On Gnutella topology dynamics by studying leaf and ultra connection jointly in phase space

In this paper, the topology dynamics of Gnutella are studied in phase space. The dynamic progress of peer degree is studied as a time series in two dimensional phase space, which is consisted of numbers of connected leaves and ultras. The reported degrees concentrate on three special software related regions which we name as ultra stable region, leaf stable region and transition belt. A method is proposed on how to classify degree traces in phase space into different categories. Then the connection churn and the degree churn are studied. It shows that the topological structure of Gnutella is more stable in its connection degree than in the topology itself. The connection drop rate is estimated and the lifetime of connections is deduced afterwards. M/M/m/m loss queue system is introduced to model the degree keeping process in Gnutella. This model reveals that the degree stability is ensured by mass new connection efforts. In other words, the stability in topological structure of Gnutella is the results of many essential unstable factors in its topology. We think it raises a challenge to the basic design philosophy for such networks.