An $L^p$ 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 sparse graphs based on $L^p$ graphons, which generalizes both the existing $L^{\infty }$ theory of dense graph limits and its extension by Bollobás 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 first broadly applicable limit theory for sparse graphs with unbounded average degrees. In this paper, we lay the foundations of the $L^p$ theory of graphons, characterize convergence, and develop corresponding random graph models, while we prove the equivalence of several alternative metrics in a companion paper.

44 pages

References in corpus (7)

• Power-law distributions in empirical data
• On replica symmetry of large deviations in random graphs
• Extremal results in sparse pseudorandom graphs
• A relative Szemerédi theorem
• An $L^p$ theory of sparse graph convergence II: LD convergence, quotients, and right convergence
• A Model Theory Approach to Structural Limits
• Convergence of graphs with intermediate density

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• An $L^p$ theory of sparse graph convergence II: LD convergence, quotients, and right convergence
• The semilinear heat equation on sparse random graphs
• Sampling perspectives on sparse exchangeable graphs
• Network Dynamics on Graphops
• A Graphon Approach to Limiting Spectral Distributions of Wigner-type Matrices
• Bootstrapping Exchangeable Random Graphs
• Consistency of Maximum Likelihood for Continuous-Space Network Models I
• Convergence of graphs with intermediate density
• Pattern Formation in Random Networks Using Graphons
• Multigraph limits, unbounded kernels, and Banach space decorated graphs
• Mean-field limit of non-exchangeable systems
• Semantic Limits of Dense Combinatorial Objects
• Relating the cut distance and the weak* topology for graphons
• Hypergraphon Mean Field Games
• Large deviations and exact asymptotics for constrained exponential random graphs
• Differential calculus on the space of countable labelled graphs
• The Shortest-Path distance on graphons