majority consensus algorithms and spectral...
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Majority ConsensusAlgorithms and Spectral Optimisation
Moez Draief
Intelligent Systems and Networks GroupImperial College London
Workshop on Large Evolving NetworksHeilbronn Institute for Mathematical Research
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Computational power of (social) networks
Off to the movies...Friends who want to watch one of two movies (together)They interact in pairs in order to come to a common choicebefore the end of the dayMajority prefers one theater
What protocol should they run to decide to go to the latter?
Biased voting [Kearns et al ’09]Players paid if consensus conforms with their preferenceFor some network topologies, minority preferenceconsistently winsIndividual behavioral characteristics (stubbornness,awareness of opposing incentives) correlate with earnings
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Computational power of (social) networks
Off to the movies...Friends who want to watch one of two movies (together)They interact in pairs in order to come to a common choicebefore the end of the dayMajority prefers one theater
What protocol should they run to decide to go to the latter?
Biased voting [Kearns et al ’09]Players paid if consensus conforms with their preferenceFor some network topologies, minority preferenceconsistently winsIndividual behavioral characteristics (stubbornness,awareness of opposing incentives) correlate with earnings
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Consensus on networks
Distributed computingInformation fusion/consistency in distributed networksNetwork awareness (Computing graph properties)Multi-agent coordination and flocking
Distributed algorithmsNumerous impossibility results in the deterministic case[Lynch et al ’90s]Random walksGossiping algorithms
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Consensus on networks
Distributed computingInformation fusion/consistency in distributed networksNetwork awareness (Computing graph properties)Multi-agent coordination and flocking
Distributed algorithmsNumerous impossibility results in the deterministic case[Lynch et al ’90s]Random walksGossiping algorithms
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Binary majority consensus
Desired outcome and metricsNodes end with opinion held by majority of nodesNode can probe neighbours and update opinionaccordingly using little (constant) memoryProbability of error (convergence to incorrect consensus)Time to convergence
ApplicationsOccurrence of a given event in cooperative decisionmakingVoting in distributed systemsRoutine to solve more elaborate distributed decisionmaking instances
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Binary majority consensus
Desired outcome and metricsNodes end with opinion held by majority of nodesNode can probe neighbours and update opinionaccordingly using little (constant) memoryProbability of error (convergence to incorrect consensus)Time to convergence
ApplicationsOccurrence of a given event in cooperative decisionmakingVoting in distributed systemsRoutine to solve more elaborate distributed decisionmaking instances
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Outline
1 Majority consensus: algorithmsVoter ModelAveraging ProcessBinary consensus
2 ExamplesComplete graphStar, ER, Ring, Line
3 Optimization of Majority ConsensusFaster convergence timeImplementationExamples
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Outline
1 Majority consensus: algorithmsVoter ModelAveraging ProcessBinary consensus
2 ExamplesComplete graphStar, ER, Ring, Line
3 Optimization of Majority ConsensusFaster convergence timeImplementationExamples
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Model
Interaction ModelConnected undirected graph G = (V ,E), |V | = nαn nodes hold 0 and (1− α)n nodes hold 1, α ∈ (1/2,1)
Nodes i and j interact at rate qij = qji , qij 6= 0 iff (i , j) ∈ E
Markov chain(Xt )t≥0 with rate matrix Q, qii = −
∑i 6=j qij
(πi)i∈V stationary distribution is uniform on V . Mixing time:∣∣Pj(Xt = i)− 1/n∣∣ = O
(e−λ2(Q)t
)where λ2(Q) = inf{
∑i,j qij(xi − xi)
2, ||x || = 1, xT 1 = 0}
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Voter Model
Probability of error
Interaction ModelConnected undirected graph G = (V ,E), |V | = nαn nodes hold 0 and (1− α)n nodes hold 1, α ∈ (1/2,1)
Nodes i and j interact at rate qij and i updates to j ’s statew.p. 1/2
Theorem [Liggett ’85, Hassin-Peleg ’01]The number of nodes in state 1 is a martingale.Probability of reaching (wrong) consensus at 1 is 1− α .
