# LA - Local Assortativity

#### Definition

Assortativity is the tendency in networks where nodes mostly connected with similar nodes based on degree of
nodes. In other words, highly-connected nodes have connection with other highly-connected nodes in the network,
and vice versa.

Local assortativity value for every node in the network can be defined as: where j, M and k are degree of v, number of links in the network and the average degree of node v’s neighbors, respectively; μ

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 r

By including the scaling terms σ

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.

Assortativity is high when high-degree nodes tend to connect to other high-degree nodes; it is low (i.e., negative) when high-degree nodes are linked to low-degree nodes.

Correlations between nodes of similar degree are often found in the mixing patterns of many observable networks. For instance, in social networks, highly connected nodes tend to be connected with other high degree nodes. 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.

Local assortativity value for every node in the network can be defined as: where j, M and k are degree of v, number of links in the network and the average degree of node v’s neighbors, respectively; μ

_{q}and σ_{q}are mean and standard deviation of the remaining degree distribution, q(k), respectively. q(k),is defined as: where p(k) is network degree distribution.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 r

_{d}is defined as, Where*j*is the out-degree of the node under consideration and_{out}*j*is the in-degree, is the average in-degree of its neighbors (to which node_{in}*v*} has an edge) and is the average out-degree of its neighbors (from which node*v*has an edge). , .By including the scaling terms σ

^{in}_{q}and σ^{out}_{q}, we ensure that the equation for local assortativity for a directed network satisfies the condition .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.

Assortativity is high when high-degree nodes tend to connect to other high-degree nodes; it is low (i.e., negative) when high-degree nodes are linked to low-degree nodes.

Correlations between nodes of similar degree are often found in the mixing patterns of many observable networks. For instance, in social networks, highly connected nodes tend to be connected with other high degree nodes. 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.

#### Software

#### References

- PIRAVEENAN, M., PROKOPENKO, M. & ZOMAYA, A. Y. 2008. Local assortativeness in scale-free networks. EPL (Europhysics Letters), 84, 28002. DOI: 10.1209/0295-5075/84/28002
- PIRAVEENAN, M., PROKOPENKO, M. & ZOMAYA, A. 2012. Assortative mixing in directed biological networks. IEEE/ACM Transactions on Computational Biology and Bioinformatics (TCBB), 9, 66-78. DOI: 10.1109/TCBB.2010.80