noisy rumor spreading and plurality consensus · 35/29 ≈ 1.2 trivial strategy. breathe before...
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Noisy Rumor Spreading and Plurality Consensus
Emanuele Natale†
joint work withPierre Fraigniaud∗
ACM Symposium onPrinciples of Distributed Computing
July 25-29, 2016Chicago, Illinois
†∗
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Rumor-Spreading Problem
FOOD!
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Rumor-Spreading Problem
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Plurality Consensus Problem
This way! This way!
THIS WAY!
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Plurality Consensus Problem
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Some examples (Plurality Consensus)Flocks of birds [Ben-Shahar et al. ’10]
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Some examples (Plurality Consensus)
Schools of fish [Sumpter et al. ’08]
Flocks of birds [Ben-Shahar et al. ’10]
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Some examples (Plurality Consensus)
Schools of fish [Sumpter et al. ’08]
Insects colonies [Franks et al. ’02]
Flocks of birds [Ben-Shahar et al. ’10]
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Some examples (Plurality Consensus)
Schools of fish [Sumpter et al. ’08]
Eukaryotic cells [Cardelli et al. ’12]
Insects colonies [Franks et al. ’02]
Flocks of birds [Ben-Shahar et al. ’10]
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Animal Communication Despite Noise
Noise affects animal communication,but animals cannot use coding theory...
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Animal Communication Despite Noise
O. Feinerman, B. Haeupler and A. Korman.Breathe before speaking: efficient informationdissemination despite noisy, limited and anonymouscommunication. (PODC ’14)
Noise affects animal communication,but animals cannot use coding theory...
Natural rules efficiently solve rumor spreading andplurality consensus despite noise.
=⇒
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Animal Communication Despite Noise
O. Feinerman, B. Haeupler and A. Korman.Breathe before speaking: efficient informationdissemination despite noisy, limited and anonymouscommunication. (PODC ’14)
Noise affects animal communication,but animals cannot use coding theory...
They only consider the binary-opinion case.Our contribution: generalize to many opinions.
Natural rules efficiently solve rumor spreading andplurality consensus despite noise.
=⇒
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Binary Case - Model
n agents. One agent has one bit to spread.
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Binary Case - Model
Communication model: PUSH model [Pittel ’87]:at each round each agent can send a bit to anotherone chosen uniformly at random.
????????
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Binary Case - Model
Communication model: PUSH model [Pittel ’87]:at each round each agent can send a bit to anotherone chosen uniformly at random.
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Binary Case - Model
Noise: before being received, each bit is flipped withprobability 1/2− ε (ε = n−const).
!
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Binary Case - Model
Noise: before being received, each bit is flipped withprobability 1/2− ε (ε = n−const).
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Binary Case - Model
Noise: before being received, each bit is flipped withprobability 1/2− ε (ε = n−const).
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Binary Case - Model
Noise: before being received, each bit is flipped withprobability 1/2− ε (ε = n−const).
!
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Breathe Before Speaking
blue vs red:1/0
trivialstrategy
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Breathe Before Speaking
blue vs red:2/0
trivialstrategy
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Breathe Before Speaking
blue vs red:3/1
trivialstrategy
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Breathe Before Speaking
blue vs red:9/6 = 1.5
trivialstrategy
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Breathe Before Speaking
blue vs red:18/13 ≈ 1.4
trivialstrategy
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Breathe Before Speaking
blue vs red:35/29 ≈ 1.2
trivialstrategy
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Breathe Before Speaking
blue vs red:35/29 ≈ 1.2
trivialstrategy
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Breathe Before Speaking
“[...] ants effectively self-restrict their own tendency to engage infurther interactions that would excite further nest-mates.”
(Razin et al. ’13)
Stage 1: Spreading
blue vs red:1/0
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Breathe Before Speaking
“[...] ants effectively self-restrict their own tendency to engage infurther interactions that would excite further nest-mates.”
(Razin et al. ’13)
Stage 1: Spreading
blue vs red:1/0
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Breathe Before Speaking
“[...] ants effectively self-restrict their own tendency to engage infurther interactions that would excite further nest-mates.”
(Razin et al. ’13)
Stage 1: Spreading
blue vs red:1/0
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Breathe Before Speaking
“[...] ants effectively self-restrict their own tendency to engage infurther interactions that would excite further nest-mates.”
(Razin et al. ’13)
Stage 1: Spreading
blue vs red:1/0
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Breathe Before Speaking
“[...] ants effectively self-restrict their own tendency to engage infurther interactions that would excite further nest-mates.”
(Razin et al. ’13)
Stage 1: Spreading
blue vs red:3/1
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Breathe Before Speaking
“[...] ants effectively self-restrict their own tendency to engage infurther interactions that would excite further nest-mates.”
(Razin et al. ’13)
Stage 1: Spreading
blue vs red:3/1
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Breathe Before Speaking
“[...] ants effectively self-restrict their own tendency to engage infurther interactions that would excite further nest-mates.”
(Razin et al. ’13)
Stage 1: Spreading
blue vs red:3/1
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Breathe Before Speaking
“[...] ants effectively self-restrict their own tendency to engage infurther interactions that would excite further nest-mates.”
(Razin et al. ’13)
Stage 1: Spreading
blue vs red:8/4
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Breathe Before Speaking
“[...] ants effectively self-restrict their own tendency to engage infurther interactions that would excite further nest-mates.”
