# Weight Neighborhood Centrality

#### Definition

The logic behind this centrality comes from this concept that for the node set with same centrality, the nodes with influential neighbors will have higher spread influences. Besides, in most of unweighted networks, the edges are treated equally, however, each edge may have underlying different significances in network structures and functions. So in this method a weight neighborhood centrality method which considers the centrality of a node and its neighbors’ centrality to generate the influential ranking list, where the diffusion importance of an edge is used to adjust the influence of its neighbors’ centrality.

The weight neighborhood centrality of node $i$, is defined as:

$$C_i(\varphi)={\varphi}_i+ {\underset{j\in \Gamma(i)}{\sum}} {w_{ij}\over \langle w \rangle}.\varphi_i$$

where $\varphi$ is the benchmark centrality, $\Gamma(i)$ is the set of nearest neighbors of node $i$, $w_{ij}$ is the diffusion importance of edge $e_{ij}$ and $\langle w \rangle$ is the average diffusion importance of all edges.

In unweighted networks, the edges are treated equally in many works. In fact, the edges are different if the influences of their corresponding connection nodes are different. Thus, we can use $w_{ij}$, weighting method based on the power-low function of degree that modified the above relation to quantify the diffusion importance of links, which is defined as:

$$w_{ij}=(k_i.k_j)^{\alpha}$$

where $k_i$ and $k_j$ are the degrees of node $i$ and node $j$ respectively and $\alpha$ is a tuning parameter that set to 1. $\alpha = 1$ leads to the strongest robustness on various networks.

One common shortcoming among most of centrality measures is that the diffusion importance of links in spreading process is completely ignored, which may lead to identify the spreading influence of nodes not very well, so the authors of Weight Neighborhood Centrality showed that their purposed model could overcome this problem.

The weight neighborhood centrality of node $i$, is defined as:

$$C_i(\varphi)={\varphi}_i+ {\underset{j\in \Gamma(i)}{\sum}} {w_{ij}\over \langle w \rangle}.\varphi_i$$

where $\varphi$ is the benchmark centrality, $\Gamma(i)$ is the set of nearest neighbors of node $i$, $w_{ij}$ is the diffusion importance of edge $e_{ij}$ and $\langle w \rangle$ is the average diffusion importance of all edges.

In unweighted networks, the edges are treated equally in many works. In fact, the edges are different if the influences of their corresponding connection nodes are different. Thus, we can use $w_{ij}$, weighting method based on the power-low function of degree that modified the above relation to quantify the diffusion importance of links, which is defined as:

$$w_{ij}=(k_i.k_j)^{\alpha}$$

where $k_i$ and $k_j$ are the degrees of node $i$ and node $j$ respectively and $\alpha$ is a tuning parameter that set to 1. $\alpha = 1$ leads to the strongest robustness on various networks.

One common shortcoming among most of centrality measures is that the diffusion importance of links in spreading process is completely ignored, which may lead to identify the spreading influence of nodes not very well, so the authors of Weight Neighborhood Centrality showed that their purposed model could overcome this problem.

#### References

- Wang J., Hou X., Li K., Ding Y., 2017. A novel weight neighborhood centrality algorithm for identifying influential spreaders in complex networks. Physica A: Statistical Mechanics and its Applications, 475, pp.88-105. DOI: 10.1016/j.physa.2017.02.007