Milan VojnovićMicrosoft Research Cambridge

Collaborators: E. Perron and D. Vasudevan

1 Consensus – with Limited Processing and Signalling

This Talk Based on

MSR Technical Report – MSR-TR-2008-114 – Aug 2008

2

Binary Consensus Problem

0

1

0

11

1

10

0

Goal: each node wants to correctly decide whether 0 or 1 was initially held by majority of nodes

3

Consensus Problem (Cont’d)

1

1

1

11

1

11

1

Correct decision

4

Consensus Problem (Cont’d)

0

0

0

00

0

00

0

Incorrect decision

5

Applications

0

0

0

1

1

1

10

0

Ex. Opinion formation in social networks

6

Applications (Cont’d)

01101

Ex. Distributed databases Top-k query processing

Query: Is object X most preferred by majority of nodes?

7

Notation

0

0

0

11

1

10

0

8

Notation (Cont’d)

1

01

0

0

0

0

1

1

9

System’s Desiderata

Reach correct consensus – initial majority

Fast convergence

Small communication overhead

Small processing per node

Decentralized

10

Related Work – Classical Voter Model

Node takes over the state of the contacted node

Binary state per node & binary signaling

0 initially held by V nodes,1 initially held by U nodes

Complete graph node interactionsProbability of incorrect consensus

UVVU

Uf VU

for ,,

1

0

0

0

1

0

1

1

11

Related Work – m-ary Hypothesis Testing

Q: How much state does S need to decide correct hypothesis with probability going to 1 with the number of observations ?

1,,0 ),,[ : 1 miaaH iii

12

000110111110100011

Hi

i. i. d. mean S

00 a 1ma1a

A: m+1 necessary and sufficient (Koplowitz, IEEE Trans IT ’75)

Ternary Protocol

Both processing and signaling take one of three states 0 or 1 or e e = “indecisive” state

1

0

e

0

0

0

e

0

e

1

1

1

13

Binary Protocol

Processing same as for ternary protocol Binary signaling – takes one of two states 0 or 1

e e

signals 0 or 1 with equal probability

14

Binary Signaling – A Motivation

Nodes may not be able to signal indifference – by the very nature of the application

Ex. two news pieces may be equally most read but only one can be recommended to the user

15

US navy ship stems into port where Russian...

US navy ship stems into port where Russian...

Soldier forced to sleep in car after hotel...

Questions of Interest

Probability of convergence to incorrect consensus ?

Time to reach consensus ?

Dependence on the number of nodes N and initial fraction of nodes holding the majority state ?

16

This Talk Assumptions

Complete graph node interactions Each node samples a node uniformly at random

across all nodes at instances of a Poisson process with intensity 1

Arbitrary graph interactions of interest – for future work

17

Summary of Results – Talk Outline

Ternary protocol Prob of error decays exponentially with the

number of nodes N – found exact exponent log(N) convergence time

Binary protocol Prob of error worse than for ternary protocol

for a factor exponentially increasing with N, but not worse than for classical voter

Convergence time C log(N) with 2 C 3

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Ternary Protocol - Dynamics

U = number of nodes in state 0 V = number of nodes in state 1 N = total number of nodes

19

N

UVVU

N

VVUNVU

N

VUVU

N

UVUNVU

VU

: )1,(

)(: )1,(

: ),1(

)(: ),1(

),(

(U,V) Markov process:

Ternary Protocol - Probability of Error

Theorem – probability of error:

U

jjVjU

VUVU

jaf

1)()(

,, 2

)(

2

1

jU

jVjU

jVjU

UVja VU

)()(

)()()(,

(U, V) = initial point, V > U

20

Proof Outline

First-step analysis:

with

Boundary conditions:

1,1,,1,1,)2( VUVUVUVUVU UVfaVfUVfaUffUVaVaU

VUNa

0 for 10 for ,0 0,,0 U, fVf UV

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Proof Outline (Cont’d)

Lemma – solution of

Boundary conditions:

VUVUVU fff ,11,, 2

1

2

1

VUf ,

0 for 10 for ,0 0,,0 U, fVf UV

22

VUf ,

}0{

}0{

12

1:),1(

12

1:)1,(

),(

U

V

VU

VUVU

i.e. is error probability of

Proof Outline (Cont’d)23

U

VfU,U = 1/2

(U, V)

(j, j)

U

jjjVjUVU nf

1)()(, 2

1

Number of pathsfrom (U, V) to (j, j) that do not intersect the line U = V-- Ballot theorem

Probability of Error (Cont’d)

Corollary – For

H() = entropy of a Bernoulli random variable with mean

Ob. Exponential decay for large N.

