Random Metric Spaces and Universality

WWe define the notion of a random metric space and prove that with probability one such a space is isometricto the Urysohn universal metric space. The main technique is the study of universal and random distance matrices; we relate the properties of metric (in particulary universal) space to the properties of distance matrices. We show the link between those questions and classification of the Polish spaces with measure (Gromov or metric triples) and with the problem about S_{\infty}-invariant measures in the space of symmetric matrices. One of the new effects -exsitence in Urysohn space so called anarchical uniformly distributed sequences. We give examples of other categories in which the randomness and universality coincide (graph, etc.).Comment: 38 PAGE

The model theory of geometric random graphs

We study the logical properties of infinite geometric random graphs, introduced by Bonato and Janssen. These are graphs whose vertex set is a dense ``generic'' subset of a metric space, where two vertices are adjacent with probability $p>0$ provided the distance between them is bounded by some constant number. We prove that for a large class of metric spaces, including circles, spheres and the complete Urysohn space, almost all geometric random graphs on a given space are elementary equivalent. Moreover, their first-order theory can reveal geometric properties of the underlying metric space.Comment: Fixed some typos, added reference to previous work by Ackerman, Freer, Kruckman and Pate

Probabilistic Programming Interfaces for Random Graphs::Markov Categories, Graphons, and Nominal Sets

We study semantic models of probabilistic programming languages over graphs, and establish a connection to graphons from graph theory and combinatorics. We show that every well-behaved equational theory for our graph probabilistic programming language corresponds to a graphon, and conversely, every graphon arises in this way.We provide three constructions for showing that every graphon arises from an equational theory. The first is an abstract construction, using Markov categories and monoidal indeterminates. The second and third are more concrete. The second is in terms of traditional measure theoretic probability, which covers 'black-and-white' graphons. The third is in terms of probability monads on the nominal sets of Gabbay and Pitts. Specifically, we use a variation of nominal sets induced by the theory of graphs, which covers Erdős-Rényi graphons. In this way, we build new models of graph probabilistic programming from graphons

The Rado simplicial complex

In this paper we study a remarkable simplicial complex X on countably many vertexes. X is universal in the sense that any count- able simplicial complex is an induced subcomplex of X. Additionally, X is homogeneous, i.e. any two isomorphic finite induced subcomplexes are related by an automorphism of X. We prove that X is the unique simplicial complex which is both universal and homogeneous. The 1- skeleton of X is the well-known Rado graph. We show that a random simplicial complex on countably many vertexes is isomorphic to X with probability 1. We prove that the geometric realisation of X is homeo- morphic to an infinite dimensional simplex. We observe several curious properties of X, for example we show that X is robust, i.e. removing any finite set of simplexes leaves a simplicial complex isomorphic to X. The robustness of X leads to the hope that suitable finite approximations of X can serve as models for very resilient networks in real life applications. In a forthcoming paper [8] we study finite approximations to the Rado complex, they can potentially be useful in real life applications due to their structural stability