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Voter Model
Time to convergence [Aldous 2012]
Complete graphEach edge has rate 1/(n − 1). The number of agents withopinion 1 evolves as Birth and Death proces with
λk ,k+1 = λk ,k−1 =k(n − k)
2(n − 1).
Time to convergence = O(n)
General graph
Conductance η(Q) = infA⊂V
∑i∈A,j∈Ac qij
|A||Ac |/n
The Markov chain tracking the number of nodes in state 0evolves at least η(Q) times as fast as on the completegraph,Time to convergence O(n/η(Q))
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Voter Model
Time to convergence [Cooper et al 2012.]
Cheeger’s inequality
Conductance: η(Q) = infA⊂V
∑i∈A,j∈Ac qij
|A||Ac |/n
Spectral Gap:λ2(Q) = inf{
∑i,j qij(xi − xi)
2, ||x || = 1, xT 1 = 0}
λ2(Q) ≤ η(Q) .
Time to convergence of voter model O(n/(λ2(Q))).
Let S of size k be the subset realising the inf in η(Q) and let x
such that xi = −√
n−kkn , i ∈ S and xi =
√k
(n−k)n , i ∈ Sc .
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Voter Model
Time to convergence [Cooper et al 2012.]
Cheeger’s inequality
Conductance: η(Q) = infA⊂V
∑i∈A,j∈Ac qij
|A||Ac |/n
Spectral Gap:λ2(Q) = inf{
∑i,j qij(xi − xi)
2, ||x || = 1, xT 1 = 0}
λ2(Q) ≤ η(Q) .
Time to convergence of voter model O(n/(λ2(Q))).
Let S of size k be the subset realising the inf in η(Q) and let x
such that xi = −√
n−kkn , i ∈ S and xi =
√k
(n−k)n , i ∈ Sc .
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Averaging Process
Distributed averaging
At each interaction of (i , j) occuring at rate qij
xi(t) = xj(t) =xi(t−) + xj(t−)
2.
Theorem [Boyd et al ’06, Aldous-Lanoue ’12]Algorithm converges to the average value, usingO(Poly(log(n)) memory per nodeTime to convergence to up O(1/n) error of the average is
Θ(log(n)/λ2(Q)) ,
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Averaging Process
Distributed averaging: Proof
Assume that∑
i xi(0) = 0.Let Q(t) = ||x(t)||2. When an i , j interaction takes place Q(t)reduces by (xi − xj)
2/2.
E(dQ(t) | x(t) = x) = −∑i,j
qij(xi − xj)
2
2dt
≤ −λ2(Q)||x ||2/2dt
In particularE||x(t)||2 ≤ ||x(0)||2e−λ2(Q)t/2
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Binary consensus
Small memory
Could we use less memory and still guarantee small error?
Impossibility [Land-Belew ’95]
Connected undirected graph G = (V ,E), |V | = n,αn nodes in 0 and (1− α)n nodes in 1, α ∈ (1/2,1),2α− 1 is the voting margin.i contacts j at rate qij > 0 ∀(i , j) ∈ E
No 1-bit distributed algorithm can solve the majority consensusproblem.
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Binary consensus
Small memory
Could we use less memory and still guarantee small error?
Impossibility [Land-Belew ’95]
Connected undirected graph G = (V ,E), |V | = n,αn nodes in 0 and (1− α)n nodes in 1, α ∈ (1/2,1),2α− 1 is the voting margin.i contacts j at rate qij > 0 ∀(i , j) ∈ E
No 1-bit distributed algorithm can solve the majority consensusproblem.
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Binary consensus
Binary Consensus with two undecided states
Averaging-like updates: States 0 < e0 < e1 < 1.