NHfN VU large )],(1[~)log(1

,

1 1/2 ),,1(/))0(),0(( NVU

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Convergence Time

Initial state:

Limit ODE:

Time:

)0()0(

))0()0((log

)()(

))()((log

33

vu

uv

tvtu

tutvt

))(2)(1)(()(

))(2)(1)(()(

tutvtvtvdt

d

tvtututudt

d

))0(),0((/))0(),0(( vuNVU

25

Convergence Time (Cont’d)

NNNt large ),log(~)(

26

Time it takes for (u(t), v(t)) to go from (u(0), v(0)) to (u(t), v(t)) such that 1-v(t) is of order 1/N

Binary Protocol – Reminder

Processing same as for ternary protocol Binary signaling – takes one of two states 0 or 1

e e

signals 0 or 1 with equal probability

27

Binary Protocol – Dynamics

(U,V) Markov process:

N

VUVVU

N

UVVUNVU

N

UVUVU

N

VUVUNVU

VU

12

1: )1,(

1)(2

1: )1,(

12

1: ),1(

1)(2

1: ),1(

),(

28

Probability of Error – Binary Signaling

Theorem –

where

UVVU pf ,

12

12

!

!2

1N

Ni

i

N

UVNi

i

UV

iN

iN

p

))]2log(1(21[~)log(1 UVpN

29

Corollary – for large N

Probability of Error (Cont’d)

Ob. Worse than under ternary protocol for a factor exponentially increasing with N

UVN

Uf VU for ,,

30

But …

Theorem –

– Not worse than classical voter model

Probability of Error – Exponentially Bounded ?

Suggested by numerical results

31

Binary Protocol – Many-Nodes Limit

The limit ODE:

For z = u + v and w = v – u, we have

)]())(1())(1[()(

)]())(1())(1[()(

2

2

tvtututvdt

d

tutvtvtudt

d

)())(1(2

1)(

)(2

1)(

2

31)( 2

twtztwdt

d

twtztzdt

d

32

Convergence Time

Theorem – Convergence time:

A, B = constants independent on N

- Slower than ternary signaling by at least factor 2

- Not slower than factor 3

NBNNtAN large for ,)log(3)()log(2

33

Proof Basic Steps

(u(t),v(t)) in this set in a finite time independent of N

Asserted bounds follow by ODE comparisons

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Convergence Time (Cont’d)

(u(0), v(0)) = (0.3, 0.7)

35

Conclusion

“Good news” results for binary consensus on complete graphs

Ternary signaling Probability of error decays exponentially with

the number of nodes N log(N) convergence time

Binary signaling Probability of error worse than for the ternary

signaling for a factor exponentially increasing with N, but not worse than for classical voter

Convergence time C log(N) with 2 C 3

36

Future work

Arbitrary graphs ?

Top k ?

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Arbitrary graphs

There exist graphs for which ternary protocol provides no benefits over classical voter

Ex. path with initial state:

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1 01 1 1 0 0 0. . . . . .

U V

Path

Path graph evolves essentially as under voter model

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01 1 1 0 0 0

01 1 0 0 0e

01 1 0 0 00

1/2

1/2

1/2

Heterogeneous Rates of Interactions

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0

1

1

0

0

0

e

0 1

e

10

e

0

1

0

Still complete graph interactions

Two node types:

Light – small interaction rate

Heavy– large interaction rate

Q: Can initial minority prevail ?

Can Initial Minority Prevail ? – Yes.

41

Example: Node types

0.2 light 0.8 heavy

Interaction rates0.1 light2 heavy

U V

Light 0.1 0.05

Heavy 0.35 0.45

0.45 0.5

V state nodes(initial majority)