An Lp theory of sparse graph convergence I: limits, sparse random graph models, and power law distributions

We introduce and develop a theory of limits for sequences of sparse graphs based on Lp graphons, which generalizes both the existing L1 theory of dense graph limits and its extension by Bollobas and Riordan to sparse graphs without dense spots. In doing so, we replace the no dense spots hypothesis with weaker assumptions, which allow us to analyze graphs with power law degree distributions. This gives the rst broadly applicable limit theory for sparse graphs with unbounded average degrees. In this paper, we lay the foundations of the Lp theory of graphons, characterize convergence, and develop corresponding random graph models, while we prove the equivalence of several alternative metrics in a companion paper.