Links play a significant role in the functioning of a complex network. The aim of this thesis is to study the links in a weighted network by introducing two new concepts. The link betweenness centrality of a link is defined as the fraction of shortest paths between all pairs of nodes in a graph that traverses that link. Although link betweenness is a widely known measure that characterizes the link, we introduce the concept, link weight tolerance, to understand the extent to which the weight of the link can be increased or decreased such that the shortest paths in the graph are unaffected, therefore the link betweenness of the links remain the same. We develop a method to generate the positive and negative tolerance of a link. We use examples to illustrate the algorithm and discuss the results. Prior to introducing this concept, in addition to surveying existing network theory measures, we also analyse the metric, betweenness centrality and describe the methods used to generate weighted and unweighted random graphs. To extend the concept of link betweenness, we introduce the second concept, link tension. Link tension provides the information related to the ability of the link to handle transmission of data and shows us the links that are important in a network.