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Bootstrap percolation in random k -uniform hypergraphs

Mihyun Kang, Christoph Koch, Tamás Makai
<span title="">2015</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/fhi2xwpnh5gmlgof2idwu5wlgq" style="color: black;">Electronic Notes in Discrete Mathematics</a> </i> &nbsp;
We investigate bootstrap percolation with infection threshold r> 1 on the binomial k-uniform random hypergraph H_k(n,p) in the regime n^-1≪ n^k-2p ≪ n^-1/r, when the initial set of infected vertices is  ...  In addition, for k=2, we show that the probability of failure decreases exponentially.  ...  Introduction Bootstrap percolation on a hypergraph with infection threshold r ≥ 1 is a deterministic infection process which evolves in rounds.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.endm.2015.06.081">doi:10.1016/j.endm.2015.06.081</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/fv5vwgqsdna7df6zvycuhzykay">fatcat:fv5vwgqsdna7df6zvycuhzykay</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200322195201/https://eurocomb2015.w.uib.no/files/2015/08/endm1959.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/c7/02/c7029abe5db5e5b13c3fd31616e6e516f9a3ff75.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.endm.2015.06.081"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

High-order bootstrap percolation in hypergraphs [article]

Oliver Cooley, Julian Zalla
<span title="2022-01-24">2022</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Motivated by the bootstrap percolation process for graphs, we define a new, high-order generalisation to k-uniform hypergraphs, in which we infect j-sets of vertices for some integer 1≤ j ≤ k-1.  ...  We investigate the smallest possible size of an initially infected set which ultimately percolates and determine the exact size in almost all cases of k and j.  ...  Bootstrap percolation has also been studied in (random) k-uniform hypergraphs in [19] , where an infection process on the vertices of a random hypergraph was studied; by contrast, inspired by recent work  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2201.09718v1">arXiv:2201.09718v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/bd7ohqgctzblhax2z36doujbba">fatcat:bd7ohqgctzblhax2z36doujbba</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20220128040940/https://arxiv.org/pdf/2201.09718v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/99/b1/99b165676c1fc3448e4ff6ab057ba3dc3a068571.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2201.09718v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Aperiodic Non-Isomorphic Lattices with Equivalent Percolation and Random-Cluster Models

Klas Markström, John C. Wierman
<span title="2010-03-29">2010</span> <i title="The Electronic Journal of Combinatorics"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/v5dyak6ulffqfara7hmuchh24a" style="color: black;">Electronic Journal of Combinatorics</a> </i> &nbsp;
The graphs are constructed by placing a copy of a rotor gadget graph or its reflection in each hyperedge of a connected self-dual 3-uniform plane hypergraph lattice.  ...  This equivalence holds for all values of $p$ and all $q\in[0,\infty]$ for the random-cluster model.  ...  A k-uniform hypergraph is one in which every hyperedge is a k-hyperedge. In this article, we will restrict consideration to 3-uniform hypergraphs.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.37236/320">doi:10.37236/320</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/i2e7s5uyrrboxi5wjqubv2fkg4">fatcat:i2e7s5uyrrboxi5wjqubv2fkg4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200321073507/http://www.mittag-leffler.se/sites/default/files/IML-0809s-11.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/e2/c7/e2c76cf9c8113701043f25c4d8db6ae9df409a4b.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.37236/320"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

Lower bounds for graph bootstrap percolation via properties of polynomials [article]

Lianna Hambardzumyan, Hamed Hatami, Yingjie Qian
<span title="2021-05-08">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We introduce a simple method for proving lower bounds for the size of the smallest percolating set in a certain graph bootstrap process.  ...  We apply this method to determine the sizes of the smallest percolating sets in multidimensional tori and multidimensional grids (in particular hypercubes).  ...  percolation in [BBLN18] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1708.04640v2">arXiv:1708.04640v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/wge5yzcaczey3pak5u6eg3mewm">fatcat:wge5yzcaczey3pak5u6eg3mewm</a> </span>
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The threshold for stacked triangulations [article]

Eyal Lubetzky, Yuval Peled
<span title="2022-01-10">2022</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In the language of bootstrap percolation in hypergraphs, it pertains to the threshold for K_d+2^d+1, the (d+1)-uniform clique on d+2 vertices.  ...  We study the occurrence of such a triangulation in the Linial–Meshulam model, i.e., for which p does the random simplicial complex Y∼𝒴_d(n,p) contain the faces of a stacked triangulation of the d-simplex  ...  E.L. was supported in part by NSF grants DMS-1812095 and DMS-2054833.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2112.12780v2">arXiv:2112.12780v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ybnc74komfhppbahz42sftrrnm">fatcat:ybnc74komfhppbahz42sftrrnm</a> </span>
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Majority bootstrap percolation on the hypercube [article]

