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We introduce and develop a theory of limits for sequences of sparse graphsbased on $L^p$ graphons, which generalizes both the existing $L^\infty$ theoryof dense graph limits and its extension by Bollob\'as and Riordan to sparsegraphs without dense spots. In doing so, we replace the no dense spotshypothesis with weaker assumptions, which allow us to analyze graphs with powerlaw degree distributions. This gives the first broadly applicable limit theoryfor sparse graphs with unbounded average degrees. In this paper, we lay thefoundations of the $L^p$ theory of graphons, characterize convergence, anddevelop corresponding random graph models, while we prove the equivalence ofseveral alternative metrics in a companion paper.

An $L^p$ theory of sparse graph convergence I: limits, sparse random graph models, and power law distributions