# Re-defined Entropy Centrality

#### Definition

This centrality is is based on decompositions of a graph into subgraphs and analysis on the entropy of neighbor nodes.

In this method the first step is finding subgraph $G_i$ constructed by node $i$ and its one-hop neighbors. The subgraph degree centrality $(SDC)$ measure of node $i$ and its one-hop neighbor node $j$ in $G_i$, denoted as $SDC_i$, is defined as:

$$SDC_i={\underset{j}{\overset{M}{\sum}}} b_{ij}$$

where $M$ is the number of one-hop neighbors of node $i$. And $b_{ij} = 1$, if there is an edge between node $i$ and node $j$, otherwise, the value of $b_{ij}$ will be $0$.

Next step is calculation of the local influence of node $i$ on its one-hop neighbors:

$$LI_i=I_i^n=log_b ({\underset{i=1}{\overset{M+1}{\sum}}}SDC_i)-{\underset{i=1}{\overset{M+1}{\sum}}} {SDC_i\over {\sum}_i^{M+1} SDC_i} log_b SDC_i$$

the local influence of node $i$ on its one-hop neighbors, denoted as $LI_i$ which equals the entropy of neighbor nodes $I_i^n$ for node $i$.

Then it need to calculate the indirect influence of node $i$ denoted as $II_i$ is expressed as

$$II_i={{\sum}_{k=1}^{M^i}II_{ik}\over M_i}$$

where $M_i$ indicates the number of $i’$s two-hop neighbor nodes

And finally the overall influence of node $i$ calculate by the following equation:

$$I_i=\omega_1 LI_i + \omega_2II_i$$

where $\omega_1$ and $\omega_2$ the weight of the local influence and indirect influence. Note that $\omega_1 +\omega_2 = 1$.

In this method the first step is finding subgraph $G_i$ constructed by node $i$ and its one-hop neighbors. The subgraph degree centrality $(SDC)$ measure of node $i$ and its one-hop neighbor node $j$ in $G_i$, denoted as $SDC_i$, is defined as:

$$SDC_i={\underset{j}{\overset{M}{\sum}}} b_{ij}$$

where $M$ is the number of one-hop neighbors of node $i$. And $b_{ij} = 1$, if there is an edge between node $i$ and node $j$, otherwise, the value of $b_{ij}$ will be $0$.

Next step is calculation of the local influence of node $i$ on its one-hop neighbors:

$$LI_i=I_i^n=log_b ({\underset{i=1}{\overset{M+1}{\sum}}}SDC_i)-{\underset{i=1}{\overset{M+1}{\sum}}} {SDC_i\over {\sum}_i^{M+1} SDC_i} log_b SDC_i$$

the local influence of node $i$ on its one-hop neighbors, denoted as $LI_i$ which equals the entropy of neighbor nodes $I_i^n$ for node $i$.

Then it need to calculate the indirect influence of node $i$ denoted as $II_i$ is expressed as

$$II_i={{\sum}_{k=1}^{M^i}II_{ik}\over M_i}$$

where $M_i$ indicates the number of $i’$s two-hop neighbor nodes

And finally the overall influence of node $i$ calculate by the following equation:

$$I_i=\omega_1 LI_i + \omega_2II_i$$

where $\omega_1$ and $\omega_2$ the weight of the local influence and indirect influence. Note that $\omega_1 +\omega_2 = 1$.

#### References

- Qiao T., Shan W., Zhou C., 2017. How to identify the most powerful node in complex networks? A novel entropy centrality approach. Entropy, 19(11). DOI: 10.3390/e19110614