. Key elements of this theory are the notions of cut distance and subgraph densities, the definitions of convergence and limit objects, the Regularity Lemma (in its weak form due to Frieze and Kannan [5]), along with the Counting Lemma.
In the paper [11] (posted on the Arxiv, but not published; see also [9], Section 17.1), the second and third authors worked out a theory of convergence and limits of simple graphs whose edges are decorated by points from some compact space. (Ordinary simple graphs can be considered as complete graphs with edges decorated by elements of a two-point space.) One could say that the theory of undecorated simple graphs extends to this case in a rather straightforward manner (at least as soon as the appropriate formulations are found). Edge-weighted graphs and multigraphs fit in this framework, provided the edge weights/multiplicities are bounded (but for unbounded multiplicities or edge weights one has to do more, as we shall see). We note that this paper took the approach of defining convergence via weak convergence of the distribution of samples, and this gives a link to exchangeability and Aldous' representation theorem. It turns out that whilst sampling convergence is equivalent to homomorphism convergence notions for compact decorations, this does not longer hold if the compactness condition is waived (see Section 4.4).
A limit theory for convergence of multigraphs was worked out by Kolossváry and Ráth [7], and essentially the same results can be derived from the limit theory of compact decorated graphs using the one-point compactification of the set of integers to encode the edge multiplicities. The limit objects can be described by functions on whose values are probability distributions on nonnegative integers.
Let us describe in a few words the general framework for graph convergence theories. We start with defining the number of occurrences of a “small” graph F in a “big” graph G. In the case of simple graphs, one can use the number of homomorphisms (adjacency-preserving maps) from F to G. One also needs the (normalized) homomorphism density
A key notion in these theories is that of convergence of a graph sequence, which is defined by specifying an appropriate family of test graphs, and then saying that a sequence of graphs is convergent, if is convergent for every test graph F. In the theory of convergence of simple graphs, the family of simple graphs is the right (in a sense, only reasonable) choice for test graphs. The limiting values of these densities can be represented by limit objects called graphons, which in the case of simple graphs are symmetric measurable functions .
The motivation of this paper is to work out a limit theory for convergence of multigraphs. Whether or not the results of Kolossváry and Ráth [7] can be viewed as a solution of the problem of multigraph convergence depends on how we define homomorphisms between two multigraphs F and G.
One natural definition is that of node-and-edge homomorphism: this is a pair of maps and such that if connects i and j, then connects and . A different definition is that of a node-homomorphism: a map such that the multiplicity of the image of an edge is not less than the multiplicity of the edge. If both F and G consist of two nodes connected by two edges, then the number of node-homomorphisms is 2, while the number of node-and-edge homomorphisms is 8.
In this paper, we consider node-and-edge homomorphisms, and for two multigraphs, we denote by their number. We define homomorphism densities and convergence based on this definition. The results of Kolossváry and Ráth are based on node-homomorphisms. It turns out that these two notions of convergence are not equivalent (see Section 2.3, and also [9], Chapter 17).
In fact, we consider a more general model, namely a limit theory of graphs whose edges are decorated by elements from a Banach space, and where the test graphs are decorated from the pre-dual space. This will include the convergence theory of compact decorated graphs as well. Along the way, we show that with a modified notion of cut distance (which we call “jumble distance”) one can state a prove an appropriate Weak Regularity Lemma and a Counting Lemma.
Recently Borgs, Chayes, Cohn and Zhao [1] developed a theory for -graphons (unbounded symmetric functions in the space for some ), and graph sequences convergent to them. They prove appropriate versions of the Regularity and Counting Lemmas. Not every graph has a finite density in such a kernel, and accordingly, they limit the set of test-graphs to simple graphs with degrees bounded by p. Their set-up is more general than ours in the sense that we work with a more restricted family of unbounded kernels, namely kernels in . On the other hand, we allow arbitrary multigraphs as test graphs (more generally, decorating by elements of a Banach space). So the two theories don't seem to contain each other (but perhaps a common generalization is possible).
Using random graphs generated by Banach space valued graphons, we show that every element of the space of limit objects that we define arises as a limit of a convergent sequence of decorated graphs, and that this space is closed under our convergence notion.
Although we do not in this paper investigate the question of uniqueness of Banach space graphons (we refer to [8] by the first author for details on that subject), we remark here that Example 4.9, Example 4.11 do show that indeterminacies in the Stieltjes/Hamburger moment problems are an extra natural obstacle to uniqueness for unbounded graphons, beyond the usual weak isomorphism equivalence (see, e.g., [9, Sections 7.3 and 10.7]).
Section snippets
Decorated graphs and graphons
If is any set, an -decorated graph is a graph where every edge ij is decorated by an element . An -decorated graph will be denoted by , where G is a simple graph (possibly with loops), and . We will see several examples in Section 2.3 how decorations can be used to express weights, multiple edges, and more.
In our setup, we will consider decorations by elements of Banach spaces. Let be a separable Banach space, let denote its dual. The elements of act on as bounded
Dense sets
In this preliminary section we prove two lemmas which will allow us to use countable “test sets” for certain properties. The first lemma lets us prove weak-* measurability using only a countable dense subset of .
Lemma 3.1 Let be a countable dense subset, and let be a function such that is measurable for each . Then W is weak-* measurable. Proof Since is countable and dense in , for any there exists a sequence that converges to b in norm. But then is the pointwise
Convergence of multigraphs
The main application of the above theory pertains to sequences of multigraphs where there is no global bound on edge multiplicities, and we are interested in convergence of node-and-edge homomorphism numbers.
Concluding remarks
5.1 An almost identical construction as in Section 4 can be used to define and study convergent sequences of edge-weighted graphs with no universal bound on the weights. In this case we use as a weight function an appropriate function , and we replace the set of natural numbers used in the previous example by the set of nonnegative reals, summation by integral etc. Caution: one has to distinguish more carefully functions and signed measures (which were interchangeable in the discrete case
Acknowledgements
The authors would like to thank S. Janson and the anonymous referee for their thorough read of the original manuscript. Their valuable comments and suggestions have led to a more self-contained final paper.
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Research supported by ERC Advanced Research Grant No. 227701, the Bolyai Research Grant of the Hungarian Academy of Sciences and ERC Consolidator Grant No. 648017.
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Research supported by ERC Consolidator Grant No. 617747 and the Hungarian National Excellence Grant 2018-1.2.1-NKP-00008.