Assortativity

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 disassortativity, as high degree nodes tend to attach to low degree nodes.

Measurement

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.

Assortativity coefficient

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

For undirected networks, the assortativity coefficient can be written as

${\displaystyle r=\frac{\sum _{j,k}jk(e_{jk}-q_j{q}_k)}{{\sigma }_q^2}}$.

In this expression, the indices ${\displaystyle j}$ and ${\displaystyle k}$ denote possible values of the remaining degree, which is defined as the degree of a node minus one (i.e., the number of edges leaving a node other than the edge currently under consideration).

The quantity ${\displaystyle q_k}$ is the probability that a randomly chosen end of an edge is attached to a node with remaining degree ${\displaystyle k}$. Because nodes with higher degree contribute proportionally more edge ends, ${\displaystyle q_k}$ is derived from the degree distribution ${\displaystyle p_k}$ as

${\displaystyle q_k=\frac{(k+1)p_{k+1}}{\sum _{j}j{p}_j}}$.

The term ${\displaystyle e_{jk}}$ is the joint probability distribution of the remaining degrees at the two ends of a randomly chosen edge. It is obtained by examining all edges in the network, counting how often the pair ${\displaystyle (j,k)}$ of remaining degrees occurs, and normalizing these counts so that

${\displaystyle \sum _{j,k}{e}_{jk}=1}$.

By construction, the marginals of ${\displaystyle e_{jk}}$ reproduce the distribution ${\displaystyle q_k}$

${\displaystyle \sum _j{e}_{jk}=q_k}$.

With these definitions, the expression above is exactly the Pearson correlation coefficient of the remaining degrees at the endpoints of edges.

In a directed graph, in-assortativity (${\displaystyle r(\text{in},\text{in})}$) and out-assortativity (${\displaystyle r(\text{out},\text{out})}$) measure the tendencies of nodes to connect with other nodes that have similar in and out degrees as themselves, respectively. Extending this further, four types of assortativity can be considered (see ). Adopting the notation of that article, it is possible to define four metrics ${\displaystyle r(\text{in},\text{in})}$, ${\displaystyle r(\text{in},\text{out})}$, ${\displaystyle r(\text{out},\text{in})}$, and ${\displaystyle r(\text{out},\text{out})}$. Let ${\displaystyle (\alpha ,\beta )}$, be one of the in/out word pairs (e.g. ${\displaystyle (\alpha ,\beta )=(\text{out},\text{in})}$). Let ${\displaystyle E}$ be the number of edges in the network. Suppose we label the edges of the network ${\displaystyle 1,\dots ,E}$. Given edge ${\displaystyle i}$, let ${\displaystyle j_i^{\alpha }}$ be the ${\displaystyle \alpha }$-degree of the source (i.e. tail) node vertex of the edge, and ${\displaystyle k_i^{\beta }}$ be the ${\displaystyle \beta }$-degree of the target (i.e. head) node of edge ${\displaystyle i}$. We indicate average values with bars, so that ${\displaystyle \overline{j^{\alpha }}}$, and ${\displaystyle \overline{k^{\beta }}}$ are the average ${\displaystyle \alpha }$-degree of sources, and ${\displaystyle \beta }$-degree of targets, respectively; averages being taken over the edges of the network. Finally, we have

${\displaystyle r(\alpha ,\beta )=\frac{\sum _i(j_i^{\alpha }-\overline{j^{\alpha }})(k_i^{\beta }-\overline{k^{\beta }})}{\sqrt{\sum _i(j_i^{\alpha }-\overline{j^{\alpha }}{)}^2}\sqrt{\sum _i(k_i^{\beta }-\overline{k^{\beta }}{)}^2}}.}$

Neighbor connectivity

Another means of capturing the degree correlation is by examining the properties of ${\displaystyle ⟨k_{nn}⟩}$, or the average degree of neighbors of a node with degree k. This term is formally defined as: ${\displaystyle ⟨k_{nn}⟩=\sum _{k^{\prime }}{k}^{\prime }P(k^{\prime }|k)}$, where ${\displaystyle P(k^{\prime }|k)}$ is the conditional probability that an edge of node with degree k points to a node with degree k'. If this function is increasing, the network is assortative, since it shows that nodes of high degree connect, on average, to nodes of high degree. Alternatively, if the function is decreasing, the network is disassortative, since nodes of high degree tend to connect to nodes of lower degree.

Local assortativity

In assortative networks, there could be nodes that are disassortative and vice versa. A local assortative measure is required to identify such anomalies within networks. Local assortativity is defined as the contribution that each node makes to the network assortativity. Local assortativity in undirected networks is defined as,

${\displaystyle \rho =\frac{j\left(j+1\right)\left(\overline{k}-{\mu }_q\right)}{2M{\sigma }_q^2}}$

Where ${\displaystyle j}$ is the excess degree of a particular node and ${\displaystyle \overline{k}}$ is the average excess degree of its neighbors and M is the number of links in the network.

Respectively, local assortativity for directed networks is a node's contribution to the directed assortativity of a network. A node's contribution to the assortativity of a directed network ${\displaystyle r_d}$ is defined as, ${\displaystyle {\rho }_d=\frac{{j_{out}}^2\left({\overline{k}}_{in}-{\mu }_q^{in}\right)+{j_{in}}^2\left({\overline{k}}_{out}-{\mu }_q^{out}\right)}{2M{\sigma }_q^{in}{\sigma }_q^{out}}}$

Where ${\displaystyle j_{out}}$ is the out-degree of the node under consideration and ${\displaystyle j_{in}}$ is the in-degree, ${\displaystyle {\overline{k}}_{in}}$ is the average in-degree of its neighbors (to which node ${\displaystyle v}$} has an edge) and ${\displaystyle {\overline{k}}_{out}}$ is the average out-degree of its neighbors (from which node ${\displaystyle v}$ has an edge).${\displaystyle {\sigma }_q^{in}\ne 0}$,${\displaystyle {\sigma }_q^{out}\ne 0}$.

By including the scaling terms ${\displaystyle {\sigma }_q^{in}}$ and ${\displaystyle {\sigma }_q^{out}}$ , we ensure that the equation for local assortativity for a directed network satisfies the condition ${\displaystyle r_d=\sum _{i=1}^N{\rho }_d}$.

Further, based on whether the in-degree or out-degree distribution is considered, it is possible to define local in-assortativity and local out-assortativity as the respective local assortativity measures in a directed network.