Milan VojnovićMicrosoft Research

Joint work with Moez Draief, Kyomin Jung, Bo Young Kim, Etienne Perron and Dinkar Vasudevan

1 Consensus

Lecture series ACiD – Algorithms and Complexity in Durham, 2012

Abstract

In this talk, I will consider the problem of distributed ranking of alternatives in a network of nodes under limited memory per node and limited information communicated between nodes. In particular, for the case of ranking of two alternatives, each node in the network is assumed to prefer one of the alternatives, and the goal for each node is to correctly identify one of the two alternatives that is preferred by majority of the nodes. This type of a problem has been studied under various names such as consensus, k-selection and quantile computation. The model is an abstraction that underlies various systems such as ranking of items in distributed peer-to-peer systems, databases and may also capture dynamics of opinion formation in social networks.

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This Talk Based on

M. Draief and M. V., Convergence Speed of Binary Interval Consensus, SIAM Journal on Control and Optimization, 2012

K. Jung, B. Y. Kim, and M. V., Distributed Ranking in Networks with Limited Memory and Communication, IEEE Int’l Symposium on Information Theory, 2012

E. Perron, D. Vasudevan, and M. V., Using Three States for Binary Consensus on Complete Graphs, IEEE Infocom 2009

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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

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Consensus Problem (Cont’d)

1

1

1

11

1

11

1

Correct decision

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Consensus Problem (Cont’d)

0

0

0

00

0

00

0

Incorrect decision

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Applications

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0

0

1

1

1

10

0

Ex. Opinion formation in social networks

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Applications (Cont’d)

01101

Ex. Distributed databases Top-k query processing

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

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Notation

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11

1

10

0

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Notation (Cont’d)

1

01

0

0

0

0

1

1

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Questions of Interest

Correctness: probability that each node identifies the initial majority alternative ?

Convergence time: time to reach consensus ?

Dependence on the number of nodes n and initial fraction of nodes (voting margin) holding the majority state ?

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Desiderata

Reach correct consensus – initial majority

Fast convergence

Small communication overhead

Small processing per node

Decentralized

12

Outline

Related Work

Ternary Protocol

Quaternary Protocol

Conclusion

13

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

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m-ary Hypothesis Testing

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

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

15

000110111110100011

Hi

i. i. d. mean S

00 a 1ma1a

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

Outline

Related Work

Ternary Protocol

Quaternary Protocol

Conclusion

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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

17

e

1

1

e

Binary Signalling

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

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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

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US navy ship stems into port where Russian...

Soldier forced to sleep in car after hotel...

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

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Summary of Results

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

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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

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Proof Outline

First-step analysis:

with

Boundary conditions:

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

VUna

0for 10for ,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 ,

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

25

VUf ,

}0{

}0{

12

1:),1(

12

1:)1,(

),(

U

V

VU

VUVU

i.e. is error probability of

Proof Outline (Cont’d)26

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

Ob. Exponential decay for large

nDfn VU large ),||(~)log(1

21

,

1 1/2 ),,1(/))0(),0(( nVU

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Convergence Time Lower Bound Lower bound:

Example: pathreduction to classical voter model

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

U V

ConvergenceTime Lower … (cont’d) Ternary protocol on a path

corresponds to a classical voter model dynamics

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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

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

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Binary Protocol – Dynamics

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(

),(

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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

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Corollary – for large n

But …

Theorem –

– Not worse than classical voter model

Probability of Error (Cont’d)

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

UVn

Uf VU for ,,

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Probability of Error – Exponentially Bounded ?

Suggested by numerical results

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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

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Convergence Time

Theorem – Convergence time:

= constants independent on

Slower than ternary signaling by at least factor 2

Not slower than factor 3

nBnntAn largefor ,)log(3)()log(2

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Proof Basic Steps

in this set in a finite time independent of

Asserted bounds follow by ODE comparisons

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Convergence Time (Cont’d)

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Extension to Plurality Protocol

alternatives Binary consensus as a special case:

Goal: each node to correctly identify an alternative that is initially a plurality winner

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Plurality Protocol

For each alternative two states: strong and weak

At each communication instance between two nodes: If the observer node is in strong state j and

the contacted node is in a different strong state, then the observer node switches to weak state j

If the observer node is in weak state j, it switches to the state of the contacted node

bits of memory per node and communication between nodes

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Plurality Protocol (cont’d)

m alternatives

2m states: weak strong

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1 2 m…

s s

s’

s

s’

s

s’

s

s’

s’

s’

s

s’

s’

observer

Dynamics

Markov process:

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Dynamics: The Limit ODE

For every and

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Convergence Time44

The ODE Dynamics45

The ODE Dynamics (cont’d)46

The ODE Dynamics (cont’d)

Exponential diminishing of non-plurality states

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Convergence Time Upper Bound

Linear in the number of alternatives Logarithmic in the voting margin

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Convergence Time Lower Bound

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Convergence Time Lower … (cont’d)

Take, for example:

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Correctness

Let the fraction of non-plurality nodes be:

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Ternary Protocol Can Fail52

0

1

1

0

0

0

e

0 1

e

10

e

0

1

0

Complete graph with asymmetric communication rates

Two node types:Light – small interaction rate

Heavy– large interaction rate

Q: Can initial minority prevail ?

