Assortativity

Though the specific measure of similarity may vary, network theorists often examine assortativity in terms of a node's degree.

Correlations between nodes of similar degree are often found in the mixing patterns of many observable networks.

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.

The two most prominent measures are the assortativity coefficient and the neighbor connectivity.

This captures the number of edges leaving the node, other than the one that connects the pair.

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

Adopting the notation of that article, it is possible to define four metrics

-degree of the source (i.e. tail) node vertex of the edge, and

-degree of targets, respectively; averages being taken over the edges of the network.

Another means of capturing the degree correlation is by examining the properties of

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.

In assortative networks, there could be nodes that are disassortative and vice versa.

A local assortative measure[7] is required to identify such anomalies within networks.

is the average excess degree of its neighbors and M is the number of links in the network.

, we ensure that the equation for local assortativity for a directed network satisfies the condition

[4] The assortative patterns of a variety of real world networks have been examined.

Note that the social networks (the first five entries) have apparent assortative mixing.

On the other hand, the technological and biological networks (the middle six entries) all appear to be disassortative.

It has been suggested that this is because most networks have a tendency to evolve, unless otherwise constrained, towards their maximum entropy state—which is usually disassortative.

[8] The table also has the value of r calculated analytically for two models of networks: In the ER model, since edges are placed at random without regard to vertex degree, it follows that r = 0 in the limit of large graph size.

The scale-free BA model also holds this property.

For the BA model in the special case of m=1 (where each incoming node attaches to only one of the existing nodes with a degree-proportional probability), a more precise result is known: as

(the number of vertices) tends to infinity, r approaches 0 at the same rate as

[2] The properties of assortativity are useful in the field of epidemiology, since they can help understand the spread of disease or cures.

For instance, the removal of a portion of a network's vertices may correspond to curing, vaccinating, or quarantining individuals or cells.

Since social networks demonstrate assortative mixing, diseases targeting high degree individuals are likely to spread to other high degree nodes.

Alternatively, within the cellular network—which, as a biological network is likely dissortative—vaccination strategies that specifically target the high degree vertices may quickly destroy the epidemic network.

Special caution must be taken to avoid this structural disassortativity.