# Juan M. Alonso: Graphs associated to finite metric spaces

**Time: **
Tue 2022-04-12 10.15

**Location: **
KTH, 3721, Lindstedtsvägen 25, and Zoom

**Video link: **
Meeting ID: 659 3743 5667

**Participating: **
Juan M. Alonso ((BIOS) IMASL - CONICET and Universidad Nacional de San Luis)

### Abstract

Many concrete problems are formulated in terms of a finite set of points in some \(\mathbb{R}^N\) which, via the ambient Euclidean metric, becomes a finite metric space \((M,d)\). This situation arises, for instance, when studying the glass transition from simulations.

To obtain information from *M* it is not always possible to use your favorite mathematical tool directly on *M*. The “favorite tool” in my case is finite dimension which, although defined on finite metric spaces, is not effectively computable when *M* is large and unstructured. In such cases, a useful alternative is to associate a graph to *M*, and do mathematics directly on the graph, rather than on the space. One should think of this graph as an approximation to *M*.

Among the many graphs that can be associated to *M*, I first considered \({MC} = {MC}(M)\), the Minimum Connected graph, a version — adapted to our situation — of the Vietoris complex of a metric space. Unfortunately *MC* is usually a rather dense graph. I then introduce \(CS = CS(M)\), the Connected Sparse graph, a streamlined version of *MC*. *CS* encodes the local information of *M*; in fact, it is almost a definition of what local structure of *M* means.

Despite its name, *CS* can be dense, even a complete graph. However, in our application to glass, we computed *CS* for more than 700 spaces with about 2200 points each. All of them turned out to be trees.

To understand this “coincidence”, I considered \(\mathfrak{M}_k\), the set of all subsets of *k* elements contained in some fixed \(\mathbb{R}^N\), and defined a metric on it. In the talk I will describe subsets *D* of \(\mathfrak{M}_k\) such that *D* contains open and dense subsets of \(\mathfrak{M}_k\), and have, moreover, the property that *CS*(*M*) is a tree for all *M* in *D*. In particular, the “general case” is for *CS* to be a tree.

- About CS:
*Metric spaces and sparse graphs*, arXiv:2103.16471. - Application to the glass transition (see section 3) of:
*Finite dimension unravels the structural features at the glass transition*, Eur. Phys. J. E (2021) 44:88.