Initial Minority Can Prevail53

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)

Outline

Related Work

Ternary Protocol

Quaternary Protocol

Conclusion

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Quaternary Protocol55

Four states

Update rules: swap or annihilate

0 1e0 e1

e0

0

e0

0

e1

0

e0

0

0 1

e0

e1

e0

e1

e0

e1

e0

e1

1

1e1

1

e1

1

Convergence

For any given connected graph, the binary interval consensus converges to the correct state with probability 1. [Benezit et al, 2010]

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Convergence (cont’d)

Each edge activated at instances of a Poisson point process of intensity

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Let for every nonempty set of nodes matrix :

Convergence Time58

Phase 1 Dynamics

Phase 1

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1 if node i in state 1 1 if node i in state 0

Phase 1 dynamics (cont’d)60

Dynamics:

Sk = set of nodes in state 0

• The result follows by using a “spectral bound” on the expected number of nodes in state 1

Complete Graph61

Each edge activated at rate 1/(n-1)

• Inversely proportional to the voting margin• Can be made arbitrarily large ! 61

Complete Graph (cont’d)62

• The general bound is tight

• 0 and 1 state nodes annihilate after a random time that has exponential distribution with parameter cut

Star-shaped Graphs63

Each edge activated at rate 1/(n-1)

Star-shaped Graphs (cont’d)64

By first step analysis:

Same scaling, different constant

Erdos-Renyi Graphs65

Each edge activated at where

Erdos-Renyi Grahps (cont’d)66

For sufficiently large expected degree, the bound is approximately as for the complete graph as intuition would suggest

Outline

Related Work

Ternary Protocol

Quaternary Protocol

Conclusion

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Conclusion

The ternary protocol has appealing properties for complete graphs:

Exponentially decreasing probability of error with Logarithmic convergence time in

The quaternary protocol features: Guarantees convergence to the correct state with

probability 1 Provided a tight bound on the expected

convergence time Instantiated to particular graphs including

complete graph, path, cycle, star-shaped and Erdos-Renyi

Critical parameters: the number of nodes and voting margin

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Open Problems

State of the Art: consider an algorithm and then analyze the probability of error and convergence time

Suggests a trade-off between accuracy and speed

Q: upper and lower bounds for the expected convergence time, some classes of input graphs, subject to a bound on the probability of error ?

Q: accuracy and speed vs. memory and communication constraints ?

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Some References

S. Shang, P. W. Cuff, S. R. Kulkarni and P. Hui, An Upper Bound on the Convergence Time for Distributed Binary Consensus, 15th Int’l Conf. on Information Fusion, 2012

M. A. Abdullah and M. Draief, Majority Consensus on Random Graphs of a Given Degree Sequence, ArXiv, 2012

E. Mossel, J. Neeman and O. Tamuz, Majority Dynamics and Aggregation of Information in Social Networks, 2012

F. Chierichetti and J. Kleinberg, Voting with Limited Information and Many Alternatives, ACM SODA 2012

F. Benezit, P. Thiran and M. Vetterli, The Distributed Multiple Voting Problem, IEEE Journal on Selected Topics in Signal Processing, Vol 5, No. 4, 2011

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Some References (cont’d)

J. Cruise and A. Ganesh, Probabilistic Consensus via Polling and Majority Rules, Proc. of Allerton Conference, 2010

D. Acemoglu, M. A. Dahleh, I. Lobel and A. Ozdaglar, Bayesian Learning in Social Networks, forthcoming Review of Economic Studies, 2011

F. Benzit, P. Thiran and M. Vetterli, Interval Consensus: From Quantized Gossip to Voting, IEEE Int’l Conf. on Acoustics, Speech, and Signal Processing, 2009

A. Nedic, A. Olshevsky, A. Ozdaglar and J. N. Tsitsiklis, Distributed Averaging Algorithms and Quantization Effects, IEEE Conf. on Decision and Control, 2008

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Some References (cont’d)

W. P. Tay, J. N. Tsitsiklis and M. Z. Win, On the Subexponential Decay of Detection Error Probabilities in Long Tandems, IEEE Trans. on Info. The., Vol 54, No 10, 2008

F. Kuhn, T. Locher, R. Wattenhofer, Tight Bounds for Distributed Selection, ACM SPAA 2007

S. Boyd, A. Ghosh, B. Prabhakar and D. Shah, Randomized gossip algorithms, IEEE Trans. on Information Theory, Vol 52, No 6, 2006

T. M. Liggett, Interacting Particle Systems, Springer, 2006

M. Greenwald and S. Khanna, Power-conserving Computation of Order-Statistics over Sensor Networks, ACM PODS 2004

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Some References (cont’d)

D. Kempe, J. Kleinberg and E. Tardos, Maximizing Influence through a Social Network, ACM KDD 2003

Y. Hassin and D. Peleg, Distributed Probabilistic Polling and Applications to Proportionate Agreement, Information and Computation, 171, 2001

M. Greenwald and S. Khanna, Space-efficient Online Computation of Quantile Summaries, ACM SIGMOD 2001

J. Koplowitz, Necessary and Sufficient Memory Size for m-hypothesis Testing, IEEE Trans. on Information Theory, Vol 21, No 1, 1975

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