3Introduction Peer to peer network Unstructured network Gnutella, NapsterStructured networkDHT-based systemssuch as Pastry, Chord, Tepastry, CANAdvantages of DHT-based systemsFast: O (log n)Can exactly find a publishing object in a gigantic network space
4Gossip-based Algorithm But if we want to get the aggregation information for the whole networkSuch as sum value, average valueOur objective is to calculate the average value of Xavg =(x1+x2+x3…+x12)/12Disadvantage of DHT-based systemsGossip-based algorithmObjective: let the estimation average value close to Xavg for every nodeX2X3X1X11X4X10X12X5X9X6X8X7
5Gossip-based Algorithm Xavg = (X1+X2+X3+X4)/4 is a real average value in a peer to peer networkXeavg is the estimated average value for the P2P network in a node(X4+x2)/2time=0,Xeavg1=X1, Xeavg2=x2, Xeavg3=x3, Xeavg4=x4Time=1, Randomly pick up another nodeXeavg1=X1/ 2, Xeavg2=(X4+x2)/ 2 Xeavg3=(X2+X3)/ 2 Xeavg4= (X1+X3+X4)/ 2X1X2/2(X1+x1+x3+x4)/4X1/2X2/2X2X1/2X1/2(X2+x2+x3+x4)/4X3X4/2(X2+x3)/2X3/2X4X4/2(X1+x3+x4)/4(X1+x3+x4)/2X3/2(X2+x2+x3+x4)/4Time = 2,Xeavg1=(X1+X1+X3+X4)/ 4, Xeavg2=(X2+X2+X3+X4)/4, Xeavg3=(X2+X2+X3+X4)/ 4, Xeavg4=(X1+X3+X4)/ 4,
6Gossip-based Algorithm After m rounds/iterations, Xeavg is very close to XavgWe can see Xeavg as Xavg
7Define a variance error= | Xeavg-Xavg | Converge SpeedDefine a variance error= | Xeavg-Xavg |Our objective is to make the variance close to 0Calculate the converge speed of this varianceIn every round, the variance drops to less than half its previous valuevar(t+1) = ( ) var(t)XeavgXavg
8Analyze Gossip-based Algorithm Gossip-based algorithm is an approximation methodWe can control the accuracyXeavg never = Xavg, but Xeavg can be very close to XavgWhen variance error=| Xeavg – Xavg| <= ε, we can say Xeavg is Xavg.
9Analyze Gossip-based Algorithm Roughly say, after O(logn+log(1/ ε)) rounds, can we say variance error <= ε in every nodeMaybe there are broken network connections
10Analyze Gossip-based Algorithm We have to control the percentage of nodes who obtain err<=εWe say with probability at least 1-δ,after O(logn+log(1/ε)+log(1/δ)) rounds,The err=|Xeavg – Xavg| <= εTheir contribution:The diffusion speed of uniform gossip is O(logn+log(1/ε)+log(1/δ)) , with probability at least 1- δ, and variance error <= ε
11Advantages of Gossip Algorithm Algorithm is very simpleConverge speed is very fastCan automatically adjust itselfNodes join the networkNodes leave the network
12Disadvantages of Gossip Algorithm From their theory, we know after O(logn+log(1/ε)+ log(1/δ)) rounds,the estimation average value in a local node can be see as a global average value.But in practice, If we do not know the size of the network, how do we know how many rounds a estimation average value is close enough to the real average value.