Rules: Swaps + Annihilation
e0 0
e0 0
e1 0
e0 0
0 1
e0 e1
e0 e1
e0 e1
e0
e1 1
1 e1 1
e1 1
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Binary consensus
Binary Consensus with two undecided states
Averaging-like updates: States 0 < e0 < e1 < 1.Rules: Swaps + Annihilation
e0 0
e0 0
e1 0
e0 0
0 1
e0 e1
e0 e1
e0 e1
e0
e1 1
1 e1 1
e1 1
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Binary consensus
Mean-field analysis (Complete graph)
Let qij = 1n−1 , i 6= j and X(t) = (|S0(t)|, |Se0(t)|, |Se1(t)|, |S1(t)|)
is a Markov process with the following transition rates
→
(|S0(t)| − 1, |Se0 (t)|+ 1, |Se1 (t)|+ 1, |S1(t)| − 1) : |S0(t)||S1(t)|
n−1
(|S0(t)|, |Se0 (t)| − 1, |Se1 (t)|+ 1, |S1(t)|) :|Se0 (t)||S1(t)|
n−1
(|S0(t)|, |Se0 (t)|+ 1, |Se1 (t)| − 1, |S1(t)|) :|S0(t)||Se1 (t)|
n−1 .
By Kurtz, X(t)/n converges to (s0(t), se0(t), se1(t), s1(t)),
se1(t) ∼ (2α− 1)1− αα
te−(2α−1)t
s1(t) ∼ (2α− 1)1− αα
e−(2α−1)t .
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Binary consensus
Mean-field analysis (Complete graph)
Let qij = 1n−1 , i 6= j and X(t) = (|S0(t)|, |Se0(t)|, |Se1(t)|, |S1(t)|)
is a Markov process with the following transition rates
→
(|S0(t)| − 1, |Se0 (t)|+ 1, |Se1 (t)|+ 1, |S1(t)| − 1) : |S0(t)||S1(t)|
n−1
(|S0(t)|, |Se0 (t)| − 1, |Se1 (t)|+ 1, |S1(t)|) :|Se0 (t)||S1(t)|
n−1
(|S0(t)|, |Se0 (t)|+ 1, |Se1 (t)| − 1, |S1(t)|) :|S0(t)||Se1 (t)|
n−1 .
By Kurtz, X(t)/n converges to (s0(t), se0(t), se1(t), s1(t)),
se1(t) ∼ (2α− 1)1− αα
te−(2α−1)t
s1(t) ∼ (2α− 1)1− αα
e−(2α−1)t .
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Binary consensus
Minority states
0 20 40 600
0.2
0.4
0.6
0.8
t
s1(t)
se1(t)
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Binary consensus
General bound
Theorem [Draief-Vojnovic ’12]Let T be the time until there are only nodes in state 0 and e0.
E(T ) = O (log n/δ(G, α))
whereδ(Q, α) = min
S⊂V ,|S|=(2α−1)nmin
λ∈Spec(QS)|λ|
Generalised conductance lemma [Babaee-Draief ’13+]We have
δ(Q, α) ≥ cαλ2(Q)
In particular,E(T ) = O(log(n)/λ2(Q))
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Binary consensus
Proof: Depletion of nodes in state 1
Let Ai and Zi indicator node in 0 and 1 resp. The transitions ofthe Markov process (Z ,A) is given by
(Z ,A)→
(Z − ei ,A− ej ) : qi,jZiAj(Z − ei + ej ,A) : qi,jZi (1− Aj − Zj )(Z ,A− ei + ej ) : qi,jAi (1− Aj − Zj )
For t ∈ [tk , tk+1) where {S0(t) = Sk}
ddt
Ek (Ai(t)) = −
(∑l∈V
qi,l
)Ek (Ai(t))+
{ ∑j∈V qi,jEk
(Aj(t)
), i /∈ Sk
0, i ∈ Sk
where Ek is the expectation conditional on {S0(t) = Sk}.
