Homological Connectivity Of Random 2-Complexes

By N. Linial, R. Meshulam. Comb., 2006

355 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/b004b2c1cfd7c27e625d6c09b03ea486b1c3c672

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Citations in the corpus

Sharp vanishing thresholds for cohomology of random flag complexes (2012, result): “I thank Noga Alon, Eric Babson, Chris Hoffman, Roy Meshulam, Elliot Paquette, and Uli Wagner for many helpful conversations.”
On the topology of random complexes built over stationary point processes. (2012, background): “Another approach has been to consider random subgraphs of complete graphs, leading to a number of papers dealing with the topology of random complexes generalising Erdős–Rényi graphs, as in, for example, [1, 14, 21, 27, 29].”
Coboundary expanders (2010, background): “The main result of [17] (for k = 1) and [22] (for k ≥ 2) is the following.”
Random geometric complexes in the thermodynamic regime (2014, background): “ random geometric complexes seems to have originated in [19], it is worth noting that Betti numbers of a random complex model were Random complexes 5 Yogeshwaran, Subag, Adler already investigated in [24], where a higher-dimensional version of the Erd¨osRenyi random graph model was constructed. The ”
Limit theorems for Betti numbers of random simplicial complexes (2010, background): “Several papers have recently appeared concerning the topology of random simplicial complexes [11, 2, 10, 12, 13, 16, 9].”
The topology of probability distributions on manifolds (2013, background): “In topological data analysis there has been work on understanding properties of random abstract simplicial complexes generated from stochastic processes [1, 2, 5, 13, 28, 29, 30, 32, 46, 47] and non-asymptotic bounds on the convergence or consistency of topological summaries as the number of points ”
Spectral Gaps of Random Graphs and Applications (2012, methodology): “Linial and Meshulam [35] introduce an analogous measure Y2(n, p) to the binomial random graph for random 2D simplicial complexes.”
Maximally Persistent Cycles in Random Geometric Complexes (2015, background): “tudy of random simplicial complexes rather than random graphs, where in addition to vertices and edges the structure consists also of triangles, tetrahedra and higher dimensional simplexes (see, e.g. [4, 38, 40, 42]). The study of random simplicial complexes focuses mainly on their homology, which i”
The fundamental group of random 2-complexes (2007, background): “By the Linial-Meshulam-Wallach results ([10, 11]), H1(Y,Z) is finite and has no p-torsion for any fixed p.”
Networks beyond pairwise interactions: structure and dynamics (2020, methodology): “In this model, one begins with a connected graph on n nodes, to which some number m of triangles are added to form facets of 3 nodes [257].”
On the phase transition in random simplicial complexes (2014, methodology): “Indeed, it was shown in [20] with subsequent work in [24] that in Y d ( n, p ) the threshold for the vanishing of the ( d − 1)-th homology is p = d ln n n .”
The threshold function for vanishing of the top homology group of random $d$-complexes (2009, methodology): “Curiously, the methods of [11, 13] do not easily extend to the case of integer coefficients, and finding the threshold functions for the vanishing of Hd−1(Yn,p,d;Z) remains open even for the case d = 2.”
Large random simplicial complexes, I (2015, methodology): “special cases: When r = 1 and p= (1,p) we obtain the classical model of random graphs of Erdo¨s and R´enyi [10]. When r = 2 and p = (1,1,p) we obtain the Linial - Meshulam model of random 2-complexes [15]. When r is arbitrary and ﬁxed and p= (1,1,...,1,p) we obtain the random simplicial complexes of”