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High-dimensional bootstrap processes in evolving simplicial complexes
TL;DR: In this paper, the authors study bootstrap percolation processes on random simplicial complexes of some fixed dimension, where each vertex selects an existing (d-1)dimensional face at random, with probability proportional to some positive and symmetric function of the weights of its vertices, and attaches to it by forming a simplex.
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Abstract: We study bootstrap percolation processes on random simplicial complexes of some fixed dimension $d \geq 3$. Starting from a single simplex of dimension $d$, we build our complex dynamically in the following fashion. We introduce new vertices one by one, all equipped with a random weight from a fixed distribution $\mu$. The newly arriving vertex selects an existing $(d-1)$-dimensional face at random, with probability proportional to some positive and symmetric function $f$ of the weights of its vertices, and attaches to it by forming a $d$-dimensional simplex. After a complex on $n$ vertices is constructed, we infect every vertex independently at random with some probability $p = p(n)$. Then, in consecutive rounds, we infect every healthy vertex the neighbourhood of which contains at least $r$ disjoint $(k-1)$-dimensional, fully infected faces. Using a reduction to the generalised P\'olya urn schemes, we determine the value of critical probability $p_c = p_c (n; \mu, f)$, such that if $p \gg p_c$ then, with probability tending to 1 as $n \to \infty$, the infection spreads to the whole vertex set of the complex, while if $p \ll p_c$ then the infection process stops with healthy vertices remaining in the complex.
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Citations
Percolation on complex networks: Theory and application
TL;DR: 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.
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Percolation on complex networks: Theory and application
TL;DR: The percolation theory has already percolated into the researches of structure analysis and dynamic modeling in network science, such as robustness, epidemic spreading, vital node identification, and community detection.
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Cohomology groups of non-uniform random simplicial complexes
Oliver Cooley,N. Del Giudice,Mihyun Kang,Philipp Sprüssel +3 more
- 26 Jul 2019
Abstract: 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.
Phase Transition in Cohomology Groups of Non-Uniform Random Simplicial Complexes
29 Jul 2022
TL;DR: In this paper , the authors considered a generalised model of a random simplicial complex, which arises from a random hypergraph, and derived a higher-dimensional notion of connectedness on this new model according to the vanishing of cohomology groups over an arbitrary abelian group.
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