An $L\log L$ theory of sparse graph convergence
Borgs, Chayes, Cohn and Zhao developed a limit theory for sparse graphs based on $L^p$ upper regularity with $p>1$, where Holder's inequality controls small sets. We develop the analogous theory in the Orlicz class $L\log L$, replacing Holder by the Young-Fenchel inequality for $Ψ(u)=(1+u)\log(1+u)-u$ and $Ψ^*(v)=e^v-1-v$. We prove a weak regularity lemma for upper $Ψ$-regular graphons with $4^{O(1/\varepsilon^2)}$ parts, using the modular $\intΨ$ as the energy; that upper $Ψ$-regular sequences have cut-metric convergent subsequences; that convergent sequences with $L\log L$ limits are upper $Ψ$-regular; and that $L\log L$ balls are cut-metric compact. The class strictly contains every $L^p$ class with $p>1$, and we give sparse $W$-random graphs that are upper $Ψ$-regular but not upper $L^p$ regular for any $p>1$. For graphs with a given degree sequence we prove, by Stein's method of exchangeable pairs with switchings, that if the average degree tends to infinity, the density to zero and the degree profile converges in $L^1$ to $φ$, uniformly random graphs converge in the normalized cut metric to $φ\otimesφ$, with no condition on the maximum degree; the sequence is upper $Ψ$-regular when $φ\in L\log L$. We also show that the maximum-entropy $β$-model is entrywise Chung-Lu when $d_{\max}^2=o(M)$, and that cut-closeness to it follows from a lower bound $e^{-o(M)}$ on its probability of having the prescribed degrees. Finally, $Ψ$ is the large deviation rate function of sparse Erdos-Renyi graphs up to a factor $1+O(1/\log\frac1p)$.