Preferential Attachment: A great deal of excitement was generated when scientists began to study large complex networks (with hundreds of thousands of nodes) with the kind of computing power broadly available at the end of the 20th century. It immediately became clear that the large and complex data sets being generated in the biological sciences and being created by the rapid uptake of social media had complex and non-trivial structure; they were nothing like random networks or regular lattice networks. So much so that scientists like Alberto Barabasi - see also his impressive online lab - soon proposed a very different network generating model that produced networks with an entirely different structure and degree distribution. This new model included two new ingredients: time and the concept of preferential attachment. The results were revolutionary and prompted a urry of activity and the birth of Network Science as a recognised discipline of the mathematical sciences. Scientists proposed that real networks grow from small beginnings and that the C(N, p) model does not capture this; in reality nodes are added to a network over time. Scientists further postulated that when a new node is added then it forms links with existing nodes with a preference for nodes that already have a high degree. This is called preferential attachment. We can generate such a network using the following steps: 1. Begin with a set of 4 nodes that are a clique, i.e. where every node is joined by an edge to every other node. 2. Add a new node to the network and have it form 1 new random edge with one of the existing nodes1 but with preferential attachment such that Pr(new node links with node 1') oc kg , (1) where ha- is the current degree of node 1'. 3. Repeat Step 2 until the new node has 3 new edges but avoid repeated edges by removing from the calculations any node{s) already connected to the new node. This means that the probabilities in Step 2 need to be recalculated each time. Question 1. Normalise equation 1 (find a denominator) so that the probability a new node links with an existing node i obeys the unbreakable laws of probability