Peer$-$to$-$peer networks tend to grow through the arrival of new participants who join by linking into the existing

structure. This growth process has an intrinsic effect on the characteristics of the overall network. Recently,

people have investigated simple abstract models for network growth that might provide insight into the way

such processes behave in real networks at a qualitative level.

Here is a simple example of such a model. The system begins with a single node v$1.$ Nodes then join one at a

time; as each node joins, it executed a protocol whereby it forms a directed link to a single other node chosen

uniformly at random form those already in the system. More concretely, if the system already contains nodes

v$1,...,$ vk$1$ and node vk wishes to join, it randomly selects one of v$1,...,$ vk$1$ and links to that node.