# Geo-referenced Eigenvector Centrality

#### Definition

This method is able to measure the influence of two factors, the topology of the network and the geo-referenced data extracted from the network and associated to the nodes.The proposed centrality measure is an adaptation of the eigenvector centrality for the spatial networks, which in addition to preserving the characteristics of the original centrality, also includes in the computation process the geo-referenced data.

$$\overset{\rightarrow}{c}= {1\over \lambda}[A\overset{\rightarrow}{x}_1 (j)+\overset{\rightarrow}{x}_1]$$

where $\overset{\rightarrow}{c}$ constitutes the centrality values for the nodes of the graph, and $A$ is the adjacency of the original urban network. $\overset{\rightarrow}{x}$ is an eigenvector of the adjacency matrix $A$ associated to the eigenvalue $\lambda$. $A\overset{\rightarrow}{x}_1$ spreads the importance of neighbouring nodes in the network.

The main contribution of the proposed model is the incorporation of the geo-located data factor to the computation structure for eigenvector centrality in the urban street networks.

$$\overset{\rightarrow}{c}= {1\over \lambda}[A\overset{\rightarrow}{x}_1 (j)+\overset{\rightarrow}{x}_1]$$

where $\overset{\rightarrow}{c}$ constitutes the centrality values for the nodes of the graph, and $A$ is the adjacency of the original urban network. $\overset{\rightarrow}{x}$ is an eigenvector of the adjacency matrix $A$ associated to the eigenvalue $\lambda$. $A\overset{\rightarrow}{x}_1$ spreads the importance of neighbouring nodes in the network.

The main contribution of the proposed model is the incorporation of the geo-located data factor to the computation structure for eigenvector centrality in the urban street networks.

#### References

- Agryzkov T., Tortosa L., Vicent J.F., Wilson R., 2019. A centrality measure for urban networks based on the eigenvector centrality concept. Environment and Planning B: Urban Analytics and City Science, 46(4), pp.668-689. DOI: 10.1177/2399808317724444