DSpace@MIT (Massachusetts Institute of Technology)· 2014
An $L^p$ theory of sparse graph convergence I: limits, sparse random graph models, and power law distributions
- 93citations
- 2014year
Short summary
A new $L^p$ graphon theory provides the first broadly applicable limit theory for sparse graphs with unbounded average degrees, analyzing graphs with power law degree distributions.
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Key points
- Introduces an $L^p$ graphon theory for sparse graph convergence.
- Generalizes existing dense graph limit theories and sparse graph extensions.
- Allows analysis of graphs with power law degree distributions.
- Provides the first broadly applicable limit theory for sparse graphs with unbounded average degrees.
AI-generated from the title and abstract; the full text is not read.
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.
The authors' abstract, as published at the source. DSpace@MIT (Massachusetts Institute of Technology), 2014 · DOI ↗
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Field: Discrete Mathematics and Combinatorics
Discrete Mathematics and CombinatoricsMathematics