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Binary consensus
Proof: Depletion of nodes in state 1
Let Ai and Zi indicator node in 0 and 1 resp. The transitions ofthe Markov process (Z ,A) is given by
(Z ,A)→
(Z − ei ,A− ej ) : qi,jZiAj(Z − ei + ej ,A) : qi,jZi (1− Aj − Zj )(Z ,A− ei + ej ) : qi,jAi (1− Aj − Zj )
For t ∈ [tk , tk+1) where {S0(t) = Sk}
ddt
Ek (Ai(t)) = −
(∑l∈V
qi,l
)Ek (Ai(t))+
{ ∑j∈V qi,jEk
(Aj(t)
), i /∈ Sk
0, i ∈ Sk
where Ek is the expectation conditional on {S0(t) = Sk}.
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Binary consensus
(random) Piecewise-linear dynamical system
DynamicsThe dynamics of the system boils down to Y (t) = (Yi(t))i∈V ,
ddt
Ek (Y (t)) = QSkEk (Y (t)) ,
for t ∈ [tk , tk+1) during which {S0(t) = Sk} and QSk is given by
QS(i , j) =
−∑
l∈V qi,l , i = jqi,j , i /∈ S, j 6= i0, i ∈ S, j 6= i .
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Binary consensus
Solution
PropositionSolving the above differential equation and using the strongMarkov property
E(Y (t)) = E[eλ(t)Y (0)
]where λ(t) = QSk (t − tk ) +
∑k−1l=0 QSl (tl+1 − tl).
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Binary consensus
Proof: Spectrum of QS
For any finite graph G, there exists δ(G, α) > 0 such that, forany non-empty subset of vertices S with |S| ∈ [(2α− 1)n, αn], ifλ is an eigenvalue of the the matrix QS defined in, then
λ ≤ −δ(G, α) < 0.
First(−∑
l∈V qi,l), i ∈ S are eigenvalues of QS
The remaining eigenvalues correspond to eigenvectors ofthe form ( x︸︷︷︸
Sc
,0, . . . ,0︸ ︷︷ ︸S
)T . Let W ⊂ Sc such that for i ∈W ,
xi 6= 0
−λ =∑i∈W
∑j∈S
qi,jx2i +
∑i∈W ,j∈Sc\W
qi,jx2i +
12
∑i,j∈W
qi,j(xi − xj)2
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Binary consensus
Proof: Spectrum of QS
For any finite graph G, there exists δ(G, α) > 0 such that, forany non-empty subset of vertices S with |S| ∈ [(2α− 1)n, αn], ifλ is an eigenvalue of the the matrix QS defined in, then
λ ≤ −δ(G, α) < 0.
First(−∑
l∈V qi,l), i ∈ S are eigenvalues of QS
The remaining eigenvalues correspond to eigenvectors ofthe form ( x︸︷︷︸
Sc
,0, . . . ,0︸ ︷︷ ︸S
)T . Let W ⊂ Sc such that for i ∈W ,
xi 6= 0
−λ =∑i∈W
∑j∈S
qi,jx2i +
∑i∈W ,j∈Sc\W
qi,jx2i +
12
∑i,j∈W
qi,j(xi − xj)2
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Binary consensus
Proof: The End
Note that
E(Y (t)) = E[eλ(t)Y (0)
]where λ(t) = QSk (t − tk ) +
∑k−1l=0 QSl (tl+1 − tl)
Hence
||E(Y (t))||2 ≤ E
[||eQSk
(t−tk )||k−1∏l=0
||eQSl(tl+1−tl )|| ||Y (0)||2
]≤√
ne−δ(G,α)t
Therefore, by Cauchy-Schwartz, we have
P(Y (t) 6= 0) ≤∑i∈V
E(Yi(t)) ≤ n e−δ(G,α)t
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Binary consensus
Proof: The End
Note that
E(Y (t)) = E[eλ(t)Y (0)
]where λ(t) = QSk (t − tk ) +
∑k−1l=0 QSl (tl+1 − tl)
Hence
||E(Y (t))||2 ≤ E
[||eQSk
(t−tk )||k−1∏l=0
||eQSl(tl+1−tl )|| ||Y (0)||2
]≤√
ne−δ(G,α)t
Therefore, by Cauchy-Schwartz, we have
P(Y (t) 6= 0) ≤∑i∈V
E(Yi(t)) ≤ n e−δ(G,α)t
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Binary consensus
