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Keywords = probabilistic method in combinatorics

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18 pages, 4699 KB  
Article
Navigability, Walkability, and Perspicacity Associated with Canonical Ensembles of Walks in Finite Connected Undirected Graphs—Toward Information Graph Theory
by Dimitri Volchenkov
Information 2023, 14(6), 338; https://doi.org/10.3390/info14060338 - 15 Jun 2023
Cited by 1 | Viewed by 2555
Abstract
Canonical ensembles of walks in a finite connected graph assign the properly normalized probability distributions to all nodes, subgraphs, and nodal subsets of the graph at all time and connectivity scales of the diffusion process. The probabilistic description of graphs allows for introducing [...] Read more.
Canonical ensembles of walks in a finite connected graph assign the properly normalized probability distributions to all nodes, subgraphs, and nodal subsets of the graph at all time and connectivity scales of the diffusion process. The probabilistic description of graphs allows for introducing the quantitative measures of navigability through the graph, walkability of individual paths, and mutual perspicacity of the different modes of the (diffusion) processes. The application of information theory methods to problems about graphs, in contrast to geometric, combinatoric, algorithmic, and algebraic approaches, can be called information graph theory. As it involves evaluating communication efficiency between individual systems’ units at different time and connectivity scales, information graph theory is in demand for a wide range of applications, such as designing network-on-chip architecture and engineering urban morphology within the concept of the smart city. Full article
(This article belongs to the Special Issue Trends in Computational and Cognitive Engineering)
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10 pages, 265 KB  
Article
A Meeting Point of Probability, Graphs, and Algorithms: The Lovász Local Lemma and Related Results—A Survey
by András Faragó
Algorithms 2021, 14(12), 355; https://doi.org/10.3390/a14120355 - 8 Dec 2021
Cited by 2 | Viewed by 4722
Abstract
A classic and fundamental result, known as the Lovász Local Lemma, is a gem in the probabilistic method of combinatorics. At a high level, its core message can be described by the claim that weakly dependent events behave similarly to independent ones. A [...] Read more.
A classic and fundamental result, known as the Lovász Local Lemma, is a gem in the probabilistic method of combinatorics. At a high level, its core message can be described by the claim that weakly dependent events behave similarly to independent ones. A fascinating feature of this result is that even though it is a purely probabilistic statement, it provides a valuable and versatile tool for proving completely deterministic theorems. The Lovász Local Lemma has found many applications; despite being originally published in 1973, it still attracts active novel research. In this survey paper, we review various forms of the Lemma, as well as some related results and applications. Full article
(This article belongs to the Special Issue Surveys in Algorithm Analysis and Complexity Theory)
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