# Complete Neighbourhood Centrality

#### Definition

The impact of a node within a network is not depending only on the direct connections but also depending on the neighbourhood connections or the whole system. Hence considering neighbourhood nodes for calculating the measure of neighbourhood centrality is more accurate. Sometimes there also need to take the importance of all nodes when calculating proposed centrality. To overcome the above problems, a complete neighbourhood centrality measure introduced.

The proposed measure considers the degree of a node and its neighbourhood in weighted network. Since the node’s significant depends on its direct neighbours and hence depends on neighbour of neighbours. Also, the significant of neighbour of a node decreases as the distance increases from that node.

Hence complete neighbourhood centrality $C_{CN}(v_i)$ of a node $v_i$ is given by:

$$C_{CN}(v_i)=d_w (v_i)$$

$$+P {\sum} _{v_j \in N(v_i)} d_w(v_i)$$

$$+P^2 {\sum} _{v_k \in N(v_i)\\ v_i, N(v_i)} d_w(v_k)$$

$$+P^3 {\sum} _{v_l \in N(v_k)\\ v_i, N(v_k)} d_w(v_l)+ ... .. ...$$

$$+P^t {\sum} _{v_s \in N(v_r)\\ v_r, N(v_q)} d_w(v_s) $$

where $d_w (v_i)$ denote the degree of the node $v_i$, $N(v_i)$ is the set of neighbour nodes of the node $v_i$, $p$ is a parameter $\in(0,1)$. This parameter is called an adjustable parameter and its value is chosen by the nature and problem of the network.

This method is more acceptable as all nodes are taken into account when finding the importance of a node. Taking link strength is also gives more practical results for measuring centrality.However, the proposed technique is focusing on the degree and position of the nodes and its neighbour. This method is straightforward to calculate and gives a clear idea about the spreading capability of a node.

The proposed measure considers the degree of a node and its neighbourhood in weighted network. Since the node’s significant depends on its direct neighbours and hence depends on neighbour of neighbours. Also, the significant of neighbour of a node decreases as the distance increases from that node.

Hence complete neighbourhood centrality $C_{CN}(v_i)$ of a node $v_i$ is given by:

$$C_{CN}(v_i)=d_w (v_i)$$

$$+P {\sum} _{v_j \in N(v_i)} d_w(v_i)$$

$$+P^2 {\sum} _{v_k \in N(v_i)\\ v_i, N(v_i)} d_w(v_k)$$

$$+P^3 {\sum} _{v_l \in N(v_k)\\ v_i, N(v_k)} d_w(v_l)+ ... .. ...$$

$$+P^t {\sum} _{v_s \in N(v_r)\\ v_r, N(v_q)} d_w(v_s) $$

where $d_w (v_i)$ denote the degree of the node $v_i$, $N(v_i)$ is the set of neighbour nodes of the node $v_i$, $p$ is a parameter $\in(0,1)$. This parameter is called an adjustable parameter and its value is chosen by the nature and problem of the network.

This method is more acceptable as all nodes are taken into account when finding the importance of a node. Taking link strength is also gives more practical results for measuring centrality.However, the proposed technique is focusing on the degree and position of the nodes and its neighbour. This method is straightforward to calculate and gives a clear idea about the spreading capability of a node.

#### References

- Das K., Samanta S., De K., Pal M., 2020. Complete neighbourhood centrality and its application. 4th International Conference on Computational Intelligence and Networks, CINE 2020, . DOI: 10.1109/CINE48825.2020.234386