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Graphons, cut norm and distance, couplings and rearrangements
, 2010
"... We give a survey of basic results on the cut norm and cut metric for graphons (and sometimes more general kernels), with emphasis on the equivalence problem. The main results are not new, but we add various technical complements. We allow graphons on general probability spaces whenever possible. We ..."
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Cited by 13 (5 self)
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We give a survey of basic results on the cut norm and cut metric for graphons (and sometimes more general kernels), with emphasis on the equivalence problem. The main results are not new, but we add various technical complements. We allow graphons on general probability spaces whenever possible. We also give some new results for {0,1}valued graphons.
Graph limits and hereditary properties
, 2011
"... 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). ..."
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Cited by 5 (4 self)
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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).
ON VERTEX, EDGE, AND VERTEXEDGE RANDOM GRAPHS
, 2008
"... We consider three classes of random graphs: edge random graphs, vertex random graphs, and vertexedge random graphs. Edge random graphs are ErdősRényi random graphs [5, 6], vertex random graphs are generalizations of geometric random graphs [16], and vertexedge random graphs generalize both. The ..."
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Cited by 1 (1 self)
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We consider three classes of random graphs: edge random graphs, vertex random graphs, and vertexedge random graphs. Edge random graphs are ErdősRényi random graphs [5, 6], vertex random graphs are generalizations of geometric random graphs [16], and vertexedge random graphs generalize both. The names of these three types of random graphs describe where the randomness in the models lies: in the edges, in the vertices, or in both. We show that vertexedge random graphs, ostensibly the most general of the three models, can be approximated arbitrarily closely by vertex random graphs, but that the two categories are distinct.
Limits of interval orders and semiorders
, 2011
"... We study poset limits given by sequences of finite interval orders or, as a special case, finite semiorders. In the interval order case, we show that every such limit can be represented by a probability measure on the space of closed subintervals of [0, 1], and we define a subset of such measures ..."
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Cited by 1 (0 self)
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We study poset limits given by sequences of finite interval orders or, as a special case, finite semiorders. In the interval order case, we show that every such limit can be represented by a probability measure on the space of closed subintervals of [0, 1], and we define a subset of such measures that yield a unique representation. In the semiorder case, we similarly find unique representations by a class of distribution functions.
ON VERTEX, EDGE, AND VERTEXEDGE RANDOM GRAPHS (EXTENDED ABSTRACT)
"... We consider three classes of random graphs: edge random graphs, vertex random graphs, and vertexedge random graphs. Edge random graphs are ErdősRényi random graphs [8, 9], vertex random graphs are generalizations of geometric random graphs [20], and vertexedge random graphs generalize both. The n ..."
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We consider three classes of random graphs: edge random graphs, vertex random graphs, and vertexedge random graphs. Edge random graphs are ErdősRényi random graphs [8, 9], vertex random graphs are generalizations of geometric random graphs [20], and vertexedge random graphs generalize both. The names of these three types of random graphs describe where the randomness in the models lies: in the edges, in the vertices, or in both. We show that vertexedge random graphs, ostensibly the most general of the three models, can be approximated arbitrarily closely by vertex random graphs, but that the two categories are distinct.
POSET LIMITS AND EXCHANGEABLE RANDOM POSETS
, 2009
"... We develop a theory of limits of finite posets in close analogy to the recent theory of graph limits. In particular, we study representations of the limits by functions of two variables on a probability space, and connections to exchangeable random infinite posets. ..."
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We develop a theory of limits of finite posets in close analogy to the recent theory of graph limits. In particular, we study representations of the limits by functions of two variables on a probability space, and connections to exchangeable random infinite posets.