Thematische Bibliographien / Probability-Graphons

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Auswahl der wissenschaftlichen Literatur zum Thema „Probability-Graphons“

Autor: Grafiati

Veröffentlicht am 14. Dezember 2024

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Zeitschriftenartikel zum Thema "Probability-Graphons"

Ackerman, Nate, Cameron E. Freer, Younesse Kaddar, et al. "Probabilistic Programming Interfaces for Random Graphs: Markov Categories, Graphons, and Nominal Sets." Proceedings of the ACM on Programming Languages 8, POPL (2024): 1819–49. http://dx.doi.org/10.1145/3632903.

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We study semantic models of probabilistic programming languages over graphs, and establish a connection to graphons from graph theory and combinatorics. We show that every well-behaved equational theory for our graph probabilistic programming language corresponds to a graphon, and conversely, every graphon arises in this way. We provide three constructions for showing that every graphon arises from an equational theory. The first is an abstract construction, using Markov categories and monoidal indeterminates. The second and third are more concrete. The second is in terms of traditional measure theoretic probability, which covers 'black-and-white' graphons. The third is in terms of probability monads on the nominal sets of Gabbay and Pitts. Specifically, we use a variation of nominal sets induced by the theory of graphs, which covers Erdős-Rényi graphons. In this way, we build new models of graph probabilistic programming from graphons.

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McMillan, Audra, and Adam Smith. "When is non-trivial estimation possible for graphons and stochastic block models?‡." Information and Inference: A Journal of the IMA 7, no. 2 (2017): 169–81. http://dx.doi.org/10.1093/imaiai/iax010.

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Abstract Block graphons (also called stochastic block models) are an important and widely studied class of models for random networks. We provide a lower bound on the accuracy of estimators for block graphons with a large number of blocks. We show that, given only the number $k$ of blocks and an upper bound $\rho$ on the values (connection probabilities) of the graphon, every estimator incurs error ${\it{\Omega}}\left(\min\left(\rho, \sqrt{\frac{\rho k^2}{n^2}}\right)\right)$ in the $\delta_2$ metric with constant probability for at least some graphons. In particular, our bound rules out any non-trivial estimation (that is, with $\delta_2$ error substantially less than $\rho$) when $k\geq n\sqrt{\rho}$. Combined with previous upper and lower bounds, our results characterize, up to logarithmic terms, the accuracy of graphon estimation in the $\delta_2$ metric. A similar lower bound to ours was obtained independently by Klopp et al.

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ZHAO, YUFEI. "On the Lower Tail Variational Problem for Random Graphs." Combinatorics, Probability and Computing 26, no. 2 (2016): 301–20. http://dx.doi.org/10.1017/s0963548316000262.

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We study the lower tail large deviation problem for subgraph counts in a random graph. Let XH denote the number of copies of H in an Erdős–Rényi random graph $\mathcal{G}(n,p)$. We are interested in estimating the lower tail probability $\mathbb{P}(X_H \le (1-\delta) \mathbb{E} X_H)$ for fixed 0 < δ < 1.Thanks to the results of Chatterjee, Dembo and Varadhan, this large deviation problem has been reduced to a natural variational problem over graphons, at least for p ≥ n−αH (and conjecturally for a larger range of p). We study this variational problem and provide a partial characterization of the so-called ‘replica symmetric’ phase. Informally, our main result says that for every H, and 0 < δ < δH for some δH > 0, as p → 0 slowly, the main contribution to the lower tail probability comes from Erdős–Rényi random graphs with a uniformly tilted edge density. On the other hand, this is false for non-bipartite H and δ close to 1.

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Braides, Andrea, Paolo Cermelli та Simone Dovetta. "Γ-limit of the cut functional on dense graph sequences". ESAIM: Control, Optimisation and Calculus of Variations 26 (2020): 26. http://dx.doi.org/10.1051/cocv/2019029.

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A sequence of graphs with diverging number of nodes is a dense graph sequence if the number of edges grows approximately as for complete graphs. To each such sequence a function, called graphon, can be associated, which contains information about the asymptotic behavior of the sequence. Here we show that the problem of subdividing a large graph in communities with a minimal amount of cuts can be approached in terms of graphons and the Γ-limit of the cut functional, and discuss the resulting variational principles on some examples. Since the limit cut functional is naturally defined on Young measures, in many instances the partition problem can be expressed in terms of the probability that a node belongs to one of the communities. Our approach can be used to obtain insights into the bisection problem for large graphs, which is known to be NP-complete.

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HATAMI, HAMED, and SERGUEI NORINE. "The Entropy of Random-Free Graphons and Properties." Combinatorics, Probability and Computing 22, no. 4 (2013): 517–26. http://dx.doi.org/10.1017/s0963548313000175.

