Identifiability for Graphexes and the Weak Kernel Metric

Borgs, Christian; Chayes, Jennifer; Cohn, Henry; Lovász, László

doi:10.1007/978-3-662-59204-5_3

Cited by 10 publications

(10 citation statements)

References 25 publications

(115 reference statements)

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“…Remark 2. Many definitions of random graphs allow for {ξ i } to be a random sample from some probability density function on [0, 1] (e.g., [5,23,21,26], and references therein) With the right distribution defined on [0, 1], many of the results would be identical when taking an expectation over the random sample {ξ i }.…”

Section: Kernel Probability Measuresmentioning

confidence: 99%

Approximate Fréchet Mean for Data Sets of Sparse Graphs

Ferguson,

Meyer

2021

Preprint

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To characterize the location (mean, median) of a set of graphs, one needs a notion of centrality that is adapted to metric spaces, since graph sets are not Euclidean spaces. A standard approach is to consider the Fréchet mean. In this work, we equip a set of graph with the pseudometric defined by the 2 norm between the eigenvalues of their respective adjacency matrix . Unlike the edit distance, this pseudometric reveals structural changes at multiple scales, and is well adapted to studying various statistical problems on sets of graphs. We describe an algorithm to compute an approximation to the Fréchet mean of a set of undirected unweighted graphs with a fixed size.

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Section: Kernel Probability Measuresmentioning

confidence: 99%

Approximate Fréchet Mean for Data Sets of Sparse Graphs

Ferguson,

Meyer

2021

Preprint

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“…Once graph theory can describe the empirically relevant, asymptotic behavior of sparse graph sequences, these results will find applications in network science and complex systems forecasting. Promising limit objects include "graphings" (for degree bounded graphs) 14 and "graphexes" (for graphs as random measures) 33,34 .…”

Section: Learning From Mathematical Physicsmentioning

confidence: 99%

Bridging the gap between graphs and networks

Iñiguez

Battiston²,

Karsai³

2020

Commun Phys

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Network science has become a powerful tool to describe the structure and dynamics of real-world complex physical, biological, social, and technological systems. Largely built on empirical observations to tackle heterogeneous, temporal, and adaptive patterns of interactions, its intuitive and flexible nature has contributed to the popularity of the field. With pioneering work on the evolution of random graphs, graph theory is often cited as the mathematical foundation of network science. Despite this narrative, the two research communities are still largely disconnected. In this commentary, we discuss the need for further crosspollination between fieldsbridging the gap between graphs and networksand how network science can benefit from such influence. A more mathematical network science may clarify the role of randomness in modeling, hint at underlying laws of behavior, and predict yet unobserved complex networked phenomena in nature.The mathematics of a networked reality Behind the history of mankind's scientific progress there is a story of mathematical theory building. One where scientists not only uncover the behavior of empirical phenomena through experiments and data analysis, but develop theories that abstract such behavior mathematically 1 . This process amounts to scientific understanding when mathematics is "sufficiently isomorphic" to reality 2 ; when theory includes the mechanisms deemed most important and, despite ignoring some features, still allows for logical inferences that accurately describe the phenomena of interest 3 . A sufficient similarity in structure between mathematics and reality allows scientists to forecast events, control natural systems, and even predict behavior that has not yet been confirmed empirically 4 .Up until recent times, scientific efforts towards a mathematical representation of reality have mostly focused on elements of the "simplest" natural systems at both the smallest and largest scales: the physical, chemical, and biological entities comprising the microscopic world, as well as thermodynamic systems and the large-scale structure of the Universe. The success of this

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“…The jumble-norm distance turns out to be a more useful tool when introducing a metric for generalized graphons, in part due to the fact that it can be considered as an L 2 -version of the cut-norm distance (since corresponds to the L 2 -norm of the characteristic function of the set S × T ), leading to simpler expressions for bounds (cf. [5] and [14]).…”

Section: Motivationmentioning

confidence: 99%

On the dense preferential attachment graph models and their graphon induced counterpart

Backhausz

Kunszenti-Kovács

2019

J. Appl. Probab.

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Letting M denote the space of finite measures on N, and µ λ ∈ M denote the Poisson distribution with parameter λ, the function W : [0, 1] 2 → M given by W (x, y) = µ c log x log y is called the PAG graphon with density c. It is known that this is the limit, in the multigraph homomorphism sense, of the dense Preferential Attachment Graph (PAG) model with edge density c. This graphon can then in turn be used to generate the socalled W-random graphs in a natural way. The aim of this paper is to compare the dense PAG model with the W-random graph model obtained from the corresponding graphon. Motivated by the multigraph limit theory, we investigate the expected jumble norm distance of the two models in terms on the number of vertices n. We present a coupling for which the expectation can be bounded from above by O(log 2 n · n −1/3 ), and provide a universal lower bound that is coupling independent, but with a worse exponent.

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Identifiability for Graphexes and the Weak Kernel Metric

Cited by 10 publications

References 25 publications

Approximate Fréchet Mean for Data Sets of Sparse Graphs

Approximate Fréchet Mean for Data Sets of Sparse Graphs

Bridging the gap between graphs and networks

On the dense preferential attachment graph models and their graphon induced counterpart

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