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Analyzing research collaborations within the School of Mathematical Sciences, UKM using Graph Theory

A graph (comprised of vertices and edges) mapping out the collaboration between 38 lecturers in the School of Mathematical Sciences is obtained for years 2012-2014. The graph is then studied to understand the nature of collaborations within the school as well as identifying the importance of different vertices in the network. Methods from social network analysis and network theory will also be applied to the graph in addition to classic graph theory measures.


Introduction
Graph theory is the study of graphs, which are mathematical abstractions called vertices (also called nodes or actors). Each related pairs of vertices is connected by an edge (also called link or relations). Essentially, graph are structures used to model relations between objects. Typically, a graph is depicted in diagrammatic form as a set of dots for the vertices, joined by lines or curves for the edges. In more recent literatures graphs are also known as networks. Graph theory is utilized in to analyse real networks.
One particularly popular form of this analysis is the social network analysis which is the analysis of social structure consisting of a set of players (vertices) and a set of relationships (edges) between the players. Social network analysis examines the structure of social relationships in a group to reveal the structure and key players of the group. The ubiquitous-ness of Facebook, which is a clear social network formed by users (vertices) and their friendships (edges) have resulted in targeted ads for certain groups and even political manipulation in the case of Cambridge Analytica [1]. Faceboook itself, uses social network analysis to closely monitor Facebook users and their relationships.
In this article, we shall do a simple network analysis of research collaborations quantified by joint publications (edges) amongst researchers (vertices) within the School of Mathematical Sciences (PPSM), Faculty of Science and Technology, Universiti Kebangsaan Malaysia (UKM) between years 2012 and 2014.

Graph Theory and Networks
A graph or network can be written as = ( , ), where is the set of vertices and is the set of edges. A simple graph is a graph that has neither self-edges nor multiedges (multiple edges between two vertices). Figure 1 is an example of a simple graph. A graph can be represented by a matrix where each row and column represents a vertex. In a simple graph, if two vertices has an edge connecting them, then the corresponding element of the matrix will have value of 1. Otherwise the corresponding element of the matrix will be zero. This matrix is called an adjacency matrix. Table 1 displays an adjacency matrix that represents the graph in Figure 1.   Figure 1 represented the network of research collaboration within PPSM, then one could say that researcher 1 (represented by vertex 1) has joint publications with all other researchers in the network. This can also be observed by looking at column or row 1 in Table 1 which has the value 1 for the all elements except for the first row. Note that for a simple matrix the diagonals must all be zero since there are no self-loops. Observe that researcher 5 only has a single joint publication with researcher 1 since entry (1,5) and (5,1) are the only nonzero entries in row and column 5 of Table 1. The degree of a vertex in a graph is the number of edges connected to it. We will denote the degree of vertex ∈ by . can be obtain from the adjacency matrix in Table 1 by summing the values of any single row or column. By summing up rows in Table 1, we get 1 = 5, 2 = 3, 3 = 3, 4 = 2, 5 = 1 and 6 = 2, which can easily be verified in Figure 1.
A path in a network is any sequence of vertices such that every consecutive pair of vertices in the sequence is connected by an edge in the network [2]. The distance between two vertices, ∈ and ∈ , is defined as the number of edges along the shortest path connecting them. Every two vertices connected by an edge has distance 1, therefore the graph in Figure 1 has 12 = 13 = 36 = 1 and so on. However there are no direct edges between quite a few vertices therefore the shortest path between these vertices will be longer. For example to connect vertex 5 to any other vertices excluding 1, we need to go through vertex 1 (since 15 = 1), therefore the distance becomes 2 such that 25 = 35 = 45 = 65 = 2. The diameter of a graph is the length of the longest shortest path between any pair of vertices in the graph for which a path actually exists [2][3][4][5]. The diameter of the graph in Figure 1 is 2.
The concept of centrality originated from the discipline of social network analysis. Centralities measures rank vertices according to their perceived importance. Perhaps the simplest centrality measure in a network is just the degree of a vertex [2,3]. For the graph in Figure 1, vertex 1 will have the highest ranking since it has the largest degree ( 1 = 5), in second place will be vertices 2 and 3 ( 2 = 3 = 3). This will be followed by vertices 4 and 6 ( 4 = 6 = 2) and the lowest ranking will go to vertex 5 since Some centralities depend on the distance between vertices such as the closeness centrality. The closeness centrality of vertex ∈ can be defined as the reciprocal of farness [4], such that (1) It uses the inverse sum of distances from a vertex to all other vertices in the graph to rank the vertex. The larger the closeness centrality value the 'closer' the vertex is to all other vertices. The betweenness centrality of vertex ∈ can be defined [3,5] as where ( ) is the number of shortest paths from to that passes through vertex . is the total number of shortest paths from to (since there may be more than one shortest path between any two vertices). Other definitions and formulas are also used to define closeness and betweenness centrality [2,3,6].