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

Abstract

We introduce and develop a theory of limits for sequences of sparsegraphs based on Lp graphons, which generalizes both the existing L∞ theory ofdense graph limits and its extension by Bollob ́as and Riordan to sparse graphswithout dense spots. In doing so, we replace the no dense spots hypothesiswith weaker assumptions, which allow us to analyze graphs with power lawdegree distributions. This gives the first broadly applicable limit theory forsparse graphs with unbounded average degrees. In this paper, we lay the foun-dations of the Lp theory of graphons, characterize convergence, and developcorresponding random graph models, while we prove the equivalence of severalalternative metrics in a companion paper.