# EWD - Extended Weighted Degree Centrality

#### Definition

This method is according to the remaining minimum degree $(RMD)$ decomposition by removing the nodes with the minimum degree iteratively. Based on the RMD decomposition, a weighted degree (WD) is presented by utilizing the $RMD$ indices of the nearest neighbors of a node. $WD$ assigns a weight to each degree of this node, which can distinguish the contribution of each degree to the spreading influence. Further, an extended weighted degree $(EWD)$ centrality is proposed by extending the $WD$ of the nearest neighbors of a node.

The extended weighted degree $(EWD)$ centrality defined by extending the $WD$ of the nearest neighbors of a node as folloes:

$$EWD(v)={\underset{u \in neighbors(v)}{\sum}} WD(u) $$

where $(WD)$ is the weighted degree and defined as follows:

$$WD_{(v)}={\underset{u \in neighbors(v)}{\sum}} {Iter(u) \over MaxIter} $$

where, the array $Iterand$ variable $MaxIter$ can be obtained from the $RMD$ decomposition algorithm. Each item in $\sum$ corresponds to assign a weight to each degree of a node. Therefore, the contribution of each degree to the spreading influence of this node is distinguished.

On one hand, implementing $RMD$ decomposition must utilize the global topological structure of a network. Thus, $RMD$ index reflects the global position of a node in this network. On the other hand, the calculation of $WD$ is based on the nearest neighbors of a node, which utilizes the local topological structure of a network. Overall, $WD$ considers both of the global and the local structure simultaneously. So, extended weighted degree $(EWD)$ centrality extended by the $WD$ of the nearest neighbors of a node.

Given $ G(V, E)$ as an undirected and unweighted network, where $V$ and $E$ are the nodes set and edges set, respectively. $RMD$ decomposition is as follows,

$RMD$ decomposition: given a $G$, find the minimum degree $(md)$ in current $G$, remove all nodes with degree=$md$ and generate a new $G$, the removed nodes are assigned with a $RMD$ index. In the similar way, find the $md$ in current $G$, remove all nodes with degree=$md$, generate a new $G$, the removed nodes are assigned with a $RMD$ index. Repeat the process and assign a corresponding $RMD$ index to the removed nodes in each iteration. Finish the process when $G$ is empty.

In summary, the $EWD$ is based on $RMD$ decomposition, a $WD$ is presented to distinguish the contribution of each degree to the influence of a node. Further, by extending the $WD$ of the nearest neighbors of a node, an $EWD$ centrality is proposed to rank the spreading influence of nodes.

The extended weighted degree $(EWD)$ centrality defined by extending the $WD$ of the nearest neighbors of a node as folloes:

$$EWD(v)={\underset{u \in neighbors(v)}{\sum}} WD(u) $$

where $(WD)$ is the weighted degree and defined as follows:

$$WD_{(v)}={\underset{u \in neighbors(v)}{\sum}} {Iter(u) \over MaxIter} $$

where, the array $Iterand$ variable $MaxIter$ can be obtained from the $RMD$ decomposition algorithm. Each item in $\sum$ corresponds to assign a weight to each degree of a node. Therefore, the contribution of each degree to the spreading influence of this node is distinguished.

On one hand, implementing $RMD$ decomposition must utilize the global topological structure of a network. Thus, $RMD$ index reflects the global position of a node in this network. On the other hand, the calculation of $WD$ is based on the nearest neighbors of a node, which utilizes the local topological structure of a network. Overall, $WD$ considers both of the global and the local structure simultaneously. So, extended weighted degree $(EWD)$ centrality extended by the $WD$ of the nearest neighbors of a node.

Given $ G(V, E)$ as an undirected and unweighted network, where $V$ and $E$ are the nodes set and edges set, respectively. $RMD$ decomposition is as follows,

$RMD$ decomposition: given a $G$, find the minimum degree $(md)$ in current $G$, remove all nodes with degree=$md$ and generate a new $G$, the removed nodes are assigned with a $RMD$ index. In the similar way, find the $md$ in current $G$, remove all nodes with degree=$md$, generate a new $G$, the removed nodes are assigned with a $RMD$ index. Repeat the process and assign a corresponding $RMD$ index to the removed nodes in each iteration. Finish the process when $G$ is empty.

In summary, the $EWD$ is based on $RMD$ decomposition, a $WD$ is presented to distinguish the contribution of each degree to the influence of a node. Further, by extending the $WD$ of the nearest neighbors of a node, an $EWD$ centrality is proposed to rank the spreading influence of nodes.

#### References

- Yang F., Li X., Xu Y., Liu X., Wang J., Zhang Y., Zhang R., Yao Y., 2018. Ranking the spreading influence of nodes in complex networks: An extended weighted degree centrality based on a remaining minimum degree decomposition. Physics Letters, Section A: General, Atomic and Solid State Physics, 382(34), pp.2361-2371. DOI: 10.1016/j.physleta.2018.05.032