An $L^{p}$ theory of sparse graph convergence II: LD convergence, quotients and right convergence

@article{Borgs2014AnT,
  title={An \$L^\{p\}\$ theory of sparse graph convergence II: LD convergence, quotients and right convergence},
  author={Christian Borgs and Jennifer T. Chayes and Henry Cohn and Yufei Zhao},
  journal={Annals of Probability},
  year={2014},
  volume={46},
  pages={337-396}
}
We extend the LpLp theory of sparse graph limits, which was introduced in a companion paper, by analyzing different notions of convergence. Under suitable restrictions on node weights, we prove the equivalence of metric convergence, quotient convergence, microcanonical ground state energy convergence, microcanonical free energy convergence and large deviation convergence. Our theorems extend the broad applicability of dense graph convergence to all sparse graphs with unbounded average degree… 

Figures from this paper

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

A theory of limits for sequences of sparse graphs based on graphons is introduced, 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.

On limits of sparse random graphs

  • L. Vena
  • Mathematics
    Electron. Notes Discret. Math.
  • 2016

Convergence of graphs with intermediate density

We propose a notion of graph convergence that interpolates between the Benjamini--Schramm convergence of bounded degree graphs and the dense graph convergence developed by Laszlo Lovasz and his

Large Deviations for Sparse Graphs

The results of the previous chapters were derived using tools from graph limit theory. This theory, however, is inadequate for understanding the behavior of sparse graphs. The goal of this chapter is

A unified view of graph regularity via matrix decompositions

The core of the approach is an abstracted matrix decomposition, which can be computed by a simple algorithm by Charikar and implies new PTASes for MAX‐CUT, MAX‐BISECTION, MIN‐BisECTION for a significantly expanded class of input graphs.

Action convergence of operators and graphs

Abstract We present a new approach to graph limit theory that unifies and generalizes the two most well-developed directions, namely dense graph limits (even the more general $L^p$ limits) and

Continuum limit of p-Laplacian evolution problems on graphs: Lq graphons and sparse graphs

This paper derives a bound on the distance between two continuous-in-time trajectories defined by two different evolution systems so-called L q-graphons, and provides rate of convergence of solutions for the discrete models to the solution of the continuous problem as the number of vertices grows.

Remarks on power-law random graphs

  • Mei Yin
  • Mathematics
    Stochastic Processes and their Applications
  • 2022

Cut norm discontinuity of triangular truncation of graphons

Matrix Decompositions and Sparse Graph Regularity

It turns out that cut pseudorandomness unifies several important pseudorRandomness concepts in prior work, and it is shown that upper regularity and a version of low threshold rank are both special cases, thus implying weak and strong regularity lemmas for these graph classes where only weak ones were previously known.
...

References

SHOWING 1-10 OF 16 REFERENCES

Left and right convergence of graphs with bounded degree

A similar equivalence is established in the bounded degree case, if the set of graphs in the definition of right-convergence is appropriately restricted.

Recurrence of Distributional Limits of Finite Planar Graphs

Suppose that $G_j$ is a sequence of finite connected planar graphs, and in each $G_j$ a special vertex, called the root, is chosen randomly-uniformly. We introduce the notion of a distributional

Metrics for sparse graphs

This paper deals mainly with graphs with $o(n^2)$ but $\omega(n)$ edges: a companion paper [arXiv:0812.2656] will discuss the (more problematic still) case of {\em extremely sparse} graphs, with O( n) edges.

Moments of Two-Variable Functions and the Uniqueness of Graph Limits

AbstractFor a symmetric bounded measurable function W on [0, 1]2 and a simple graph F, the homomorphism density $$t(F,W) = \int _{[0,1]^{V (F)}} \prod_ {i j\in E(F)} W(x_i, x_j)dx .$$ can be thought

Quick Approximation to Matrices and Applications

The matrix approximation is generalized to multi-dimensional arrays and from that derive approximation algorithms for all dense Max-SNP problems and the Regularity Lemma is derived.

Szemerédi’s Lemma for the Analyst

Abstract.Szemerédi’s regularity lemma is a fundamental tool in graph theory: it has many applications to extremal graph theory, graph property testing, combinatorial number theory, etc. The goal of

On the completeness of a certain metric space with an application to Blaschke’s selection theorem

2. Preliminaries. Let K be a metric space with elements x, y, • • • and distance function d(x, y). A sequence xi, x2, • • • in K such that ^2T^(Xiy Xi+i) converges has been called an absolutely

Counting Graph Homomorphisms

Counting homomorphisms between graphs (often with weights) comes up in a wide variety of areas, including extremal graph theory, properties of graph products, partition functions in statistical

An Lp theory of sparse graph convergence I: limits

  • sparse random graph models, and power law distributions, preprint
  • 2014