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I can use software to calculate the Betti numbers $\beta_0,\beta_1,\beta_2,\dots$ of a finite simplicial complex.

This is prohibitive for large complexes, built on say > 100,000 nodes.

Is there some way to computationally approximate the ranks of the first $n$ homology groups? Results e.g. Carlson here seem to work only for data points in Euclidean space, where the relations on which the complex is built are interpretations of the data (i.e. persistent homology). I have a set of fixed, deterministic relations on a set of vertices i.e. a graph, and the corresponding clique complex.

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    $\begingroup$ Could you clarify what you mean by 'approximate' here? $\endgroup$ – Neil Hoffman Nov 27 '18 at 17:20
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    $\begingroup$ I mean to get the Betti numbers, but only “nearly”. Some error is allowed. $\endgroup$ – Alexander Kartun-Giles Nov 27 '18 at 17:22
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    $\begingroup$ Surely some numerical approximation is available which is inaccurate but quick, converging in some limit to the actual situation? $\endgroup$ – Alexander Kartun-Giles Nov 27 '18 at 19:13
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    $\begingroup$ You can do mod p homology computations on data of large size. It is a kind of approximation. $\endgroup$ – Dima Pasechnik Nov 27 '18 at 20:23
  • $\begingroup$ Ok I will look at mod p homology. I'm thinking of getting the Euler characteristic of the clique complex of a sparse complex network. $\endgroup$ – Alexander Kartun-Giles Nov 28 '18 at 17:38
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You have to find a way to reduce the size of your simplicial complex. Some algorithms based e.g. on discrete Morse theory can do that fairly rapidly, but they don't have guarantees on the amount of size reduction. I don't think there exists faster algorithms for approximate Betti numbers in general, but I believe it can be done if your simplicial complex is "low-dimensional", meaning that for any point, the size of the sub complex formed by vertices at distance less than $r$ from that point grows slowly with $r$.

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  • $\begingroup$ OK I will look at this "lossy compression" idea. $\endgroup$ – Alexander Kartun-Giles Nov 28 '18 at 17:41
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    $\begingroup$ Those reduction methods typically won't change the homotopy type, so they'll give you exact Betti numbers in fact $\endgroup$ – alesia Nov 28 '18 at 18:55
  • $\begingroup$ So you can reduce the computation time, without losing information? Like lossless compression? $\endgroup$ – Alexander Kartun-Giles Dec 1 '18 at 13:25
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    $\begingroup$ Actually the volume condition I was mentioning was for a different algorithm that approximates Betti numbers. It shouldn't affect the complex reduction algorithms $\endgroup$ – alesia Dec 1 '18 at 18:00
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    $\begingroup$ web.math.ku.dk/~moller/students/brian_brost.pdf seems to have a good state of the art (section 7) $\endgroup$ – alesia Dec 1 '18 at 21:08

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