Large random simplicial complexes, I

Costa, A.; Farber, Michael

doi:10.1142/s179352531650014x

Cited by 40 publications

(62 citation statements)

References 17 publications

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“…The intersections of the domains D k with the plane α 0 = α 3 = · · · = 0 is shown on the Figure 1. Like in our previous publications [8], [9], [10], we shall consider here the multi-parameter random simplicial complexes with the multi-parameters p i = n −αi where in general the vector of exponents α = (α 0 , . .…”

Section: The Notion Of a Critical Dimensionmentioning

confidence: 99%

Large random simplicial complexes, III the critical dimension

Costa

Farber

2017

J. Knot Theory Ramifications

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In this paper we study the notion of critical dimension of random simplicial complexes in the general multi-parameter model described in [8], [9], [10]. This model includes as special cases the Linial-MeshulamWallach model [19], [20] as well as the clique complexes of random graphs. We characterise the concept of critical dimension in terms of various geometric and topological properties of random simplicial complexes such as their Betti numbers, the fundamental group, the size of minimal cycles and the degrees of simplexes. We mention in the text a few interesting open questions.

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Section: The Notion Of a Critical Dimensionmentioning

confidence: 99%

Large random simplicial complexes, III the critical dimension

Costa

Farber

2017

J. Knot Theory Ramifications

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“…In the present paper we study two very general probabilistic models generating random simplicial complexes of arbitrary dimension which we call the lower and upper models. random simplicial complexes was studied in a series of papers [2], [3], [4], [5] under the name of multiparameter random simplicial complexes; the name reflects the fact that the geometric and topological properties of simplicial complexes in this model depend on the set of probability parameters p 0 , p 1 , . .…”

Section: Introductionmentioning

confidence: 99%

Random simplicial complexes, duality and the critical dimension

2019

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In this paper we discuss two general models of random simplicial complexes which we call the lower and the upper models. We show that these models are dual to each other with respect to combinatorial Alexander duality. The behaviour of the Betti numbers in the lower model is characterised by the notion of critical dimension, which was introduced by A. Costa and M. Farber in [5]: random simplicial complexes in the lower model are homologically approximated by a wedge of spheres of dimension equal the critical dimension. In this paper we study the Betti numbers in the upper model and introduce new notions of critical dimension and spread. We prove that (under certain conditions) an upper random simplicial complex is homologically approximated by a wedge of spheres of the critical dimension.In the case of the lower model one builds the random simplicial complex inductively, step by step, starting with a random set of vertices, then adding randomly edges between the selected vertices, and on the following step adding randomly 2-simplexes (triangles) to the random graph obtained on the previous stage, and so on. Examples are given by the Erdös -Rényi [7] random graphs and their high dimensional generalizations, the Linial, Meshulam, Wallach [14] , [15] random simplicial complexes, as well as random clique complexes [12], [13]. In larger generality the lower model of Michael

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“…Note that most random complexes which appear in literature are homogeneous. For example the multi-parameter random simplicial complexes of [6], [7], [8], [9] are homogeneous lower random simplicial complexes.…”

Section: 2mentioning

confidence: 99%

“…Historically the first models of random simplicial complexes were suggested by Linial and Meshulam [17] and Meshulam and Wallach [18]. More recently Costa and Farber [6,7,8,9] studied a multi-parameter generalisation of the Linial-Meshulam-Wallach models involving a sequence of probability parameters p 0 , p 1 , p 2 , . .…”

Section: Introductionmentioning

confidence: 99%

Random simplicial complexes in the medial regime

Farber

Mead

2020

Topology and its Applications

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We describe topology of random simplicial complexes in the lower and upper models in the medial regime, i.e. under the assumption that the probability parameters pσ approach neither 0 nor 1. We show that nontrivial Betti numbers of typical lower and upper random simplicial complexes in the medial regime lie in a narrow range of dimensions. For instance, an upper random simplicial complex Y on n vertices in the medial regime with high probability has non-vanishing Betti numbers bj (Y ) only for k + c < n − j < k + log 2 k + c ′ where k = log 2 ln n and c, c ′ are constants. A lower random simplicial complex on n vertices in the medial regime is with high probability (k + a)connected and its dimension d satisfies d ∼ k + log 2 k + a ′ where a, a ′ are constants. The paper develops a new technique, based on Alexander duality, which relates the lower and upper models.

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Large random simplicial complexes, I

Cited by 40 publications

References 17 publications

Large random simplicial complexes, III the critical dimension

Large random simplicial complexes, III the critical dimension

Random simplicial complexes, duality and the critical dimension

Random simplicial complexes in the medial regime

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