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Threshold and hitting time for high-order connectedness in random hypergraphs

Oliver Cooley, Mihyun Kang, Christoph Koch
This generalises the classical hitting time result of Bollobás and Thomason for graphs.  ...  We also deduce a hitting time result for the random hypergraph process-the hypergraph becomes j-connected at exactly the moment when the last isolated j-set disappears.  ...  Acknowledgement The authors would like to thank an anonymous referee for detailed feedback which greatly improved the quality of the manuscript.  ... 

Vanishing of Cohomology Groups of Random Simplicial Complexes (Keynote Speakers)

Oliver Cooley, Nicola Del Giudice, Mihyun Kang, Philipp Sprüssel, Marc Herbstritt
2018 International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms  
As a corollary, we deduce a hitting time result for a different model of random simplicial complexes introduced in [Linial and Meshulam, Combinatorica, 2006], a result which has only been known for dimension  ...  For each 1 ≤ j ≤ k − 1, we determine when all cohomology groups with coefficients in F 2 from dimension one up to j vanish and the zero-th cohomology group is isomorphic to F 2 .  ...  As a corollary, we deduce a hitting time result for Y p in general dimension, thus extending the hitting time result of Kahle and Pittel [15] .  ... 
doi:10.4230/lipics.aofa.2018.7 dblp:conf/aofa/CooleyGKS18 fatcat:b375abvdfvguza2dzm6okf6aua

Threshold and hitting time for high-order connectivity in random hypergraphs [article]

Oliver Cooley, Mihyun Kang, Christoph Koch
2015 arXiv   pre-print
We also deduce a hitting time result for the random hypergraph process -- the hypergraph becomes j-connected at exactly the moment when the last isolated j-set disappears.  ...  We determine the threshold at which the random k-uniform hypergraph with edge probability p becomes j-connected with high probability.  ...  Concluding remark In [5] , it is determined for the case j = 1 that the hitting time for d-strong 1-connectedness, i.e. the time at which the hypergraph first has the property that deleting any set of  ... 
arXiv:1502.07289v1 fatcat:dyv6opjaorg2zcnfkr4usesx4a

Vanishing of cohomology groups of random simplicial complexes [article]

Oliver Cooley, Nicola Del Giudice, Mihyun Kang, Philipp Sprüssel
2018 arXiv   pre-print
As a corollary, we deduce a hitting time result for a different model of random simplicial complexes introduced in [Linial and Meshulam, Combinatorica, 2006], a result which was previously only known for  ...  This property is not deterministically monotone for this model of random complexes, but nevertheless we show that it has a single sharp threshold.  ...  An extended abstract of the case k = 2 has appeared in the Proceedings of Eurocomb 2017 [11] .  ... 
arXiv:1806.04566v1 fatcat:2l62ywhfefh2tjy7rcfso57bom

Homological connectivity of random hypergraphs [article]

Oliver Cooley, Penny Haxell, Mihyun Kang, Philipp Sprüssel
2016 arXiv   pre-print
Although this is not intrinsically a monotone property, we show that it nevertheless has a single sharp threshold, and indeed prove a hitting time result relating the connectedness to the disappearance  ...  We consider simplicial complexes that are generated from the binomial random 3-uniform hypergraph by taking the downward-closure.  ...  Bollobás and Thomason [2] subsequently proved a hitting time result: With high probability the random graph process, in which edges are added one at a time in random order, becomes connected at exactly  ... 
arXiv:1604.00842v1 fatcat:72cc2cjxhnac7e7nsy6r3letim

Positional Games [article]

Michael Krivelevich
2014 arXiv   pre-print
and hypergraphs.  ...  We survey the basic notions of the field, its approaches and tools, as well as numerous recent advances, standing open problems and promising research directions.  ...  The author wishes to thank Asaf Ferber, Dan Hefetz, Miloš Stojaković and Tibor Szabó for careful reading of the manuscript and many helpful comments, and also for an extensive and fruitful cooperation  ... 
arXiv:1404.2731v1 fatcat:hlv7jjt4vbgt5kwhr7pc4fgyue

Phase transition in cohomology groups of non-uniform random simplicial complexes [article]

Oliver Cooley, Nicola Del Giudice, Mihyun Kang, Philipp Sprüssel
2020 arXiv   pre-print
We also prove a hitting time result for a natural process interpretation, in which simplices and their downward-closure are added one by one.  ...  We prove that this notion of connectedness displays a phase transition and determine the threshold.  ...  The vertex-connectedness threshold for uniform random hypergraphs, which we quoted from [9] for the proof of (b), also follows as a special case of earlier and much stronger results from [10] and from  ... 
arXiv:2005.07103v3 fatcat:vgrsvopfefbuvejew2oz2ez2p4

On Weak Hamiltonicity of a Random Hypergraph [article]

Daniel Poole
2014 arXiv   pre-print
In particular, for d ≥ 3 fixed and p=(d-1)!  ...  As a consequence, the probability that the random hypergraph H_d(n, m=n( n + c)/d), where m potential edges are chosen uniformly at random to be present, is weak Hamiltonian also tends to e^-e^-c.  ...  If true, this conjecture would generalize to hypergraphs the weak Hamilton version of Bollobás' [3] hitting time result.  ... 
arXiv:1410.7446v1 fatcat:lozx7u3vtnc25okh2mbqyjevdi

