Gossip-Based Computation of Aggregation Information

1. Gossip-Based Computation of Aggregation Information David Kempe Alin Dobra Johannes Gehrke Presented by Hao Zhou

2. Content Introduction Gossip-based Algorithm Analyze Gossip-based Algorithm

3. Introduction Peer to peer network Unstructured network Gnutella, Napster Structured network DHT-based systems such as Pastry, Chord, Tepastry, CAN Advantages of DHT-based systems Fast: O (log n) Can exactly find a publishing object in a gigantic network space

4. Gossip-based Algorithm • But if we want to get the aggregation information for the whole network • Such as sum value, average value • Our objective is to calculatethe average value of Xavg =(x1+x2+x3…+x12)/12 • Disadvantage of DHT-based systems • Gossip-based algorithm • Objective: let the estimation average value close to Xavg for every node X2 X3 X1 X11 X4 X10 X12 X5 X9 X6 X8 X7

5. Gossip-based Algorithm • Xavg = (X1+X2+X3+X4)/4 is a real average value in a peer to peer network • Xeavg is the estimated average value for the P2P network in a node (X4+x2)/2 • time=0, • Xeavg1=X1, Xeavg2=x2, Xeavg3=x3, Xeavg4=x4 • Time=1, Randomly pick up another node • Xeavg1=X1/ 2, Xeavg2=(X4+x2)/ 2 Xeavg3=(X2+X3)/ 2 Xeavg4= (X1+X3+X4)/ 2 X1 X2/2 (X1+x1+x3+x4)/4 X1/2 X2/2 X2 X1/2 X1/2 (X2+x2+x3+x4)/4 X3 X4/2 (X2+x3)/2 X3/2 X4 X4/2 (X1+x3+x4)/4 (X1+x3+x4)/2 X3/2 (X2+x2+x3+x4)/4 • Time = 2, • Xeavg1=(X1+X1+X3+X4)/ 4, Xeavg2=(X2+X2+X3+X4)/4, Xeavg3=(X2+X2+X3+X4)/ 4, Xeavg4=(X1+X3+X4)/ 4,

6. Gossip-based Algorithm • After m rounds/iterations, Xeavg is very close to Xavg • We can see Xeavg as Xavg

7. Converge Speed Define a variance error= | Xeavg-Xavg | Our objective is to make the variance close to 0 Calculate the converge speed of this variance In every round, the variance drops to less than half its previous value var(t+1) = ( ) var(t) Xeavg Xavg

8. Analyze Gossip-based Algorithm Gossip-based algorithm is an approximation method We can control the accuracy Xeavg never = Xavg, but Xeavg can be very close to Xavg When variance error=| Xeavg – Xavg| <= ε, we can say Xeavg is Xavg.

9. Analyze Gossip-based Algorithm Roughly say, after O(logn+log(1/ ε)) rounds, can we say variance error <= ε in every node Maybe there are broken network connections

10. Analyze Gossip-based Algorithm We have to control the percentage of nodes who obtain err<=ε We say with probability at least 1-δ, after O(logn+log(1/ε)+log(1/δ)) rounds, The err=|Xeavg – Xavg| <= ε Their 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

11. Advantages of Gossip Algorithm Algorithm is very simple Converge speed is very fast Can automatically adjust itself Nodes join the network Nodes leave the network

12. Disadvantages 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.

13. Thank you !