  In this paper, the topology dynamic of Gnutella is studied through phase
space. The dynamic changes in peer degree are studied as a time series in two
dimensional phase space which is defined as the number of connected leaves and
the number of connected ultras. The reported degrees are concentrated in three
special Software related regions that we named as Ultra Stable Region, Leaf
Stable Region and Transition Belt. A method is proposed to classify degree
traces in phase space into different classes. Connection churn then is studied
along with the churn in degree. It shows that the topological structure of
Gnutella is rather stable in its connection degree but not the topology itself.
The connection drop rate is estimated and the live time of connections is
inferred afterwards. M/M/m/m loss queue system is introduced to model the
degree keeping process in Gnutella. This model revealed that the degree stable
is ensured by large new connection efforts. In other words the stable in
topological structure of Gnutella is a results of essential unstable in its
topology. That opens a challenge to the basic design philosophy of this
network.
