30 research outputs found

    LIPIcs, Volume 258, SoCG 2023, Complete Volume

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    LIPIcs, Volume 258, SoCG 2023, Complete Volum

    Dynamical models for random simplicial complexes

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    Notes in Pure Mathematics & Mathematical Structures in Physics

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    These Notes deal with various areas of mathematics, and seek reciprocal combinations, explore mutual relations, ranging from abstract objects to problems in physics.Comment: Small improvements and addition

    Errata and Addenda to Mathematical Constants

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    We humbly and briefly offer corrections and supplements to Mathematical Constants (2003) and Mathematical Constants II (2019), both published by Cambridge University Press. Comments are always welcome.Comment: 162 page

    Dynamical Models for Random Simplicial Complexes

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    We study a general model of random dynamical simplicial complexes and derive a formula for the asymptotic degree distribution. This asymptotic formula encompasses results for a number of existing models, including random Apollonian networks and the weighted random recursive tree. It also confirms results on the scale-free nature of Complex Quantum Network Manifolds in dimensions d>2d > 2, and special types of Network Geometry with Flavour models studied in the physics literature by Bianconi, Rahmede [Sci.Rep.  5, 13979 (2015) and Phys.Rev.E  93, 032315 (2016)\mathit{Sci. Rep.} \; \mathbf{5},\text{ 13979 (2015) and }\mathit{Phys. Rev. E} \; \mathbf{93},\text{ 032315 (2016)}].Comment: 45 pages (main body 37 pages), 4 figure

    Complex network view of evolving manifolds

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    We study complex networks formed by triangulations and higher-dimensional simplicial complexes representing closed evolving manifolds. In particular, for triangulations, the set of possible transformations of these networks is restricted by the condition that at each step, all the faces must be triangles. Stochastic application of these operations leads to random networks with different architectures. We perform extensive numerical simulations and explore the geometries of growing and equilibrium complex networks generated by these transformations and their local structural properties. This characterization includes the Hausdorff and spectral dimensions of the resulting networks, their degree distributions, and various structural correlations. Our results reveal a rich zoo of architectures and geometries of these networks, some of which appear to be small worlds while others are finite-dimensional with Hausdorff dimension equal or higher than the original dimensionality of their simplices. The range of spectral dimensions of the evolving triangulations turns out to be from about 1.4 to infinity. Our models include simplicial complexes representing manifolds with evolving topologies, for example, an h-holed torus with a progressively growing number of holes. This evolving graph demonstrates features of a small-world network and has a particularly heavy-tailed degree distribution.Comment: 14 pages, 15 figure
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