Abstract
Banks that provide loans to clients face the problem that the client’s repayment behaviour can differ significantly. Predicting the repayment behaviour is imperative for banks for planning purposes, and for deciding on the credit worthiness of prospective clients. A common approach is to group clients in different categories depending on their credit risk profile which is based on past behaviour.
In this thesis, we investigate an approach that uses graph theory tools to divide clients into different groups of individuals with similar repayment behaviour. We define a graph whose vertices are the clients, with two vertices being adjacent if the repayment behaviour of the clients they represent is similar. Then we use a method based on the eigenvalues of a matrix related to the adjacency matrix of this graph to find groups of clients with similar repayment behaviour. This method is often used for community detection in large real world networks.
This thesis consists of six chapters. In Chapter 1, we define the most significant terms used throughout the thesis. We also consider the relevant theory pertaining to graphs that will be used in the study and finally, provide an underlying motivation for our research.
In Chapter 2, we explore the theory behind matrices. We define the basic concepts of matrix theory focusing on adjacency matrix, eigenvalues, and eigenvectors.
In Chapter 3, we look at the theory of probability and random graphs. We also consider the derivation and application of the configuration model, which will be used to define the modularity of a cut in the following chapter.
xv
Upon exploring and understanding the above-mentioned theory, in Chapter 4 we proceed to define the modularity and compile the relevant theory and algorithms to partition the vertex set of the given graph into two or more subsets (often called communities). These algorithms first consider the leading eigenvalues of a modular matrix to determine whether the graph is divisible or not. This method of splitting complex graphs with edges connecting between different vertices as described in [14] and has proved to be a better approach for detecting natural community structure in real-world networks when compared to other traditional graph partitioning methods. Unlike other traditional graph partitioning methods, this method is unconstrained by the need to find graphs of any particular order but rather determines its own maximum number of subgraphs.
In chapter 5, we focus on the application of the modularity theory using leading eigenvalue. Using real world data from a bank, we then create a graph, which will be used for our analysis by firstly finding the similarity matrix of the clients then creating its modularity matrix. We then split the graph based on the leading eigenvalues and the corresponding eigenvectors of the modularity matrix. Then we compare the grouping of clients corresponding to the split we obtained to the traditional grouping, finding that the quality of both groupings is comparable.
Finally, in chapter 6, we give the conclusion to the thesis.