Price's model

Price introduced this directed network model in $1976,$ based on the concept of preferential attachment.

The model starts by assuming a graph that has an average node out$-$degree of $.$ During the generation of subsequent nodes, will be kept constant. Each subsequent node is generated one at a time using the following process.

A new node is inserted into the graph. This node is connected to$,$ on average, already existing nodes $($the exact mechanism for choosing the out$-$degree is not important as long as the average is maintained$).$ The target node for each one of these connections is selected in proportion to the in$-$degree of that node. Because new nodes will have an in$-$degree of zero, this needs to be modified to $$in proportion to the in$-$degree plus $"$ to ensure that new nodes can be connected to subsequent nodes. Therefore, if the in$-$degree of node is $,$ then node is selected as the target for each connection with probability

This is called a $$preferential attachment" model, as new nodes are preferentially attached to already well connected nodes. The result is a kind of $$rich get richer" process, and, as the number of new nodes overwhelms the existing nodes in the graph, the node in$-$degree distribution starts to follow a power$-$law: