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13
AN Lp THEORY OF SPARSE GRAPH CONVERGENCE I: LIMITS, SPARSE RANDOM GRAPH MODELS, AND POWER LAW
"... Abstract. We introduce and develop a theory of limits for sequences of sparse graphs based on Lp graphons, which generalizes both the existing L ∞ theory of dense graph limits and its extension by Bollobás and Riordan to sparse graphs without dense spots. In doing so, we replace the no dense spots ..."
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Abstract. We introduce and develop a theory of limits for sequences of sparse graphs based on Lp graphons, which generalizes both the existing L ∞ theory of dense graph limits and its extension by Bollobás and Riordan to sparse graphs without dense spots. In doing so, we replace the no dense spots hypothesis with weaker assumptions, which allow us to analyze graphs with power law degree distributions. This gives the first broadly applicable limit theory for sparse graphs with unbounded average degrees. In this paper, we lay the foundations of the Lp theory of graphons, characterize convergence, and develop corresponding random graph models, while we prove the equivalence of several alternative metrics in a companion paper.
Interval graph limits
, 2011
"... We work out the graph limit theory for dense interval graphs. The theory developed departs from the usual description of a graph limit as a symmetric function W (x, y) on the unit square, with x and y uniform on the interval (0, 1). Instead, we fix a W and change the underlying distribution of the c ..."
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We work out the graph limit theory for dense interval graphs. The theory developed departs from the usual description of a graph limit as a symmetric function W (x, y) on the unit square, with x and y uniform on the interval (0, 1). Instead, we fix a W and change the underlying distribution of the coordinates x and y. We find choices such that our limits are continuous. Connections to random interval graphs are given, including some examples. We also show a continuity result for the chromatic number and clique number of interval graphs. Some results on uniqueness of the limit description are given for general graph limits.
INVARIANT MEASURES CONCENTRATED ON COUNTABLE STRUCTURES
"... Abstract. Let L be a countable language. We say that a countable infinite Lstructure M admits an invariant measure when there is a probability measure on the space of Lstructures with the same underlying set as M that is invariant under permutations of that set, and that assigns measure one to the ..."
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Abstract. Let L be a countable language. We say that a countable infinite Lstructure M admits an invariant measure when there is a probability measure on the space of Lstructures with the same underlying set as M that is invariant under permutations of that set, and that assigns measure one to the isomorphism class of M. We show that M admits an invariant measure if and only if it has trivial definable closure, i.e., the pointwise stabilizer in Aut(M) of an arbitrary finite tuple of M fixes no additional points. When M is a Fraïssé limit in a relational language, this amounts to requiring that the age of M have strong amalgamation. Our results give rise to new instances of structures that admit
GRAPH LIMITS AND HEREDITARY PROPERTIES
"... Abstract. We collect some general results on graph limits associated to hereditary classes of graphs. As examples, we consider some classes of intersection graphs (interval graphs, unit interval graphs, threshold graphs, chordal graphs). 1. ..."
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Abstract. We collect some general results on graph limits associated to hereditary classes of graphs. As examples, we consider some classes of intersection graphs (interval graphs, unit interval graphs, threshold graphs, chordal graphs). 1.
Monotone graph limits and quasimonotone graphs
, 2011
"... Abstract. The recent theory of graph limits gives a powerful framework for understanding the properties of suitable (convergent) sequences (Gn) of graphs in terms of a limiting object which may be represented by a symmetric function W on [0, 1], i.e., a kernel or graphon. In this context it is natur ..."
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Abstract. The recent theory of graph limits gives a powerful framework for understanding the properties of suitable (convergent) sequences (Gn) of graphs in terms of a limiting object which may be represented by a symmetric function W on [0, 1], i.e., a kernel or graphon. In this context it is natural to wish to relate specific properties of the sequence to specific properties of the kernel. Here we show that the kernel is monotone (i.e., increasing in both variables) if and only if the sequence satisfies a ‘quasimonotonicity ’ property defined by a certain functional tending to zero. As a tool we prove an inequality relating the cut and L 1 norms of kernels of the form W1 − W2 with W1 and W2 monotone that may be of interest in its own right; no such inequality holds for general kernels.
ASYMPTOTIC STRUCTURE AND SINGULARITIES IN CONSTRAINED DIRECTED GRAPHS
, 1405
"... Abstract. We study the asymptotics of large directed graphs, constrained to have certain densities of edges and/or outward pstars. Our models are close cousins of exponential random graph models (ERGMs), in which edges and certain other subgraph densities are controlled by parameters. The idea of d ..."