József Balogh, Béla Bollobás, Robert Morris
<span title="2007-02-13">2007</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In majority bootstrap percolation on a graph G, an infection spreads according to the following deterministic rule: if at least half of the neighbours of a vertex v are already infected, then v is also  ...  Percolation occurs if eventually every vertex is infected.  ...  Let G be a labelled 3-uniform hypergraph with n vertices.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/0702373v1">arXiv:math/0702373v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/uq7ejwrxgfgvbcxlrlmdeij7y4">fatcat:uq7ejwrxgfgvbcxlrlmdeij7y4</a> </span>
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Majority Bootstrap Percolation on the Hypercube

JÓZSEF BALOGH, BÉLA BOLLOBÁS, ROBERT MORRIS
<span title="2008-08-15">2008</span> <i title="Cambridge University Press (CUP)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/domxx2wewzae3e2m2ziqaqbyhm" style="color: black;">Combinatorics, probability &amp; computing</a> </i> &nbsp;
In majority bootstrap percolation on a graph G, an infection spreads according to the following deterministic rule: if at least half of the neighbours of a vertex v are already infected, then v is also  ...  Percolation occurs if eventually every vertex is infected.  ...  We shall need the following lemma about counting 3-uniform hypergraphs.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1017/s0963548308009322">doi:10.1017/s0963548308009322</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ujwklau7gjb7retgwdxked4h7u">fatcat:ujwklau7gjb7retgwdxked4h7u</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20080221170049/http://www.math.uiuc.edu/~jobal/cikk/bootmaj_sub.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/59/81/59814572f2ee276cff59ebca014304c49c554942.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1017/s0963548308009322"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> cambridge.org </button> </a>

A Sharp Threshold for Bootstrap Percolation in a Random Hypergraph [article]

Natasha Morrison, Jonathan A. Noel
<span title="2020-10-07">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Given a hypergraph ℋ, the ℋ-bootstrap process starts with an initial set of infected vertices of ℋ and, at each step, a healthy vertex v becomes infected if there exists a hyperedge of ℋ in which v is  ...  We show that this process exhibits a sharp threshold when ℋ is a hypergraph obtained by randomly sampling hyperedges from an approximately d-regular r-uniform hypergraph satisfying some mild degree and  ...  This work was initiated while the authors were visiting Rob Morris at IMPA in 2016. We are grateful to Rob and IMPA for their hospitality and for providing a stimulating research environment.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1806.02903v2">arXiv:1806.02903v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zvp6gf2z5ncffi5qsbfds6pptm">fatcat:zvp6gf2z5ncffi5qsbfds6pptm</a> </span>
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A sharp threshold for bootstrap percolation in a random hypergraph

Natasha Morrison, Jonathan A. Noel
<span title="2021-01-01">2021</span> <i title="Institute of Mathematical Statistics"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/4f7vg4vyabgf7fcdx6hl7zjorq" style="color: black;">Electronic Journal of Probability</a> </i> &nbsp;
Given a hypergraph H, the H-bootstrap process starts with an initial set of infected vertices of H and, at each step, a healthy vertex v becomes infected if there exists a hyperedge of H in which v is  ...  We show that this process exhibits a sharp threshold when H is a hypergraph obtained by randomly sampling hyperedges from an approximately d-regular r-uniform hypergraph satisfying some mild degree and  ...  hyperedges to grow faster than they would if the infection was uniform.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1214/21-ejp650">doi:10.1214/21-ejp650</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6cwgi3dl7vfnnbhuawfzn7zuoi">fatcat:6cwgi3dl7vfnnbhuawfzn7zuoi</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20220125155651/https://projecteuclid.org/journals/electronic-journal-of-probability/volume-26/issue-none/A-sharp-threshold-for-bootstrap-percolation-in-a-random-hypergraph/10.1214/21-EJP650.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/a2/e9/a2e9f0959568a4abebf2bccec79c118d7ffc94ce.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1214/21-ejp650"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

The Component Graph of the Uniform Spanning Forest: Transitions in Dimensions 9,10,11,.. [article]

