Random Graph Theory and P2P

Transkript

1 Graph Theory and PP GI Arbeitsgespräch 00 September th 00, Darmstadt

2 Impacts of PP.00E+.80E+ on PP Traffic per week in byte.60e+.0e+.0e+.00e+ 8.00E E+0.00E+0.00E E Date (yyyymmdd) Data_Transfers Unidentified File_Sharing Only Signaling!! Data source:

3 The Gnutella Network on PP Measurements taken at the LKN in May 00

4 PP-Networks as Graph on PP Properties of PP networks: Degree varies between and 50 (Ultrapeers) Connections are established randomly, without any restrictions to physical topology Goal Establish an analytical model for PP networks Compute number of neighbors according to hop distance Analyze signaling loads Analyze availability Compare the performance of different PP networks

5 Graph A random graph is a collection of points or vertices With lines or edges connecting pairs of them at random Each edge has at least degree on PP 5

6 Approach based on Generating Functions (I) Generating Functions Probability distribution of vertex degree k: p k on PP Generating Function: First Moment: k 0( ) k 0 k = 0 G x = p x, G () = dg ( x) 0 k = kipk = = G0 k = 0 dx x= '() First Moment = Average degree of a vertex = number of first hop neighbors = k = G ' 0 ( ) 6

7 Approach based on Generating Functions (II) on PP Probability distribution of vertex degree of the first hop neighbors: Probability distribution of vertex degree of the nd hop neighbors: Average number of second nearest neighbors: G ( x) k kipk ix G k = 0 = = G kip k = 0 i k i 0 k = 0 k 0 0 ' ( x) () ' ( ) k ( ) = ( ) k p G x G G x d z = G G x ( ( )) 0 dx x= Probability distribution of vertex degree of the nd hop neighbors: ( ( ( ))) G G G x 0 7

8 Measurements on PP M.A. Jovanovic, F.S. Annexstein and K.A. Berman, "Scalability issues in large peer to peer networks -- A case study of Gnutella", Tech. Report, Univ. of Cincinati, 00. Number of Servents/Pings ,99 direct connections,9 direct connections Theory (average Nr. of Con. with,99 direct) Theory (average Nr. of Con. with,9 direct) Theory:,9 conn. Theory:,99 conn Hop-Count Measurements taken at LKN, 00.,9 conn.,99 conn. Power Law with τ =. Measured Growth is small 8

9 Number of reachable nodes on PP Nr. of reachable nodes Nr. of reachable nodes Content availability Content availability Replication rate: Power Law with τ =. z =.0 Exponential Distribution with κ=. z=.0

10 Basis for the Data Rates on PP TX PING() and QUERY() messages initialized by one node PING() and QUERY() forwarded on all but one of its virtual vertices PONG() and QUERY_HIT() forwarded on one of its virtual vertices. QUERY_HIT()-messages, which has to be sent on one of its virtual vertices. PONG()-message as an answer to a received PING()-message, which has to be sent on one of its virtual vertices RX PING() and QUERY() messages initialized by other nodes PONG() and QUERY_HIT() received on one of its virtual vertices. PONG() and QUERY_HIT()-messages, as a response to a previously initialized PING() or QUERY()- message. 0

11 Data rates TX in kbit/s RX in kbit/s on PP TX in kbit/s RX in kbit/s Power Law with τ =. z =.0 Exponential Distribution with κ=. z=.0

12 Powerlaw with cut off on PP Probability Power Law with τ =. z=. d_max=7 degree Nr. of reachable nodes TX in kbit/s RX in kbit/s

13 on PP PP-networks can be described with analytic models Evaluate the traffic Evaluate the availability Evaluate the reach Gnutella 0. with powerlaw (τ =., d_max=7) Other networks, e.g. Gnutella 0.6 with other parameters and further probabilities

14 References B. Bollobàs. Modern Graph Theory. Springer, 998 B. Bollobàs.. Academic Press. 985 P. Erdös, A. Rényi. On the Eveloution of random graphs. Publ. Math. Inst. Hungar. Acad. Sci., vol. 5, pp.7-6, 960. M. E. J. Newman, S. H. Strogatz,D. J. Watts. graphs with arbitrary degree distributions and their applications. Phys. Rev. E 6, 068, (00) H. S. Wilf. Generatingfunctionology. Academic Press, 99. A. Feldmann, A. C. Gilbert, W. Willinger, and T. G. Kurtz. The Changing Nature of Network Traffic: Scaling Phenomena. ACM Computer Communication Review, vol. 8, pp. 5-9, Apr E. Markatos. Tracing a large-scale Peer-to-Peer System: an hour in the lifetime of Gnutella. nd IEEE/ACM Int. Symp. On Cluster Computing and the Grid 00 B. Krishnamurthy, J. Wang, Y. Xie. Early Measurements of a Cluster-based Architecture for PP Systems. ACM SIGCOMM Internet Workshop 00. S. Saroui, P. Gummadi, S. Gribble. A Measurement Study of Peer-to-Peer File Sharing Systems. Technical Report #UW-CSE K. Anderson. Analysis of the Traffic on the Gnutella Network. CSE Project report. K. Kant. An Analytic Model for Peer to Peer File Sharing Networks. Proc. of International Communications Conference, May 00, Anchorage, AL R. Schollmeier, G. Kunzmann. GnuViz Mapping the Gnutella Network to its Geographical Locations. Special Issue on PP networking of the PIK Journal to appear in Spring 00. R. Schollmeier, A. Dumanois. Peer-to-Peer Traffic Characteristics. EUNICE 00. Budapest, Hungary. Sepetember 00. R. Schollmeier, F. Hermann. Topology-Analysis of Pure Peer-to-Peer Networks. In Proceedings of the Fachtagung "Kommunikation in Verteilten Systemen" (KiVS 00). Leipzig, Germany, February 6-8, 00 Gnutella Protocol Specification, version 0.; available at M.A. Jovanovic, F.S. Annexstein and K.A. Berman, "Scalability issues in large peer to peer networks - - A case study of Gnutella", Tech. Report, Univ. of Cincinati, 00.