# Network constraints on the mixing patterns of binary node metadata

## TL;DR

This paper investigates how network properties like degree distribution and node attribute counts constrain the possible values of the assortativity coefficient for binary node attributes, affecting its interpretability.

## Contribution

It introduces methods to determine bounds on assortativity values considering network constraints, highlighting limitations in its interpretability.

## Key findings

- Network properties impose significant bounds on assortativity values.
- Certain network constraints can severely limit the range of attainable assortativity.
- Implications for interpreting assortativity in constrained networks.

## Abstract

We consider the network constraints on the bounds of the assortativity coefficient, which measures the tendency of nodes with the same attribute values to be interconnected. The assortativity coefficient is the Pearson's correlation coefficient of node attribute values across network edges and ranges between -1 and 1. We focus here on the assortativity of binary node attributes and show that properties of the network, such as degree distribution and the number of nodes with each attribute value place constraints upon the attainable values of the assortativity coefficient. We explore the assortativity in three different spaces, that is, ensembles of graph configurations and node-attribute assignments that are valid for a given set of network constraints. We provide means for obtaining bounds on the extremal values of assortativity for each of these spaces. Finally, we demonstrate that under certain conditions the network constraints severely limit the maximum and minimum values of assortativity, which may present issues in how we interpret the assortativity coefficient.