# ETC - Evidence Theory Centrality

#### Definition

Evidence theory can consider the results from different sources comprehensively and the Shannon entropy can measure the uncertainty of information. Evidence Theory Centrality $(ETC)$ uses these two methods to rate the results gained from different measures and combine them to generate a new ranking result.

The method works in this way that, Choose the methods participating the fusion based on real need (DC, BC, and CC,....). In the next step it obtains their ranking value and do the following to get the $BPA$ distribution of each method:

$$Mi_j= {Mi_j-min(M_i)\over sum(M_i) - n\times min(M_i)}$$

where $M_i$ refers to value distribution of the selected method $i$, and $Mi_j$ refers to the value of node $j$ in method $M_i$. $n$ is the number of nodes in graph.

Next step is calculating the Shannon entropy and do the following to determine the weight of each method (The Shannon entropy reflects the chaotic degree of distribution so the weights and Shannon entropy are inversely proportional):

$$S={\underset{i=1}{\overset{N}{\sum}}} S_{Mi}$$

$$W_{Mi}={S - S_{Mi}\over N_1S}$$

where $M_i$ is the chosen method, $S_{Mi}$ is its Shannon entropy. $S$ is the sum of Shannon entropy and $W_{Mi}$ is the weight of method $M_i$.

Finally, the last step is Integrating them to generate a new $BPA$ distribution $m$

$$m_j={\underset{i=1}{\overset{N_1}{\sum}}} {\underset{j=1}{\overset{N_2}{\sum}}} W_{Mi} \star Mi_j$$

where $m_j$ is integrated value of node $j$, $N_1$ is the number of methods involved with fusion and $N_2$ denotes the number of nodes in graph. Afterwards evidence theory is used to fuse for $N_1-1$ times.

Rather than specifying some fixed methods to fuse, this method provides a flexible framework for selecting the required method to fuse according to the actual situation. The advantage of the porposed method is its flexibility, it can adjust its result by changing the methods participating fusion based on real demand. if the method performs bad under specific evaluation criteria, the $ETC$ would reduce its weight. The main disadvantage of $ETC$ is that it needs know the value of other methods first, which would cost more time.

The method works in this way that, Choose the methods participating the fusion based on real need (DC, BC, and CC,....). In the next step it obtains their ranking value and do the following to get the $BPA$ distribution of each method:

$$Mi_j= {Mi_j-min(M_i)\over sum(M_i) - n\times min(M_i)}$$

where $M_i$ refers to value distribution of the selected method $i$, and $Mi_j$ refers to the value of node $j$ in method $M_i$. $n$ is the number of nodes in graph.

Next step is calculating the Shannon entropy and do the following to determine the weight of each method (The Shannon entropy reflects the chaotic degree of distribution so the weights and Shannon entropy are inversely proportional):

$$S={\underset{i=1}{\overset{N}{\sum}}} S_{Mi}$$

$$W_{Mi}={S - S_{Mi}\over N_1S}$$

where $M_i$ is the chosen method, $S_{Mi}$ is its Shannon entropy. $S$ is the sum of Shannon entropy and $W_{Mi}$ is the weight of method $M_i$.

Finally, the last step is Integrating them to generate a new $BPA$ distribution $m$

$$m_j={\underset{i=1}{\overset{N_1}{\sum}}} {\underset{j=1}{\overset{N_2}{\sum}}} W_{Mi} \star Mi_j$$

where $m_j$ is integrated value of node $j$, $N_1$ is the number of methods involved with fusion and $N_2$ denotes the number of nodes in graph. Afterwards evidence theory is used to fuse for $N_1-1$ times.

Rather than specifying some fixed methods to fuse, this method provides a flexible framework for selecting the required method to fuse according to the actual situation. The advantage of the porposed method is its flexibility, it can adjust its result by changing the methods participating fusion based on real demand. if the method performs bad under specific evaluation criteria, the $ETC$ would reduce its weight. The main disadvantage of $ETC$ is that it needs know the value of other methods first, which would cost more time.

#### References

- Zhao, J., Song, Y. and Deng, Y., 2020. A Novel Model to Identify the Influential Nodes: Evidence Theory Centrality. IEEE Access, 8, pp.46773-46780. DOI: 10.1109/ACCESS.2020.2978142