Aggregation models on hypergraphs

TL;DR

This paper explores a method to infer topological features of systems modeled on hypergraphs from observable data, applying it to polymer models and extending known relations to identify additional interactions.

Contribution

It introduces a novel approach to reconstruct topological information from correlation data on hypergraphs, generalizing existing models and identifying criteria for peer-to-peer interactions.

Findings

Derived iterative relations for hypergraph partition functions.

Validated the inverse problem's robustness and precision.

Proposed a criterion to detect peer-to-peer interactions.

Abstract

Following a newly introduced approach by Rasetti and Merelli we investigate the possibility to extract topological information about the space where interacting systems are modelled. From the statistical datum of their observable quantities, like the correlation functions, we show how to reconstruct the activities of their constitutive parts which embed the topological information. The procedure is implemented on a class of polymer models on hypergraphs with hard-core interactions. We show that the model fulfils a set of iterative relations for the partition function that generalise those introduced by Heilmann and Lieb for the monomer-dimer case. After translating those relations into structural identities for the correlation functions we use them to test the precision and the robustness of the inverse problem. Finally the possible presence of a further interaction of peer-to-peer type is considered and a criterion to discover it is identified.