Proof: The End
Note that
E(Y (t)) = E[eλ(t)Y (0)
]where λ(t) = QSk (t − tk ) +
∑k−1l=0 QSl (tl+1 − tl)
Hence
||E(Y (t))||2 ≤ E
[||eQSk
(t−tk )||k−1∏l=0
||eQSl(tl+1−tl )|| ||Y (0)||2
]≤√
ne−δ(G,α)t
Therefore, by Cauchy-Schwartz, we have
P(Y (t) 6= 0) ≤∑i∈V
E(Yi(t)) ≤ n e−δ(G,α)t
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Binary consensus
Summary
Upper bound on the expected convergence time for anumber of distributed candidate dynamics for solvingMajority consensus
Bounds based on the location of the spectral gap of ratematrix (generalised-cut: quick for expander graphs).For binary consensus, expected convergence time criticallydepends on the voting margin
Application to particular network topologies: completegraphs, stars, ER graph, paths, cycles.
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Binary consensus
Summary
Upper bound on the expected convergence time for anumber of distributed candidate dynamics for solvingMajority consensus
Bounds based on the location of the spectral gap of ratematrix (generalised-cut: quick for expander graphs).For binary consensus, expected convergence time criticallydepends on the voting margin
Application to particular network topologies: completegraphs, stars, ER graph, paths, cycles.
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Outline
1 Majority consensus: algorithmsVoter ModelAveraging ProcessBinary consensus
2 ExamplesComplete graphStar, ER, Ring, Line
3 Optimization of Majority ConsensusFaster convergence timeImplementationExamples
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Complete graph
Upper Bounds
Corollary
An application of the theorem to complete graph qi,j = 1n−1 for
all i 6= j , yields
E(Ti) ≤1
2α− 1log(n).
Exact asymptoticsA direct analysis of the dynamics of the 1st phase tracking theinteractions of nodes in state 1 and nodes in state 0 implies that
E(T1) =n − 1
|S0| − |S1|(H|S1| + H|S0|−|S1| − H|S0|
)where Hk =
∑ki=1
1i
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Complete graph
Various initial conditions
|S0| − |Sn| = (2α− 1)n, α a constant larger than 1/2
E(T1) =1
2α− 1log(n) + O(1).
If |S0| = |S1|
E(T1) =π2
6n(1 + o(1)).
µn = (|S0| − |S1|)/n is strictly positive but small (o(1)),
E(T1) =1µn
log(nµn) + O(1).
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Complete graph
Complete Graph: Theory v. Simulation
100
101
102
103
104
105
0
5
10
15
20
n
E(T
1)
100
101
102
103
104
105
0
5
10
15
20
n
E(T
2)
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Star, ER, Ring, Line
Star
Star Network: q1,i = qi,1 = 1n−1 , i 6= 1 and qi,j = 0, i , j 6= 1.
E(Ti ) ≤ 12α−1 n log(n). Using, direct calculation
E(T1) =1
(2α− 1)(3− 2α)n log(n) + O(n)
ER-graph: qi,j = 1npn
Xi,j Xi,j i.i.d. Bernoulli r.v. with mean
c log(n)n , c > 2
2α−1 , for h−1 the inverse of h(x) = x log(x) + 1− x ,
E(Ti ) ≤1
(2α− 1)h−1(
2c(2α−1)
) log(n) + O(1)
Path: E(Ti ) ≤ 16(1−α)2
π2 n2 log(n) + O(1)
Ring: E(Ti ) ≤ 4(1−α)2
π2 n2 log(n) + O(1).