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Every graphon defines a random graph on any given number n of vertices. It was known that the graphon is random-free if and only if the entropy of this random graph is subquadratic. We prove that for random-free graphons, this entropy can grow as fast as any subquadratic function. However, if the graphon belongs to the closure of a random-free hereditary graph property, then the entropy is O(n log n). We also give a simple construction of a non-step-function random-free graphon for which this entropy is linear, refuting a conjecture of Janson.

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Keliger, Dániel, Illés Horváth, and Bálint Takács. "Local-density dependent Markov processes on graphons with epidemiological applications." Stochastic Processes and their Applications 148 (June 2022): 324–52. http://dx.doi.org/10.1016/j.spa.2022.03.001.

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Backhausz, Ágnes, and Dávid Kunszenti-Kovács. "On the dense preferential attachment graph models and their graphon induced counterpart." Journal of Applied Probability 56, no. 2 (2019): 590–601. http://dx.doi.org/10.1017/jpr.2019.34.

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AbstractLetting ℳ denote the space of finite measures on ℕ, and μλ ∊ ℳ denote the Poisson distribution with parameter λ, the function W : [0, 1]2 → ℳ given by W(x, y) = μc 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 so-called W-random graphs in a natural way, and similar constructions also work in the slightly more general context of the so-called PAGκ models. The aim of this paper is to compare these dense PAGκ models with the W-random graph models obtained from the corresponding graphons. Motivated by the multigraph limit theory, we investigate the expected jumble-norm distance of the two models in terms of the number of vertices n. We present a coupling for which the expectation can be bounded from above by O(log3/2n · n−1/2), and provide a universal lower bound that is coupling-independent, but without the logarithmic term.

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Backhausz, Ágnes, and Balázs Szegedy. "Action convergence of operators and graphs." Canadian Journal of Mathematics, September 17, 2020, 1–50. http://dx.doi.org/10.4153/s0008414x2000070x.

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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 Benjamini–Schramm limits (even in the stronger local-global setting). We illustrate by examples that this new framework provides a rich limit theory with natural limit objects for graphs of intermediate density. Moreover, it provides a limit theory for bounded operators (called P-operators) of the form $L^\infty (\Omega )\to L^1(\Omega )$ for probability spaces $\Omega $ . We introduce a metric to compare P-operators (for example, finite matrices) even if they act on different spaces. We prove a compactness result, which implies that, in appropriate norms, limits of uniformly bounded P-operators can again be represented by P-operators. We show that limits of operators, representing graphs, are self-adjoint, positivity-preserving P-operators called graphops. Graphons, $L^p$ graphons, and graphings (known from graph limit theory) are special examples of graphops. We describe a new point of view on random matrix theory using our operator limit framework.

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Markering, Maarten. "The Large Deviation Principle for Inhomogeneous Erdős–Rényi Random Graphs." Journal of Theoretical Probability, June 14, 2022. http://dx.doi.org/10.1007/s10959-022-01181-1.

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AbstractConsider the inhomogeneous Erdős-Rényi random graph (ERRG) on n vertices for which each pair $$i,j\in \{1,\ldots ,n\}$$ i , j ∈ { 1 , … , n } , $$i\ne j,$$ i ≠ j , is connected independently by an edge with probability $$r_n(\frac{i-1}{n},\frac{j-1}{n})$$ r n ( i - 1 n , j - 1 n ) , where $$(r_n)_{n\in \mathbb {N}}$$ ( r n ) n ∈ N is a sequence of graphons converging to a reference graphonr. As a generalisation of the celebrated large deviation principle (LDP) for ERRGs by Chatterjee and Varadhan (Eur J Comb 32:1000–1017, 2011), Dhara and Sen (Large deviation for uniform graphs with given degrees, 2020. arXiv:1904.07666) proved an LDP for a sequence of such graphs under the assumption that r is bounded away from 0 and 1, and with a rate function in the form of a lower semi-continuous envelope. We further extend the results by Dhara and Sen. We relax the conditions on the reference graphon to $$(\log r, \log (1- r))\in L^1([0,1]^2)$$ ( log r , log ( 1 - r ) ) ∈ L 1 ( [ 0 , 1 ] 2 ) . We also show that, under this condition, their rate function equals a different, more tractable rate function. We then apply these results to the large deviation principle for the largest eigenvalue of inhomogeneous ERRGs and weaken the conditions for part of the analysis of the rate function by Chakrabarty et al. (Large deviation principle for the maximal eigenvalue of inhomogeneous Erdoös-Rényi random graphs, 2020. arXiv:2008.08367).

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Janssen, Jeannette, and Aaron Smith. "Reconstruction of line-embeddings of graphons." Electronic Journal of Statistics 16, no. 1 (2022). http://dx.doi.org/10.1214/21-ejs1940.

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