High-dimensional bootstrap processes in evolving simplicial complexes

Fountoulakis, Nikolaos; Przykucki, Michał

doi:10.48550/arxiv.1910.10139

Cited by 2 publications

(2 citation statements)

References 25 publications

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“…Moreover, the mixture pattern often refers to the high-order structure of networks. This is a new research hotspot of network science in recent years, for which the percolation theory also plays an important role [10,[541][542][543][544][545][546].…”

Section: Discussion and Outlookmentioning

confidence: 99%

Percolation on complex networks: Theory and application

Li,

Liu,

Lü

et al. 2021

Preprint

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In the last two decades, network science has blossomed and influenced various fields, such as statistical physics, computer science, biology and sociology, from the perspective of the heterogeneous interaction patterns of components composing the complex systems. As a paradigm for random and semi-random connectivity, percolation model plays a key role in the development of network science and its applications. On the one hand, the concepts and analytical methods, such as the emergence of the giant cluster, the finite-size scaling, and the mean-field method, which are intimately related to the percolation theory, are employed to quantify and solve some core problems of networks. On the other hand, the insights into the percolation theory also facilitate the understanding of networked systems, such as robustness, epidemic spreading, vital node identification, and community detection. Meanwhile, network science also brings some new issues to the percolation theory itself, such as percolation of strong heterogeneous systems, topological transition of networks beyond pairwise interactions, and emergence of a giant cluster with mutual connections. So far, the percolation theory has already percolated into the researches of structure analysis and dynamic modeling in network science. Understanding the percolation theory should help the study of many fields in network science, including the still opening questions in the frontiers of networks, such as networks beyond pairwise interactions, temporal networks, and network of networks. The intention of this paper is to offer an overview of these applications, as well as the basic theory of percolation transition on network systems.

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Section: Discussion and Outlookmentioning

confidence: 99%

Percolation on complex networks: Theory and application

Li,

Liu,

Lü

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…[17,22,[26][27][28][29]33]), and several notions of connectedness have been analysed (see e.g. [3,4,25,34,38]), as well as related concepts such as expansion [24] and bootstrap percolation [23]. In this paper, we consider a model of random simplicial complexes generated from non-uniform random hypergraphs, in which edges may have different sizes, and study cohomology groups over an arbitrary (not necessarily finite) abelian group R.…”

Section: Motivationmentioning

confidence: 99%

Phase Transition in Cohomology Groups of Non-Uniform Random Simplicial Complexes

Cooley

Giudice

Kang

et al. 2022

Electron. J. Combin.

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We consider a generalised model of a random simplicial complex, which arises from a random hypergraph. Our model is generated by taking the downward-closure of a non-uniform binomial random hypergraph, in which for each $k$, each set of $k+1$ vertices forms an edge with some probability $p_k$ independently. As a special case, this contains an extensively studied model of a (uniform) random simplicial complex, introduced by Meshulam and Wallach [Random Structures & Algorithms 34 (2009), no. 3, pp. 408–417].We consider a higher-dimensional notion of connectedness on this new model according to the vanishing of cohomology groups over an arbitrary abelian group $R$. We prove that this notion of connectedness displays a phase transition and determine the threshold. We also prove a hitting time result for a natural process interpretation, in which simplices and their downward-closure are added one by one. In addition, we determine the asymptotic behaviour of cohomology groups inside the critical window around the time of the phase transition.

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High-dimensional bootstrap processes in evolving simplicial complexes

Cited by 2 publications

References 25 publications

Percolation on complex networks: Theory and application

Percolation on complex networks: Theory and application

Phase Transition in Cohomology Groups of Non-Uniform Random Simplicial Complexes

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