Assortativity

Assortativity, or assortative mixing is a preference for a network's nodes to attach to others that are similar in some way. Though the specific measure of similarity may vary, network theorists often examine assortativity in terms of a node's degree. The addition of this characteristic to network models more closely approximates the behaviors of many real world networks.

Correlations between nodes of similar degree are often found in the mixing patterns of many observable networks. For instance, in social networks, nodes tend to be connected with other nodes with similar degree values. This tendency is referred to as assortative mixing, or assortativity. On the other hand, technological and biological networks typically show disassortative mixing, or dissortativity, as high degree nodes tend to attach to low degree nodes.

Assortativity is often operationalized as a correlation between two nodes. However, there are several ways to capture such a correlation. The two most prominent measures are the assortativity coefficient and the neighbor connectivity. These measures are outlined in more detail below.

The assortativity coefficient is the Pearson correlation coefficient of degree between pairs of linked nodes. Positive values of r indicate a correlation between nodes of similar degree, while negative values indicate relationships between nodes of different degree. In general, r lies between −1 and 1. When r = 1, the network is said to have perfect assortative mixing patterns, when r = 0 the network is non-assortative, while at r = −1 the network is completely disassortative.

The assortativity coefficient is given by ${\displaystyle r={\frac {\sum _{jk}{jk(e_{jk}-q_{j}q_{k})}}{\sigma _{q}^{2}}}}$. The term ${\displaystyle q_{k}}$ is the distribution of the remaining degree. This captures the number of edges leaving the node, other than the one that connects the pair. The distribution of this term is derived from the degree distribution ${\displaystyle p_{k}}$ as ${\displaystyle q_{k}={\frac {(k+1)p_{k+1}}{\sum _{j\geq 1}jp_{j}}}}$. Finally, ${\displaystyle e_{jk}}$ refers to the joint probability distribution of the remaining degrees of the two vertices. This quantity is symmetric on an undirected graph, and follows the sum rules ${\displaystyle \sum _{jk}{e_{jk}}=1\,}$ and ${\displaystyle \sum _{j}{e_{jk}}=q_{k}\,}$.

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