# A null model for Dunbar's circles

## TL;DR

This paper introduces a null model based on social resource constraints that explains the layered structure of ego-networks, aligning with Dunbar's circles and exhibiting Poisson degree distributions.

## Contribution

It proposes a grand-canonical ensemble model that reproduces the hierarchical social structure and scaling ratios observed in ego-networks, connecting social layers to resource costs.

## Key findings

- Layer degrees follow a Poisson distribution in the thermodynamic limit.
- Equispaced layer costs are necessary for constant group size scaling.
- The model fits and compares well with empirical social network data.

## Abstract

An individual's social group may be represented by their ego-network, formed by the links between the individual and their acquaintances. Ego-networks present an internal structure of increasingly large nested layers (or circles) of decreasing relationship intensity, whose size exhibits a precise scaling ratio. Starting from the notion of limited social bandwidth, and assuming fixed costs for the links in each layer, we propose a null model built on a grand-canonical ensemble that generates the observed hierarchical social structure. The observed internal structure of ego-networks becomes a natural outcome to expect when we assume the existence of layers demanding different amounts of resources. In the thermodynamic limit, reached when the number of ego-network copies is large, the specific layer degrees follow a Poisson distribution. We also find that, under certain conditions, equispaced layer costs are necessary to obtain a constant group size scaling. Our model presents interesting analogies to a Bose-Einstein gas, that we briefly discuss. Finally, we fit and compare the model with an empirical social network.