On Edge Exchangeable Random Graphs

Janson, Svante

doi:10.1007/s10955-017-1832-9

Cited by 23 publications

(30 citation statements)

References 37 publications

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“…Remark 5. The above Poisson-Dirichlet universality convergence can be rephrased in the theory of edge exchangeable random graphs as the convergence towards the rank 1 multigraph driven by the Poisson-Dirichlet partition, see [17,Example 7.1 and 7.8] and [12].…”

Section: Number Of Verticesmentioning

confidence: 99%

Universality for Random Surfaces in Unconstrained Genus

Budzinski¹,

Curien²,

Petri³

2019

Electron. J. Combin.

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Starting from an arbitrary sequence of polygons whose total perimeter is $2n$, we can build an (oriented) surface by pairing their sides in a uniform fashion. Chmutov & Pittel have shown that, regardless of the configuration of polygons we started with, the degree sequence of the graph obtained this way is remarkably constant in total variation distance and converges towards a Poisson–Dirichlet partition as $n \to \infty$. We actually show that several other geometric properties of the graph are universal. En route we provide an alternative proof of a weak version of the result of Chmutov & Pittel using probabilistic techniques and related to the circle of ideas around the peeling process of random planar maps. At this occasion we also fill a gap in the existing literature by surveying the properties of a uniform random map with $n$ edges. In particular we show that the diameter of a random map with $n$ edges converges in law towards a random variable taking only values in $\{2,3\}$.

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Section: Number Of Verticesmentioning

confidence: 99%

Universality for Random Surfaces in Unconstrained Genus

Budzinski¹,

Curien²,

Petri³

2019

Electron. J. Combin.

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“…Random-walk models generate networks one edge at a time; this specification of a graph may be viewed as a sequence of edges. Recent work on so-called edge exchangeable graphs (Crane and Dempsey, 2017;Williamson, 2016;Cai et al, 2016;Janson, 2017) define network models in terms of an exchangeable sequence of edges. Edges in random-walk models are not exchangeable.…”

Section: Relationship To Preferential Attachment and Other Modelsmentioning

confidence: 99%

“…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. Other models based on degrees (PA models, the model of Caron and Fox () and so‐called rank 1 edge exchangeable models (Janson, )) are similarly constrained.…”

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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“…Another class of attempts suggests to completely give up on the node label exchangeability requirement, and to consider edge exchangeability instead, e.g., using variations of Pitman–Yor processes [ 36 , 37 , 38 ]. It remains unclear at present whether these developments imply that too many network models that were found to be quite useful in practice and that do use node labels

, are statistically hopeless.…”

Section: Impasse With Sparsitymentioning

confidence: 99%

“…Already several works have addressed this problem [ 32 , 33 , 34 , 35 , 36 , 37 , 38 ], using different approaches such as relaxing the condition

but always characterizing models with average degree diverging with the network size N , considering edge exchangeable models or alternatively using an embedding space as a basic mechanism to combine sparsity with projectivity and exchangeability [ 31 , 39 ].…”

Section: Introductionmentioning

confidence: 99%

Sparse Power-Law Network Model for Reliable Statistical Predictions Based on Sampled Data

Kartun-Giles

Krioukov

Gleeson

et al. 2018

Entropy

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A projective network model is a model that enables predictions to be made based on a subsample of the network data, with the predictions remaining unchanged if a larger sample is taken into consideration. An exchangeable model is a model that does not depend on the order in which nodes are sampled. Despite a large variety of non-equilibrium (growing) and equilibrium (static) sparse complex network models that are widely used in network science, how to reconcile sparseness (constant average degree) with the desired statistical properties of projectivity and exchangeability is currently an outstanding scientific problem. Here we propose a network process with hidden variables which is projective and can generate sparse power-law networks. Despite the model not being exchangeable, it can be closely related to exchangeable uncorrelated networks as indicated by its information theory characterization and its network entropy. The use of the proposed network process as a null model is here tested on real data, indicating that the model offers a promising avenue for statistical network modelling.

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On Edge Exchangeable Random Graphs

Cited by 23 publications

References 37 publications

Universality for Random Surfaces in Unconstrained Genus

Universality for Random Surfaces in Unconstrained Genus

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

Sparse Power-Law Network Model for Reliable Statistical Predictions Based on Sampled Data

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