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Abstract. We study the asymptotics of large directed graphs, constrained to have certain densities of edges and/or outward pstars. Our models are close cousins of exponential random graph models (ERGMs), in which edges and certain other subgraph densities are controlled by parameters. The idea of directly constraining edge and other subgraph densities comes from Radin and Sadun [24]. Such modeling circumvents a phenomenon first made precise by Chatterjee and Diaconis [3]: that in ERGMs it is often impossible to independently constrain edge and other subgraph densities. In all our models, we find that large graphs have either uniform or bipodal structure. When edge density (resp. pstar density) is fixed and pstar density (resp. edge density) is controlled by a parameter, we find phase transitions corresponding to a change from uniform to bipodal structure. When both edge and pstar density are fixed, we find only bipodal structures and no phase transition. 1.
Invariant measures via inverse limits of finite structures, ArXiv eprint 1310.8147
, 2013
"... Abstract. Building on recent results regarding symmetric probabilistic constructions of countable structures, we provide a method for constructing probability measures, concentrated on certain classes of countably infinite structures, that are invariant under all permutations of the underlying se ..."
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Abstract. Building on recent results regarding symmetric probabilistic constructions of countable structures, we provide a method for constructing probability measures, concentrated on certain classes of countably infinite structures, that are invariant under all permutations of the underlying set that fix all constants. These measures are constructed from inverse limits of measures on certain finite structures. We use this construction to obtain invariant probability measures concentrated on the classes of countable models of certain firstorder theories, including measures that do not assign positive measure to the isomorphism class of any single model. We also characterize those transitive Borel Gspaces admitting a Ginvariant probability measure, when G is an arbitrary countable product of symmetric groups on a countable set.
A UNIFIED APPROACH TO STRUCTURAL LIMITS  AND LIMITS OF GRAPHS WITH BOUNDED TREEDEPTH
, 2013
"... In this paper we introduce a general framework for the study of limits of relational structures in general and graphs in particular, which is based on a combination of model theory and (functional) analysis. We show how the various approaches to graph limits fit to this framework and that they natu ..."
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In this paper we introduce a general framework for the study of limits of relational structures in general and graphs in particular, which is based on a combination of model theory and (functional) analysis. We show how the various approaches to graph limits fit to this framework and that they naturally appear as “tractable cases ” of a general theory. As an outcome of this, we provide extensions of known results. We believe that this put these into next context and perspective. For example, we prove that the sparse–dense dichotomy exactly corresponds to random free graphons. The second part of the paper is devoted to the study of sparse structures. First, we consider limits of structures with bounded diameter connected components and we prove that in this case the convergence can be “almost ” studied componentwise. We also propose the structure of limits objects for convergent sequences of sparse structures. Eventually, we consider the specific case of limits of colored rooted trees with bounded height and of graphs with bounded treedepth, motivated by their role of elementary brick these graphs play in decompositions of sparse
AN EXAMPLE OF GRAPH LIMITS OF GROWING SEQUENCES OF RANDOM GRAPHS
"... Abstract. We consider in this note a class of growing random graphs obtained by creating vertices sequentially one by one: at each step, we choose uniformly the neighbours of the newly created vertex; its degree is a random variable with a fixed but arbitrary distribution (depending on the number of ..."
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Abstract. We consider in this note a class of growing random graphs obtained by creating vertices sequentially one by one: at each step, we choose uniformly the neighbours of the newly created vertex; its degree is a random variable with a fixed but arbitrary distribution (depending on the number of existing vertices). Examples from this class turn out to be the Erdős–Rényi random graph, a natural random threshold graph, etc. By working with the notion of graph limits, we define a kernel which, under certain conditions, is the limit of the growing random graph. Moreover, for a subclass of models, the growing graph has for any given number of vertices n the same distribution as the random graph with n vertices that the kernel defines. The motivation stems from a model of graph growth whose attachment mechanism does not require information about properties of the graph at each iteration. 1.
A CLASSIFICATION OF ORBITS ADMITTING A UNIQUE INVARIANT MEASURE
"... Abstract. The group S ∞ acts via the logic action on the space of countable structures in a given countable language that have a fixed underlying set. We consider the number of ergodic probability measures on this space that are invariant under the logic action and are concentrated on the isomorph ..."
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Abstract. The group S ∞ acts via the logic action on the space of countable structures in a given countable language that have a fixed underlying set. We consider the number of ergodic probability measures on this space that are invariant under the logic action and are concentrated on the isomorphism class of a particular structure. We show that this number must be either zero, or one, or continuum. Further, such an isomorphism class admits a unique S∞invariant probability measure precisely when the structure is highly homogeneous; by a result of Peter J. Cameron, these are the structures that are interdefinable with one of the five reducts of the rational