Tom Hutchcroft, Yuval Peres
<span title="2018-10-15">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
To separate dimensions 5,6,7, and 8, we prove a similar result concerning ubiquitous subhypergraphs in the component hypergraph.  ...  In particular, we consider the graph formed by contracting each tree of the uniform spanning forest down to a single vertex, which we call the component graph.  ...  Negative association in uniform forests and connected graphs. Random Structures Algorithms, 24(4):444–460, 2004. [9] O. Häggström. Random-cluster measures and uniform spanning trees.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1702.05780v2">arXiv:1702.05780v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/tm7xikskl5eglixh3slhdzihhe">fatcat:tm7xikskl5eglixh3slhdzihhe</a> </span>
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The large deviations of the whitening process in random constraint satisfaction problems

Alfredo Braunstein, Luca Dall'Asta, Guilhem Semerjian, Lenka Zdeborová
<span title="2016-05-19">2016</span> <i title="IOP Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/o3oboaa6vjee5ismb4vsvaui2a" style="color: black;">Journal of Statistical Mechanics: Theory and Experiment</a> </i> &nbsp;
percolation dynamics.  ...  Random constraint satisfaction problems undergo several phase transitions as the ratio between the number of constraints and the number of variables is varied.  ...  The notations and parts of the strategy will be the same as in the asymptotic expansion of the T = 1 results described in Appendix B, in particular we consider the degree l of the hypergraphs to scale  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1088/1742-5468/2016/05/053401">doi:10.1088/1742-5468/2016/05/053401</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/dd3nxrch5ng4djrlydrr2j2cai">fatcat:dd3nxrch5ng4djrlydrr2j2cai</a> </span>
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On percolation in random graphs with given vertex degrees

Svante Janson
<span title="">2009</span> <i title="Institute of Mathematical Statistics"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/4f7vg4vyabgf7fcdx6hl7zjorq" style="color: black;">Electronic Journal of Probability</a> </i> &nbsp;
This is used to study existence of giant component and existence of k-core. As a variation of the latter, we study also bootstrap percolation in random regular graphs.  ...  We study the random graph obtained by random deletion of vertices or edges from a random graph with given vertex degrees.  ...  Bootstrap percolation in random regular graphs Bootstrap percolation on a graph G is a process that can be regarded as a model for the spread of an infection.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1214/ejp.v14-603">doi:10.1214/ejp.v14-603</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/demtlzmrdbdhzm35e4dtf7wjki">fatcat:demtlzmrdbdhzm35e4dtf7wjki</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20081008192126/http://www.math.uu.se/~svante/papers/sj215.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/0e/64/0e649d67b64f4b9c8c5716f3a7d2299802a9468e.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1214/ejp.v14-603"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

On percolation in random graphs with given vertex degrees [article]

Svante Janson
<span title="2008-04-10">2008</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
This is used to study existence of giant component and existence of k-core. As a variation of the latter, we study also bootstrap percolation in random regular graphs.  ...  We study the random graph obtained by random deletion of vertices or edges from a random graph with given vertex degrees.  ...  Bootstrap percolation in random regular graphs Bootstrap percolation on a graph G is a process that can be regarded as a model for the spread of an infection.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0804.1656v1">arXiv:0804.1656v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/3q7l343tkvbyff4t2lussksyi4">fatcat:3q7l343tkvbyff4t2lussksyi4</a> </span>
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A large deviation approach to super-critical bootstrap percolation on the random graph G_n,p [article]

Giovanni Luca Torrisi and Michele Garetto, Emilio Leonardi
<span title="2020-01-16">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We consider the Erdös–Rényi random graph G_n,p and we analyze the simple irreversible epidemic process on the graph, known in the literature as bootstrap percolation.  ...  More specifically, we establish large deviation principles for the sequence of random variables {n- A_n^*/f(n)}_n≥ 1 with explicit rate functions and allowing the scaling function f to vary in the widest  ...  In [27] the results of [26] were extended to k-uniform random hypergraphs. Bootstrap percolation on random graphs obtained by combining G n,pn with a regular lattice was investigated in [35] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1802.01847v3">arXiv:1802.01847v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/hsdg7wg2azac7pf77e5qv5f2n4">fatcat:hsdg7wg2azac7pf77e5qv5f2n4</a> </span>
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The component graph of the uniform spanning forest: transitions in dimensions $$9,10,11,\ldots $$9,10,11,…

Tom Hutchcroft, Yuval Peres
<span title="2018-11-23">2018</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/lvus5hrxgzgttamz7rydylvvoy" style="color: black;">Probability theory and related fields</a> </i> &nbsp;
To separate dimensions 5, 6, 7, and 8, we prove a similar result concerning ubiquitous subhypergraphs in the component hypergraph.  ...  In particular, we consider the graph formed by contracting each tree of the uniform spanning forest down to a single vertex, which we call the component graph.  ...  distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in  ... 
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