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Stationary random graphs on \(\mathbb Z\) with prescribed iid degrees and finite mean connections. (English) Zbl 1127.05093

Summary: Let \(F\) be a probability distribution with support on the non-negative integers. A model is proposed for generating stationary simple graphs on \(\mathbb Z\) with degree distribution \(F\) and it is shown for this model that the expected total length of all edges at a given vertex is finite if \(F\) has finite second moment. It is not hard to see that any stationary model for generating simple graphs on \(\mathbb Z\) will give infinite mean for the total edge length per vertex if \(F\) does not have finite second moment. Hence, finite second moment of \(F\) is a necessary and sufficient condition for the existence of a model with finite mean total edge length.

MSC:

05C80 Random graphs (graph-theoretic aspects)
60G50 Sums of independent random variables; random walks
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