Abstract
We study a recent model for edge exchangeable random graphs introduced by Crane and Dempsey; in particular we study asymptotic properties of the random simple graph obtained by merging multiple edges. We study a number of examples, and show that the model can produce dense, sparse and extremely sparse random graphs. One example yields a power-law degree distribution. We give some examples where the random graph is dense and converges a.s. in the sense of graph limit theory, but also an example where a.s. every graph limit is the limit of some subsequence. Another example is sparse and yields convergence to a non-integrable generalized graphon defined on (0,infty ).
Highlights
A model for edge exchangeable random graphs and hypergraphs was recently introduced by [11,12], who gave a representation theorem showing that every infinite edge exchangeable random hypergraph can be constructed by this model
We concentrate on the graph case, we state the definitions in Sect. 4 more generally for hypergraphs
Previous papers concentrate on the multigraph version; in contrast and as a complement, in the present paper we study mainly the simple graph version
Summary
A model for edge exchangeable random graphs and hypergraphs was recently introduced by [11,12], who gave a representation theorem showing that every infinite edge exchangeable random hypergraph can be constructed by this model. Many of the edges will be repeated many times, see e.g. Remark 6.7, and the multigraph and the simple graph versions can be expected to be quite different. The model is, as said above, based on an arbitrary distribution of edges Different choices of this distribution can give a wide range of different types of random graphs, and the main purpose of the paper is to investigate the types of random graphs that may be created by this model; for this purpose we give some general results on the numbers of vertices and edges, and a number of examples ranging from dense to very sparse graphs.
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