Network constraints on the mixing patterns of binary node metadata

We consider the network constraints on the bounds of the assortativity coefficient, which aims to
quantify the tendency of nodes with the same attribute values to be connected. The assortativity
coefficient can be considered as the Pearson’s correlation coefficient of node metadata values across
network edges and lies in the interval [−1, 1]. However, properties of the network, such as degree
distribution and the distribution of node metadata values place constraints upon the attainable
values of the assortativity coefficient. This is important as a particular value of assortativity may say
as much about the network topology as about how the metadata are distributed over the network –
a fact often overlooked in literature where the interpretation tends to focus simply on the propensity
of similar nodes to link to each other, without any regard on the constraints posed by the topology.
In this paper we quantify the effect that the topology has on the assortativity coefficient in the case
of binary node metadata. Specifically we look at the effect that the degree distribution, or the full
topology, and the proportion of each metadata value has on the extremal values of the assortativity
coefficient. We provide the means for obtaining bounds on the extremal values of assortativity for
different settings and demonstrate that under certain conditions the maximum and minimum values
of assortativity are severely limited, which may present issues in interpretation when these bounds
are not considered.