(Razin et al. ’13)
Stage 1: Spreading
blue vs red:40/24 ≈ 1.7
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Breathe Before Speaking
Stage 1: Spreading
blue vs red:40/24 ≈ 1.7
Stage 2: Amplifying majority
# > # ?# < # ?
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Mathematical Challenges
• Stochastic Dependence
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Mathematical Challenges
• Stochastic Dependence
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Mathematical Challenges
• Stochastic Dependence
• “Small Deviations”
Pr(X ≥ t) ≤ f(t)
X
Pr(X)
0
Pr(X ≥ t) ≥ 12 + g(t)
g(t), f(t) << 1
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Multivalued Case
!
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Multivalued Case
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Multivalued Case
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Multivalued Case
!
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Multivalued Case
Noise Matrix:
∼P :=
p ,
p ,
p ,
p , p ,
p ,
p , p ,
p ,
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Multivalued Case
δ-majority-biased configuration w.r.t. :
Noise Matrix:
Configuration c := (# /n,# /n,# /n)
∼P :=
p ,
p ,
p ,
p , p ,
p ,
p , p ,
p ,
# /n−# /n > δ
# /n−# /n > δ
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Majority-Preserving Matrix
Randomsenderin conf. c
Noise actingaccording tomatrix P
Messagedistributedas c · P
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Majority-Preserving Matrix
(cP ) − (cP ) > εδ
(cP ) − (cP ) > εδ
(ε, δ)-majority-preserving noise matrix:
(cP ) − (cP ) > εδ
(cP ) − (cP ) > εδ
Randomsenderin conf. c
Noise actingaccording tomatrix P
Messagedistributedas c · P
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Main Result
Theorem. Let S be the initial set of agents withopinions in [k]. Suppose that S is δ =Ω(√
logn/|S|)-majority-biased with |S| = Ω( lognε2 )
and the noise matrix P is (ε, δ)-majority-preserving.Then the plurality consensus problem can be solvedin O( logn
ε2 ) rounds w.h.p., with O(log logn+ log 1ε )
memory per node.
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Main Result
Theorem. Let S be the initial set of agents withopinions in [k]. Suppose that S is δ =Ω(√
logn/|S|)-majority-biased with |S| = Ω( lognε2 )
and the noise matrix P is (ε, δ)-majority-preserving.Then the plurality consensus problem can be solvedin O( logn
ε2 ) rounds w.h.p., with O(log logn+ log 1ε )
memory per node.
|S| = 1 =⇒ rumor spreading in O( lognε2 ) rounds
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Main Result
Theorem. Let S be the initial set of agents withopinions in [k]. Suppose that S is δ =Ω(√
logn/|S|)-majority-biased with |S| = Ω( lognε2 )
and the noise matrix P is (ε, δ)-majority-preserving.Then the plurality consensus problem can be solvedin O( logn
ε2 ) rounds w.h.p., with O(log logn+ log 1ε )
memory per node.
P =(
1/2 + ε 1/2− ε1/2− ε 1/2 + ε
)=⇒ Feinerman et al.
|S| = 1 =⇒ rumor spreading in O( lognε2 ) rounds
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Poisson Approximation
?∼ c · P
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Poisson Approximation
?∼ c · P
? ∼Poisso
n((c ·P )i)
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Poisson ApproximationLemma. balls-in-bins experiment:- h colored balls are thrown in n bins, hi balls have color1 ≤ i ≤ k,- Xu,iu∈1,...,n,i∈1,...,k number of i-colored balls that end upin bin u,- f non-negative function with Z≥0 argumentsxu,iu∈1,...,n,i∈1,...,k and z,- Yu,iu∈1,...,n,i∈1,...,k independent r.v. withYu,i ∼Poisson(hi/n) and Z integer valued r.v. independent fromXu,is and Yu,is.
E [f (X1,1, ..., Xn,1, Xn,2, ..., Xn,k, Z)]
≤ ek√∏
i hi E [f (Y1,1, ..., Yn,1, Yn,2, ..., Yn,k, Z)] .
Corollary. Given conf. c, if event E holds in process P withprob 1− n−b with b > (k log h)/(2 logn), then it holds w.h.p.also in the original process.
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Probability Amplification
A dice with k faces is thrown ` times.
prob
abili
ty
γ
1 2 . . . face
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Probability Amplification
A dice with k faces is thrown ` times.
M := most frequent face in the ` throws(breaking ties at random).
For any j 6= 1
Pr (M = 1)− Pr (M = j) ≥ const ·√` γ(1− γ2)
`−12
prob
abili
ty
γ
1 2 . . . face
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Probability Amplification
A dice with k faces is thrown ` times.
M := most frequent face in the ` throws(breaking ties at random).
For any j 6= 1
Pr (M = 1)− Pr (M = j) ≥ const ·√` γ(1− γ2)
`−12
prob
abili
ty
γ
1 2 . . . face
open problem: const ≈ e−Θ(k)
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Binomial vs Beta
Given p ∈ (0, 1) and 0 ≤ j ≤ ` it holds
Pr (Bin(n, p) ≤ j) =∑j<i≤`
(`
i
)pi (1− p)`−i
=(
`
j + 1
)(j + 1)
∫ p
0zj (1− z)`−j−1
dz
= Pr (Beta(n− k, k + 1) < 1− p).
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Binomial vs Beta
Given p ∈ (0, 1) and 0 ≤ j ≤ ` it holds
Pr (Bin(n, p) ≤ j) =∑j<i≤`
(`
i
)pi (1− p)`−i
=(
`
j + 1
)(j + 1)
∫ p
0zj (1− z)`−j−1
dz
= Pr (Beta(n− k, k + 1) < 1− p).
Multinomial vs Dirichlet?
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