Remarks on power-law random graphs

13 Aug 2020  ·  Yin Mei ·

We interpret a classical problem for the Poisson approximations for $U$-statistics in the context of random graphs. Our model is closely connected to a motivating example in Borgs et al. [Trans. Amer. Math. Soc. 372 (2019), 3019-3062], and may be termed a power-law random graph without Bernoulli edges. We examine the distinctively different structures of the limit graph in detail in all regimes, critical, super-critical, and sub-critical. Many interesting results are established. Though elementary at first sight, our model serves as an uncovered boundary case between different types of graph convergence.

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Probability Statistical Mechanics Mathematical Physics Mathematical Physics