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

Borgs, Christian; Chayes, Jennifer; Cohn, Henry; Zhao, Yufei

doi:10.1090/tran/7543

Cited by 118 publications

(190 citation statements)

References 44 publications

(81 reference statements)

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“…They also characterize the convergence of graph sequences based on graph homomorphism densities . Recently, graphon theory has been generalized for sparse graph sequences .…”

Section: Introductionmentioning

confidence: 99%

A graphon approach to limiting spectral distributions of Wigner‐type matrices

Zhu

2019

Random Struct Algorithms

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We present a new approach, based on graphon theory, to finding the limiting spectral distributions of general Wigner‐type matrices. This approach determines the moments of the limiting measures and the equations of their Stieltjes transforms explicitly with weaker assumptions on the convergence of variance profiles than previous results. As applications, we give a new proof of the semicircle law for generalized Wigner matrices and determine the limiting spectral distributions for three sparse inhomogeneous random graph models with sparsity ω(1/n): inhomogeneous random graphs with roughly equal expected degrees, W‐random graphs and stochastic block models with a growing number of blocks. Furthermore, we show our theorems can be applied to random Gram matrices with a variance profile for which we can find the limiting spectral distributions under weaker assumptions than previous results.

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“…They also characterize the convergence of graph sequences based on graph homomorphism densities . Recently, graphon theory has been generalized for sparse graph sequences .…”

Section: Introductionmentioning

confidence: 99%

A graphon approach to limiting spectral distributions of Wigner‐type matrices

Zhu

2019

Random Struct Algorithms

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“…It seems that the two core ingredients of our proof, Lemma 3.4 and the second moment argument do have sparse counterparts. The sparse counterpart to Lemma 3.4 is , Theorem 2.14] which says that if

p_{n} ≫ \frac{1}{n}

then the sequence

\frac{1}{p_{n}} \cdot G (n, p_{n} \cdot W)

(here, the factor

\frac{1}{p_{n}}

in front of the random graph

G (n, p_{n} \cdot W)

denotes edge weighting; this is the natural way to deal with the scaling in this situation) converges to W in the cut‐distance almost surely. Our second moment argument is complicated but it builds on the seminal work which works down to the range

p_{n} = Θ (\frac{1}{\sqrt{n}})

. Thus, at least when

W \in L^{\infty} (Ω^{2})

, our methods possibly extend to this range.…”

Section: Discussionmentioning

confidence: 99%

“…such that for each i ∈ N, W i × i = 0 almost everywhere. Let us turn our attention to the main subject of the paper, that is, to the behavior of the clique number in G(n, W ) scaled as in (3). As a warm-up for studying (3), we first deal with its bipartite counterpart.…”

Section: Proposition 21mentioning

confidence: 99%

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Cliques in dense inhomogeneous random graphs

Doležal

Hladký

Máthé

2017

Random Struct. Alg.

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The theory of dense graph limits comes with a natural sampling process which yields an inhomogeneous variant G(n,W) of the Erdos-Renyi random graph. Here we study the clique number of these random graphs. We establish the concentration of the clique number of G(n,W) for each fixed n, and give examples of graphons for which G(n,W) exhibits wild long-term behavior. Our main result is an asymptotic formula which gives the almost sure clique number of these random graphs. We obtain a similar result for the bipartite version of the problem. We also make an observation that might be of independent interest: Every graphon avoiding a fixed graph is countably-partite.Comment: 30 pages, 2 figure; accepted to publication in Random Structures and Algorithm

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“…Borgs et al . () and Orbanz and Roy ()): the generated graph is dense; unless it is k ‐partite or disconnected, the distance between any two vertices in an infinite graph is almost surely 1 or 2, etc. A graphon can encode any fixed pattern on some number n of vertices, but this pattern then occurs on every possible subgraph of size n with fixed probability. (b)Configuration models are popular in probability (because of their simplicity) but have limited use in statistics unless the quantity of interest is the degree sequence itself: they are ‘maximally random’ given the degrees, in a manner similar to exponential families being maximally random given a sufficient statistic, and are thus insensitive to any structure that is not captured by the degrees.…”

Section: Introductionmentioning

confidence: 99%

Random-Walk Models of Network Formation and Sequential Monte Carlo Methods for Graphs

Bloem-Reddy

Orbanz

2018

Journal of the Royal Statistical Society Series B: Statistical Methodology

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Summary We introduce a class of generative network models that insert edges by connecting the starting and terminal vertices of a random walk on the network graph. Within the taxonomy of statistical network models, this class is distinguished by permitting the location of a new edge to depend explicitly on the structure of the graph, but being nonetheless statistically and computationally tractable. In the limit of infinite walk length, the model converges to an extension of the preferential attachment model—in this sense, it can be motivated alternatively by asking what preferential attachment is an approximation to. Theoretical properties, including the limiting degree sequence, are studied analytically. If the entire history of the graph is observed, parameters can be estimated by maximum likelihood. If only the final graph is available, its history can be imputed by using Markov chain Monte Carlo methods. We develop a class of sequential Monte Carlo algorithms that are more generally applicable to sequential network models and may be of interest in their own right. The model parameters can be recovered from a single graph generated by the model. Applications to data clarify the role of the random‐walk length as a length scale of interactions within the graph.

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

Cited by 118 publications

References 44 publications

A graphon approach to limiting spectral distributions of Wigner‐type matrices

A graphon approach to limiting spectral distributions of Wigner‐type matrices

Cliques in dense inhomogeneous random graphs

Random-Walk Models of Network Formation and Sequential Monte Carlo Methods for Graphs

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