Line graph

Algorithm

Discrete mathematics

Applied mathematics

Graph theory

Theoretical computer science

Convergence (economics)

Random graph

Economics

Computer science

Limit (mathematics)

Pathwidth

Graph

Mathematics

Dense graph

Mathematical analysis

Artificial intelligence

Combinatorics

1-planar graph

Economic growth

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Transactions of the American Mathematical Society

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 Bollob\'as 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.

An 𝐿^{𝑝} theory of sparse graph convergence I: Limits, sparse random graph models, and power law distributions