Random Simplicial Complexes

Random Simplicial Complexes
Author: Lior Aronshtam
Publisher:
Total Pages: 64
Release: 2013
Genre: Random graphs
ISBN:

Random graphs are vastly researched and are of great importance in modern discrete mathematics. A graph may be viewed as a one-dimensional simplicial complex.

Random Graphs

Random Graphs
Author: Svante Janson
Publisher: John Wiley & Sons
Total Pages: 350
Release: 2011-09-30
Genre: Mathematics
ISBN: 1118030966

A unified, modern treatment of the theory of random graphs-including recent results and techniques Since its inception in the 1960s, the theory of random graphs has evolved into a dynamic branch of discrete mathematics. Yet despite the lively activity and important applications, the last comprehensive volume on the subject is Bollobas's well-known 1985 book. Poised to stimulate research for years to come, this new work covers developments of the last decade, providing a much-needed, modern overview of this fast-growing area of combinatorics. Written by three highly respected members of the discrete mathematics community, the book incorporates many disparate results from across the literature, including results obtained by the authors and some completely new results. Current tools and techniques are also thoroughly emphasized. Clear, easily accessible presentations make Random Graphs an ideal introduction for newcomers to the field and an excellent reference for scientists interested in discrete mathematics and theoretical computer science. Special features include: * A focus on the fundamental theory as well as basic models of random graphs * A detailed description of the phase transition phenomenon * Easy-to-apply exponential inequalities for large deviation bounds * An extensive study of the problem of containing small subgraphs * Results by Bollobas and others on the chromatic number of random graphs * The result by Robinson and Wormald on the existence of Hamilton cycles in random regular graphs * A gentle introduction to the zero-one laws * Ample exercises, figures, and bibliographic references

Higher Order Networks: An Introduction to Simplicial Complexes

Higher Order Networks: An Introduction to Simplicial Complexes
Author: Ginestra Bianconi
Publisher: Cambridge University Press
Total Pages: 149
Release: 2021-12-23
Genre: Mathematics
ISBN: 1108726739

This Element presents one of the most recent developments in network science in a highly accessible style. This Element will be of interest to interdisciplinary scientists working in network science, in addition to mathematicians working in discrete topology and geometry and physicists working in quantum gravity.

Torsion in Homology of Random Simplicial Complexes

Torsion in Homology of Random Simplicial Complexes
Author: J. Andrew Newman
Publisher:
Total Pages: 110
Release: 2018
Genre: Homology theory
ISBN:

During the mid-twentieth century, Paul Erdos and Alfréd Rényi developed their now-standard random graph model. Beyond being practical in graph theory to nonconstructively prove the existence of graphs with certain interesting properties, the Erdős–Rényi model is also a model for generating random (one-dimensional) topological spaces. Within the last fifteen years, this model has been generalized to the higher-dimensional simplicial complex model of Nati Linial and Roy Meshulam. As in the case of the probabilistic method more generally, there are (at least) two reasons why one might apply random methods in topology: to understand what a "typical" topological space looks like and to give nonconstructive proofs of the existence of topological spaces with certain properties. Here we consider both of these applications of randomness in topology in considering the properties of torsion in homology of simplicial complexes. For the former, we discuss experimental results that strongly suggest torsion in homology of random Linial–Meshulam complexes is distributed according to Cohen–Lenstra heuristics. For the latter, we use the probabilistic method to give an upper bound on the number of vertices required to construct d-dimensional simplicial complexes with prescribed torsion in homology. This upper bound is optimal in the sense that it is a constant multiple of a known lower bound.

Topological Signal Processing

Topological Signal Processing
Author: Michael Robinson
Publisher: Springer Science & Business Media
Total Pages: 245
Release: 2014-01-07
Genre: Technology & Engineering
ISBN: 3642361048

Signal processing is the discipline of extracting information from collections of measurements. To be effective, the measurements must be organized and then filtered, detected, or transformed to expose the desired information. Distortions caused by uncertainty, noise, and clutter degrade the performance of practical signal processing systems. In aggressively uncertain situations, the full truth about an underlying signal cannot be known. This book develops the theory and practice of signal processing systems for these situations that extract useful, qualitative information using the mathematics of topology -- the study of spaces under continuous transformations. Since the collection of continuous transformations is large and varied, tools which are topologically-motivated are automatically insensitive to substantial distortion. The target audience comprises practitioners as well as researchers, but the book may also be beneficial for graduate students.

Geometric and Topological Inference

Geometric and Topological Inference
Author: Jean-Daniel Boissonnat
Publisher: Cambridge University Press
Total Pages: 247
Release: 2018-09-27
Genre: Computers
ISBN: 1108419399

A rigorous introduction to geometric and topological inference, for anyone interested in a geometric approach to data science.