# Gravity Centrality Index

#### Definition

By applying the $k-shell$ value of each node as its mass and the shortest path distance between two nodes as their distance, then inspired by the idea of the gravity formula, gravity centrality index developed to identify the influential spreaders in complex networks. Enlighten by the idea of classical gravity formula proposed by Isaac Newton, the k-shell value of node $i$ can be consider as its mass, and the shortest path distance between two nodes in network is viewed as their distance. In this way, the influence of node $i$ is measured as fallows:

$$G(i)={\underset{j\in \psi_i}{\sum}} {ks(i)ks(j)\over d_{ij}^2}$$

where $d_{ij}$ is the shortest path distance between node $i$ and node $j$. $\psi_i$ is the neighborhood set whose distance to node $i$ is less than or equal to a given value $r$. An extended gravity index is further developed as fallows:

$$G_+(i)={\underset{j\in \Lambda_i}{\sum}}G(j)$$

$\Lambda_i$ is the nearest neighborhood of node $i$.

The idea of the gravity method comes from the well-known gravity formula, which is very dramatic and impressive. What’s more, the gravity model can reflect the facts that, on one hand, the interaction influence between two nodes is proportional to their corresponding $k-shell$ values; on the other hand, the influences of the neighbors decreases with their distance.

$$G(i)={\underset{j\in \psi_i}{\sum}} {ks(i)ks(j)\over d_{ij}^2}$$

where $d_{ij}$ is the shortest path distance between node $i$ and node $j$. $\psi_i$ is the neighborhood set whose distance to node $i$ is less than or equal to a given value $r$. An extended gravity index is further developed as fallows:

$$G_+(i)={\underset{j\in \Lambda_i}{\sum}}G(j)$$

$\Lambda_i$ is the nearest neighborhood of node $i$.

The idea of the gravity method comes from the well-known gravity formula, which is very dramatic and impressive. What’s more, the gravity model can reflect the facts that, on one hand, the interaction influence between two nodes is proportional to their corresponding $k-shell$ values; on the other hand, the influences of the neighbors decreases with their distance.

#### References

- Ma L.L., Ma C., Zhang H.F., Wang B.H., 2016. Identifying influential spreaders in complex networks based on gravity formula. Physica A: Statistical Mechanics and its Applications, 451, pp.205-212. DOI: 10.1016/j.physa.2015.12.162