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We extend the $L^p$ theory of sparse graph limits, which was introduced in acompanion paper, by analyzing different notions of convergence. Under suitablerestrictions on node weights, we prove the equivalence of metric convergence,quotient convergence, microcanonical ground state energy convergence,microcanonical free energy convergence, and large deviation convergence. Ourtheorems extend the broad applicability of dense graph convergence to allsparse graphs with unbounded average degree, while the proofs require newtechniques based on uniform upper regularity. Examples to which our theoryapplies include stochastic block models, power law graphs, and sparse versionsof $W$-random graphs.

An $L^p$ theory of sparse graph convergence II: LD convergence, quotients, and right convergence