# Quantum statistics in Network Geometry with Fractional Flavor

## TL;DR

This paper extends the Network Geometry with Flavor model to fractional values, revealing that face statistics follow Bose-Einstein or Fermi-Dirac distributions and exploring spectral properties of the resulting networks.

## Contribution

It introduces fractional flavor values in the model, showing how face statistics are governed by quantum distributions and analyzing spectral properties of the associated networks.

## Key findings

- Face statistics follow Bose-Einstein or Fermi-Dirac distributions.
- Fractional flavor extends the model's topological and statistical complexity.
- Spectral properties of the network are characterized in this new regime.

## Abstract

Growing network models have been shown to display emergent quantum statistics when nodes are associated to a fitness value describing the intrinsic ability of a node to acquire new links. Recently it has been shown that quantum statistics emerge also in a growing simplicial complex model called Network Geometry with Flavor which allow for the description of many-body interaction between the nodes. This model depend on an external parameter called flavor that is responsible for the underlying topology of the simplicial complex. When the flavor takes the value $s=-1$ the $d$-dimensional simplicial complex is a manifold in which every $(d-1)$-dimensional face can only have an incidence number $n_{\alpha}\in\{0,1\}$. In this case the faces of the simplicial complex are naturally described by the Bose-Einstein, Boltzmann and Fermi-Dirac distribution depending on their dimension. In this paper we extent the study of Network Geometry with Flavor to fractional values of the flavor $s=-1/m$ in which every $(d-1)$-dimensional face can only have incidence number $n_{\alpha}\in\{0,1,2,\dots, m\}$. We show that in this case the statistical properties of the faces of the simplicial complex are described by the Bose-Einstein or the Fermi-Dirac distribution only. Finally we comment on the spectral properties of the networks constituting the underlying structure of the considered simplicial complexes.