Thresholds in Random Motif Graphs

Michael Anastos, Peleg Michaeli, Samantha Petti, Michael Wagner
2019 International Workshop on Approximation Algorithms for Combinatorial Optimization  
Moreover, in the first three cases we give the analogous hitting time results; with high probability, the first graph in the random motif graph process that has minimum degree one (or two) is connected  ...  We establish that every monotone property has a threshold in this model, and determine the thresholds for connectivity, Hamiltonicity, the existence of a perfect matching, and subgraph appearance.  ...  The analogue hitting time result is also true. 66:4 Thresholds in Random Motif Graphs Next, we describe the threshold for the appearance of a subgraph S.  ... 
doi:10.4230/lipics.approx-random.2019.66 dblp:conf/approx/AnastosMP19 fatcat:j2tspe36zvcurm4tfiglox3jfq

The sharp threshold for jigsaw percolation in random graphs [article]

Oliver Cooley, Tobias Kapetanopoulos, Tamás Makai
2019 arXiv   pre-print
We show that this threshold is sharp, and that it lies at 1/4n n.  ...  Bollobás, Riordan, Slivken and Smith proved that when the two graphs are independent binomial random graphs, whether the jigsaw process percolates undergoes a phase transition when the product of the two  ...  For random graphs, the famous hitting time result of Bollobás and Thomason [6] relates the threshold for connectedness of a random graph to the disappearance of the last isolated vertex, implying that  ... 
arXiv:1809.01907v2 fatcat:5su4yahvrfd3neaic26gxjsb5u

Resilience for tight Hamiltonicity [article]

Peter Allen, Olaf Parczyk, Vincent Pfenninger
2021 arXiv   pre-print
Specifically, for any γ>0 and k≥3, we show that asymptotically almost surely, every subgraph of the binomial random k-uniform hypergraph G^(k)(n,n^γ-1) in which all (k-1)-sets are contained in at least  ...  We prove that random hypergraphs are asymptotically almost surely resiliently Hamiltonian.  ...  In random hypergraphs, Dudek and Frieze [11, 12] found for several different notions of 'cycle' the threshold for Hamiltonicity in the binomial random hypergraph G (k) (n, p), that is the nvertex k-graph  ... 
arXiv:2105.04513v1 fatcat:y3gyj67jknc4tkkxsxlv3ltkfy

An Introduction to Metabolic Networks and Their Structural Analysis

V. Lacroix, L. Cottret, P. Thebault, M.-F. Sagot
2008 IEEE/ACM Transactions on Computational Biology & Bioinformatics  
The literature is vast enough already, and the thread through it is difficult to follow even for the more experienced worker in the field.  ...  There has been a renewed interest for metabolism in the computational biology community, leading to an avalanche of papers coming from methodological network analysis as well as experimental and theoretical  ...  Later, the same first author and others [54] extended the distance measure to networks and based it this time on simple operations on hypergraphs, such as union, intersection, and difference of two hypergraphs  ... 
doi:10.1109/tcbb.2008.79 pmid:18989046 fatcat:iomp7dhnvndgtlwhpeyc4dnvoa

Recent advances in percolation theory and its applications

Abbas Ali Saberi
2015 Physics reports  
Recent applications of the percolation theory in natural and artificial landscapes are also reviewed.  ...  In this review we will first outline the basic features of the ordinary model and take a glimpse at a number of selective variations and modifications of the original model.  ...  Broadbent and Hammersley proposed the concept of a percolation threshold above which the links form an infinite cluster with high probability.  ... 
doi:10.1016/j.physrep.2015.03.003 fatcat:fwp76ca44vanfh2inf5a6tzjiq

Combinatorial Landscapes

Christian M. Reidys, Peter F. Stadler
2002 SIAM Review  
In this review we focus on the connections of landscape theory with algebraic combinatorics and random graph theory, where exact results are available.  ...  Fitness landscapes have proven to be a valuable concept in evolutionary biology, combinatorial optimization, and the physics of disordered systems.  ...  In the random graph Q n α,λ the probability λ * = 1 − α−1 √ α −1 is the threshold value for connectivity.  ... 
doi:10.1137/s0036144501395952 fatcat:s46zur5dnrc6rhe7xpdeb3znmi

Social physics [article]

Marko Jusup, Petter Holme, Kiyoshi Kanazawa, Misako Takayasu, Ivan Romic, Zhen Wang, Suncana Gecek, Tomislav Lipic, Boris Podobnik, Lin Wang, Wei Luo, Tin Klanjscek (+3 others)
2021 arXiv   pre-print
, cooperation as a basis for civilised life, the structure of (social) networks, and the integration of intelligent machines in such networks.  ...  Recent decades have seen a rise in the use of physics-inspired or physics-like methods in attempts to resolve diverse societal problems.  ...  In these simulations, the number of hyperlinks equals the critical number that guarantees hypergraph connectedness, l = l c .  ... 
arXiv:2110.01866v1 fatcat:ccfxyezl6zgddd6uvrxubmaxua
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