Topics of Stochastic Algebraic Topology

Costa, A.; Farber, Michael; Kappeler, Thomas

doi:10.1016/j.entcs.2012.05.005

Cited by 15 publications

(9 citation statements)

References 20 publications

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“…There are a number of accessible surveys on this subject (e.g. [6,9,12,15,45]), all of which share a common theme of studying topological invariants of simplicial complexes built on point sets. At the time of writing, another excellent review [7] by Carlsson appeared, which is longer than the earlier ones, more up to date, and which also contains a gentle introduction the topological concepts needed in the current paper.…”

Section: Some Historymentioning

confidence: 99%

Random geometric complexes in the thermodynamic regime

Yogeshwaran

Subag

Adler

2015

Probab. Theory Relat. Fields

116

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We consider the topology of simplicial complexes with vertices the points of a random point process and faces determined by distance relationships between the vertices. In particular, we study the Betti numbers of these complexes as the number of vertices becomes large, obtaining limit theorems for means, strong laws, concentration inequalities and central limit theorems. As opposed to most prior papers treating random complexes, the limit with which we work is in the so-called 'thermodynamic' regime (which includes the percolation threshold) in which the complexes become very large and complicated, with complex homology characterised by diverging Betti numbers. The proofs combine probabilistic arguments from the theory of stabilizing functionals of point processes and topological arguments exploiting the properties of Mayer-Vietoris exact sequences. The Mayer-Vietoris arguments are crucial, since homology in general, and Betti numbers in particular, are global rather than local phenomena, and most standard probabilistic arguments are based on the additivity of functionals arising as a consequence of locality.

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Section: Some Historymentioning

confidence: 99%

Random geometric complexes in the thermodynamic regime

Yogeshwaran

Subag

Adler

2015

Probab. Theory Relat. Fields

116

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“…These applications include the new theory of large networks [21]. Recent surveys covering various aspects of topology of large random spaces can be found in [5] and [18]. This paper focuses on the notion of a critical dimension of random simplicial complexes.…”

Section: Introductionmentioning

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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“…The mathematical study of large random simplicial complexes started relatively recently and several different probabilistic models of random topological objects have appeared within the last 10 years, see [4] and [14] for surveys. One may mention random surfaces [18], random 3-dimensional manifolds [9], random configuration spaces of linkages [11].…”

Section: Introductionmentioning

confidence: 99%

Large random simplicial complexes, I

Costa

Farber

2016

J. Topol. Anal.

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In this paper we introduce and develop the multi-parameter model of random simplicial complexes with randomness present in all dimensions. Various geometric and topological properties of such random simplicial complexes are characterised by convex domains in the high-dimensional parameter space (rather than by intervals, as in the usual one-parameter models). We find conditions under which a multi-parameter random simplicial complex is connected and simply connected. Besides, we give an intrinsic characterisation of the multi-parameter probability measure. We analyse links of simplexes and intersections of multi-parameter random simplicial complexes and show that they are also multi-parameter random simplicial complexes.

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Topics of Stochastic Algebraic Topology

Cited by 15 publications

References 20 publications

Random geometric complexes in the thermodynamic regime

Random geometric complexes in the thermodynamic regime

Large random simplicial complexes, III the critical dimension

Large random simplicial complexes, I

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