# SRIC - Short-Range Interaction Centrality

#### Definition

The Key Borrowers Detection by the Intensities of their Short-Range Interactions done by the key borrower index that is a modified index named preference-based power index. This index define as follows:

$$\alpha_i= {\mathscr{X}_i\over {\sum}_j \mathscr{X}_j}$$

where ${\mathscr{X}_i}$ is intensity of connection for a node $i$ which calculate as follows:

$${\mathscr{X}_i}={\underset{wl}{\sum}} f(i,w_l)/N_w$$

The $f(i,w_l)$ define the intensity of connection between a lender and a borrower and the number of borrowers in the group represents by $N_w$.

key borrower index $(\alpha_i)$ shows the most pivotal borrower. This borrower indeed is the most interconnected one in network (It borrows relatively large amounts of money from quite a few agents.

For the “many lenders/borrowers” case it needs to aggregate the index over all lenders. Therefore, the aggregation of the index for each borrower over all lenders should take into account the size of each lender’s total loans. The importance of a borrower for a large lender (who provides a large amount of loans in total) is not the same as its importance for a small one (who provides a small amount of loans in total). Therefore, the final value of the key borrower index has the following form:

$$\alpha_i= {\underset{l}{\sum}} \left ({\mathscr{X}_i\over {\sum}_j \mathscr{X}_j}\right) \times {Total{\_}loons_l\over {\sum}_l Total{\_}loons_l}$$

where $Total{\_}loans_l$ is the total amount of loans provided to all borrowers by a lender $l$.

The degree centrality, closeness, and betweenness measures lack information about the value of the links (borrowed/lent amounts of money). Applicability of the key borrower index is come from incorporation of the desired features of the existing centrality measures and at the same time lacks the deficiency noted above.

$$\alpha_i= {\mathscr{X}_i\over {\sum}_j \mathscr{X}_j}$$

where ${\mathscr{X}_i}$ is intensity of connection for a node $i$ which calculate as follows:

$${\mathscr{X}_i}={\underset{wl}{\sum}} f(i,w_l)/N_w$$

The $f(i,w_l)$ define the intensity of connection between a lender and a borrower and the number of borrowers in the group represents by $N_w$.

key borrower index $(\alpha_i)$ shows the most pivotal borrower. This borrower indeed is the most interconnected one in network (It borrows relatively large amounts of money from quite a few agents.

For the “many lenders/borrowers” case it needs to aggregate the index over all lenders. Therefore, the aggregation of the index for each borrower over all lenders should take into account the size of each lender’s total loans. The importance of a borrower for a large lender (who provides a large amount of loans in total) is not the same as its importance for a small one (who provides a small amount of loans in total). Therefore, the final value of the key borrower index has the following form:

$$\alpha_i= {\underset{l}{\sum}} \left ({\mathscr{X}_i\over {\sum}_j \mathscr{X}_j}\right) \times {Total{\_}loons_l\over {\sum}_l Total{\_}loons_l}$$

where $Total{\_}loans_l$ is the total amount of loans provided to all borrowers by a lender $l$.

The degree centrality, closeness, and betweenness measures lack information about the value of the links (borrowed/lent amounts of money). Applicability of the key borrower index is come from incorporation of the desired features of the existing centrality measures and at the same time lacks the deficiency noted above.

#### References

- Aleskerov F., Andrievskaya I., Permjakova E., 2016. Key borrowers detected by the intensities of their short-range interactions. Springer Proceedings in Mathematics and Statistics, 156, pp.267-280. DOI: 10.1007/978-3-319-29608-1_18