Attachment Mechanisms in Catalogue Networks

A network is a configuration of agents and the relationships between the agents. Agents are represented by nodes or vertices with a relationship between two objects represented by an edge linking the two. Two nodes are said to be connected if an edge exists linking the nodes. Edges can be directed, meaning they represent a relationship in one direction only between nodes, or can be weighted, meaning the edge has a value, or weight, associated with it. In a network with weighted edges the weight of the edge is considered rather than just whether it exists. The degree of a node in an unweighted network is the number of edges with one end attached to the node. Figure 1 shows an example of a basic undirected network, other examples of networks include the world wide web, social networks and food webs. Often a network cannot be considered to be fixed, and it’s evolution and growth needs to be considered. One important tool for doing this is preferential attachment, the first of application of this in a growing network is thought to have been Price[1] in 1976 although it has also been considered previously by Simon[2] in 1955 and Yule[3] in 1925. The term preferential attachement was coined by Barabasi and Albert[4] in their 1999 paper on the growth of the world wide web. In a simple application of preferential attachment, a new node with one edge is added to the network, connected to an existing node with probabilities proportional to the degree of the node. For a network with N nodes the probability of choosing node ni with degree ki is