# Entropy-Based Centrality

#### Definition

Entropy-Based Centrality a novel improved centrality by integrating interaction intimacy and confidence level to measure the total influence of an individual which can be decomposed into direct effect and indirect effect.

Each node in the set $V$ represents an individual in a social network, $E$ denotes the set of undirected edges connecting two adjacent individuals, and the set of weight $W$ corresponds to the intimacy of connection through which the influence flows. Also, $w_{ij}$ denotes the weighted value on the given edge connecting node $i$ and node $j$. To quantify the influence of a given in dividual, individual’s influence deconstruct into two components including direct influence $DI$ and indirect influence $II$, achieved by integrating the connection weight and confidence level to the information entropy.

This novel definition of entropy-based centrality proposed in order to compute on direct influence, which takes into consideration two aspects including connection intimacy of individual and confidence level among neighbors, each focusing on the personal emotion that determines the diffusion of social influence. The higher the connection weight for individual pairs, the greater the degree of mutual influence between them. Motivated by this idea, the weight influence entropy for individual $i$, denoted as $DI_i^w$, is defined as follows:

$$DI_i^w=-{\underset{j=1}{\overset{N_i}{\sum}}} {w_{ij}\over {\sum_{k=1}^{N_i}}w_{ik}} . log_{10} {w_{ij}\over {\sum_{k=1}^{N_i}}w_{ik}}$$

Where $N-i$ denotes the total number of neighbors of individual $i$, and $w_{ij}$ indicates the weight of connection bonding individual $i$ with its neighbor $j$. Generally, the probability of an individual $i$ transmitting influence to his or her neighbor $j$ will not always be consistent. Additionally, which neighbor is chosen as a recipient of information depends on how trustworthy individual $i$ think neighbor $j$. Therefore, considering the heterogeneity of confidence in neighbors, the confidence level defined according to the recipient’s relative social status measured by its degree ratio among all neighbors, where the probability of confidence level $T_{ij}$ is given as follows:

$$T_{ij}={k_j^{\beta}\over \sum_{l \in N_i}(k_l^{\beta})}$$

Where $N_i$ indicates the set of individual $i$’s neighbors, $k_j$ indicates the degree of recipient $j$, and tunable parameter $\beta$ called the confidence strength reinforces the sensitivity of recipient’s degree $k_j$ to the confidence probability $T_{ij}$ . The individual’s degree $k_j$ can refer to the level of node heterogeneity. When $\beta > 0$, it means that individual $i$ has more trend to influence those who possess a higher degree and vice versa. Specially $\beta = 0$, individual $i$ influences all neighbors with equal probability of confidence.

Based on the above discussion, the definition of the confidence influence entropy $DI_c$ for individual $i$ is stated as follows:

$$DI_i^c=-{\underset{j=1}{\overset{N_i}{\sum}}} T_{ij}. log_{10}T_{ij}= - {\underset{j=1}{\overset{N_i}{\sum}}} {k_j^{\beta}\over \sum_{l\in Ni}(k_l^{\beta})} .log_{10} {k_j^{\beta}\over \sum_{l\in Ni}(k_l^{\beta})}$$

This method provides an improved centrality measurement elaborated on the concept of entropy, which takes into account the weight of connection and confidence level. In addition this method quantifies the influence entropy for each individual, containing weight influence entropy and confidence influence entropy.

Each node in the set $V$ represents an individual in a social network, $E$ denotes the set of undirected edges connecting two adjacent individuals, and the set of weight $W$ corresponds to the intimacy of connection through which the influence flows. Also, $w_{ij}$ denotes the weighted value on the given edge connecting node $i$ and node $j$. To quantify the influence of a given in dividual, individual’s influence deconstruct into two components including direct influence $DI$ and indirect influence $II$, achieved by integrating the connection weight and confidence level to the information entropy.

This novel definition of entropy-based centrality proposed in order to compute on direct influence, which takes into consideration two aspects including connection intimacy of individual and confidence level among neighbors, each focusing on the personal emotion that determines the diffusion of social influence. The higher the connection weight for individual pairs, the greater the degree of mutual influence between them. Motivated by this idea, the weight influence entropy for individual $i$, denoted as $DI_i^w$, is defined as follows:

$$DI_i^w=-{\underset{j=1}{\overset{N_i}{\sum}}} {w_{ij}\over {\sum_{k=1}^{N_i}}w_{ik}} . log_{10} {w_{ij}\over {\sum_{k=1}^{N_i}}w_{ik}}$$

Where $N-i$ denotes the total number of neighbors of individual $i$, and $w_{ij}$ indicates the weight of connection bonding individual $i$ with its neighbor $j$. Generally, the probability of an individual $i$ transmitting influence to his or her neighbor $j$ will not always be consistent. Additionally, which neighbor is chosen as a recipient of information depends on how trustworthy individual $i$ think neighbor $j$. Therefore, considering the heterogeneity of confidence in neighbors, the confidence level defined according to the recipient’s relative social status measured by its degree ratio among all neighbors, where the probability of confidence level $T_{ij}$ is given as follows:

$$T_{ij}={k_j^{\beta}\over \sum_{l \in N_i}(k_l^{\beta})}$$

Where $N_i$ indicates the set of individual $i$’s neighbors, $k_j$ indicates the degree of recipient $j$, and tunable parameter $\beta$ called the confidence strength reinforces the sensitivity of recipient’s degree $k_j$ to the confidence probability $T_{ij}$ . The individual’s degree $k_j$ can refer to the level of node heterogeneity. When $\beta > 0$, it means that individual $i$ has more trend to influence those who possess a higher degree and vice versa. Specially $\beta = 0$, individual $i$ influences all neighbors with equal probability of confidence.

Based on the above discussion, the definition of the confidence influence entropy $DI_c$ for individual $i$ is stated as follows:

$$DI_i^c=-{\underset{j=1}{\overset{N_i}{\sum}}} T_{ij}. log_{10}T_{ij}= - {\underset{j=1}{\overset{N_i}{\sum}}} {k_j^{\beta}\over \sum_{l\in Ni}(k_l^{\beta})} .log_{10} {k_j^{\beta}\over \sum_{l\in Ni}(k_l^{\beta})}$$

This method provides an improved centrality measurement elaborated on the concept of entropy, which takes into account the weight of connection and confidence level. In addition this method quantifies the influence entropy for each individual, containing weight influence entropy and confidence influence entropy.

#### References

- Ni, C., Yang, J. and Kong, D., 2019. Sequential seeding strategy for social influence diffusion with improved entropy-based centrality. Physica A: Statistical Mechanics and its Applications, p.123659. DOI: 10.1016/j.physa.2019.123659