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**Gossip-Based Computation of Aggregation Information**

David Kempe Alin Dobra Johannes Gehrke Presented by Hao Zhou

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**Introduction Gossip-based Algorithm Analyze Gossip-based Algorithm**

Content Introduction Gossip-based Algorithm Analyze Gossip-based Algorithm

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**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

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**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

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**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,

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**Gossip-based Algorithm**

After m rounds/iterations, Xeavg is very close to Xavg We can see Xeavg as Xavg

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**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

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**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.

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**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

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**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

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**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

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**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.

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Thank you !

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