Assortativity on arvau

Wikipedia / 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 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. the network is completely disassortative.

For undirected networks, the assortativity coefficient can be written as

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).

is the joint probability distribution of remaining degrees occurs, and normalizing these counts so that

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) 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 , be one of the in/out-degree of the source (i.e. tail-degree of the target (i.e. head-degree of targets, respectively; averages being taken over the edges of the network. Finally, we have

Neighbor connectivity

, or the average degree of neighbors of a node with degree 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,

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

Respectively, local assortativity for directed networks

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.

Assortative mixing patterns of real networks

The assortative patterns of a variety of real world networks have been examined. While models of social networks tend to exhibit assortative mixing, models of technological and biological networks 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 statewhich is usually disassortative.

The table also has the value of r calculated analytically for two models of networks:

the random graph of Erdős and Rényi
BA Model (Barabási-Albert model)

Application

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 networkwhich, as a biological network is likely dissortativevaccination strategies that specifically target the high degree vertices may quickly destroy the epidemic network.

Structural disassortativity

The basic structure of a network can cause these measures to show disassortativity, which is not representative of any underlying assortative or disassortative mixing. Special caution must be taken to avoid this structural disassortativity.