Graph Theory and PP GI Arbeitsgespräch 00 September th 00, Darmstadt {Schollmeier}@lkn.ei.tum.de
Impacts of PP.00E+.80E+ on PP Traffic per week in byte.60e+.0e+.0e+.00e+ 8.00E+0 6.00E+0.00E+0.00E+0 0.00E+00 0008 00060 00060 000505 00007 0000 0000 00006 0009 Date (yyyymmdd) 00 000 00096 00089 0007 0006 00057 0009 0000 0000 Data_Transfers Unidentified File_Sharing Only Signaling!! Data source: http://netflow.internet.edu/weekly/
The Gnutella Network on PP Measurements taken at the LKN in May 00
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
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
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
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
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 5000 500 000 500 000 500 000 500 000 500 0,99 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. 5 6 7 Hop-Count Measurements taken at LKN, 00.,9 conn.,99 conn. Power Law with τ =. Measured Growth is small 8
Number of reachable nodes on PP Nr. of reachable nodes Nr. of reachable nodes Content availability Content availability Replication rate: 0.0055 9 Power Law with τ =. z =.0 Exponential Distribution with κ=. z=.0
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
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
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
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
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