Cluster Analysis of Local Convergent Sequences of Structures

TL;DR

This paper introduces a novel method for asymptotic clustering of large structures based on local convergence, combining analytic and geometric techniques to understand connectivity at the limit without explicit limit structures.

Contribution

It presents a new approach to cluster analysis of large structures using local convergence and a reinterpretation of limit representation theorems.

Findings

Derived an asymptotic clustering method for locally convergent sequences.

Connected the connectivity structure at the limit to finite structures.

Provided a new interpretation of limit representation theorems.

Abstract

The cluster analysis of very large objects is an important problem, which spans several theoretical as well as applied branches of mathematics and computer science. Here we suggest a novel approach: under assumption of local convergence of a sequence of finite structures we derive an asymptotic clustering. This is achieved by a blend of analytic and geometric techniques, and particularly by a new interpretation of the authors' representation theorem for limits of local convergent sequences, which serves as a guidance for the whole process. Our study may be seen as an effort to describe connectivity structure at the limit (without having a defined explicit limit structure) and to pull this connectivity structure back to the finite structures in the sequence in a continuous way.