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

2. Introduction Gossip-based Algorithm Analyze Gossip-based Algorithm
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 calculate the 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. Define a variance error= | Xeavg-Xavg |
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 !