  We extend the $L^p$ theory of sparse graph limits, which was introduced in a
companion paper, by analyzing different notions of convergence. Under suitable
restrictions 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. Our
theorems extend the broad applicability of dense graph convergence to all
sparse graphs with unbounded average degree, while the proofs require new
techniques based on uniform upper regularity. Examples to which our theory
applies include stochastic block models, power law graphs, and sparse versions
of $W$-random graphs.
