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9084 results in Discrete Mathematics, Information Theory and Coding

Page 53 of 455

Deep Learning on Graphs
  • Yao Ma, Jiliang Tang
  • Published online:
    02 September 2021
    Print publication:
    23 September 2021
  • Deep learning on graphs has become one of the hottest topics in machine learning. The book consists of four parts to best accommodate our readers with diverse backgrounds and purposes of reading. Part 1 introduces basic concepts of graphs and deep learning; Part 2 discusses the most established methods from the basic to advanced settings; Part 3 presents the most typical applications including natural language processing, computer vision, data mining, biochemistry and healthcare; and Part 4 describes advances of methods and applications that tend to be important and promising for future research. The book is self-contained, making it accessible to a broader range of readers including (1) senior undergraduate and graduate students; (2) practitioners and project managers who want to adopt graph neural networks into their products and platforms; and (3) researchers without a computer science background who want to use graph neural networks to advance their disciplines.

  • On the edit distance function of the random graph

  • Part of
  • Ryan R. Martin, Alex W. N. Riasanovsky
  • Journal:
    Combinatorics, Probability and Computing / Volume 31 / Issue 2 / March 2022
    Published online by Cambridge University Press:
    02 September 2021, pp. 345-367

  • Given a hereditary property of graphs $\mathcal{H}$ and a $p\in [0,1]$, the edit distance function $\textrm{ed}_{\mathcal{H}}(p)$ is asymptotically the maximum proportion of edge additions plus edge deletions applied to a graph of edge density p sufficient to ensure that the resulting graph satisfies $\mathcal{H}$. The edit distance function is directly related to other well-studied quantities such as the speed function for $\mathcal{H}$ and the $\mathcal{H}$-chromatic number of a random graph.

    Let $\mathcal{H}$ be the property of forbidding an Erdős–Rényi random graph $F\sim \mathbb{G}(n_0,p_0)$, and let $\varphi$ represent the golden ratio. In this paper, we show that if $p_0\in [1-1/\varphi,1/\varphi]$, then a.a.s. as $n_0\to\infty$,

    \begin{align*}{\textrm{ed}}_{\mathcal{H}}(p) = (1+o(1))\,\frac{2\log n_0}{n_0}\cdot\min\left\{\frac{p}{-\log(1-p_0)},\frac{1-p}{-\log p_0}\right\}.\end{align*}
    Moreover, this holds for $p\in [1/3,2/3]$ for any $p_0\in (0,1)$.

    A primary tool in the proof is the categorization of p-core coloured regularity graphs in the range $p\in[1-1/\varphi,1/\varphi]$. Such coloured regularity graphs must have the property that the non-grey edges form vertex-disjoint cliques.

Lectures on Random Lozenge Tilings
  • Vadim Gorin
  • Published online:
    31 August 2021
    Print publication:
    09 September 2021
  • Over the past 25 years, there has been an explosion of interest in the area of random tilings. The first book devoted to the topic, this timely text describes the mathematical theory of tilings. It starts from the most basic questions (which planar domains are tileable?), before discussing advanced topics about the local structure of very large random tessellations. The author explains each feature of random tilings of large domains, discussing several different points of view and leading on to open problems in the field. The book is based on upper-division courses taught to a variety of students but it also serves as a self-contained introduction to the subject. Test your understanding with the exercises provided and discover connections to a wide variety of research areas in mathematics, theoretical physics, and computer science, such as conformal invariance, determinantal point processes, Gibbs measures, high-dimensional random sampling, symmetric functions, and variational problems.

Page 53 of 455