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Star, ER, Ring, Line
Star
0 0.5 10
20
40
60
80
2 1
E(T
1) / n
0 0.5 10
20
40
60
80
2 1
E(T
2) / n
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Star, ER, Ring, Line
ER-graph
Star Network: q1,i = qi,1 = 1n−1 , i 6= 1 and qi,j = 0, i , j 6= 1.
E(Ti ) ≤ 12α−1 n log(n). Using, direct calculation
E(T1) =1
(2α− 1)(3− 2α)n log(n) + O(n)
ER-graph: qi,j = 1npn
Xi,j Xi,j i.i.d. Bernoulli r.v. with mean
c log(n)n , c > 2
2α−1 , for h−1 the inverse of h(x) = x log(x) + 1− x ,
E(Ti ) ≤1
(2α− 1)h−1(
2c(2α−1)
) log(n) + O(1)
Path: E(Ti ) ≤ 16(1−α)2
π2 n2 log(n) + O(1)
Ring: E(Ti ) ≤ 4(1−α)2
π2 n2 log(n) + O(1).
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Star, ER, Ring, Line
ER-graph
0 0.5 10
50
100
150
200
2α−1
E(T
1)
0 0.5 10
50
100
150
200
2α−1
E(T
2)
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Star, ER, Ring, Line
Path and Ring
Star Network: q1,i = qi,1 = 1n−1 , i 6= 1 and qi,j = 0, i , j 6= 1.
E(Ti ) ≤ 12α−1 n log(n). Using, direct calculation
E(T1) =1
(2α− 1)(3− 2α)n log(n) + O(n)
ER-graph: qi,j = 1npn
Xi,j Xi,j i.i.d. Bernoulli r.v. with mean
c log(n)n , c > 2
2α−1 , for h−1 the inverse of h(x) = x log(x) + 1− x ,
E(Ti ) ≤1
(2α− 1)h−1(
2c(2α−1)
) log(n) + O(1)
Path: E(Ti ) ≤ 16(1−α)2
π2 n2 log(n) + O(1)
Ring: E(Ti ) ≤ 4(1−α)2
π2 n2 log(n) + O(1).
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Outline
1 Majority consensus: algorithmsVoter ModelAveraging ProcessBinary consensus
2 ExamplesComplete graphStar, ER, Ring, Line
3 Optimization of Majority ConsensusFaster convergence timeImplementationExamples
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Faster convergence time
Convex Optimization [Boyd, Diaconis, Xiao ’04]
For technical reasons, let us assume
Q = P − In
where P is a symmetric stochastic matrix.
Eigenvalue (convex) optimizationMinimize the time it takes majority consensus to converge, i.e.
minimize λ2(P) = sup{xT Px | xT x = 1, xT 1 = 0}subject to Pij ≥ 0, Pij = 0 if i , j /∈ E
and∑
j
Pij = 1,∀i
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Implementation
Subgradient method
Let u be the eigenvector associated with λ2(P).Let E`, ` = (i , j) an edge in the graph such that
E`ij = E`
ji = 1, E`ii = E`
jj = −1
The subgradient of the objective function λ2(P) is
g(P) =(
uT E1u, . . . ,uT Emu)
In particular, for ` = (i , j)
uT E`u = (ui − uj)2
To compute eingenvector we could use Lanczos method orrecent distributed algorithms [Kempe-McSherry ’08].
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Implementation
Projected subgradient method [Bertsekas ’99]
k ← 1repeatSubgradient StepCalculate g(k) and update P ← P − βk g(k), βk step size, β → 0,∑
k βk →∞Sequential Projection
Projection onto non-negative orthantP` ←max {P`,0}, ` = 1, ...,mFor each node i = 1, ...,n, L(i) = {`| edge ` connected to i }
Projection onto half-spacesWhile
∑`∈L(i) P` > 1
L(i)← {`|` ∈ L(i),P` > 0}γ ← min
{min`∈L(i) P`,
(∑`∈L(i) P` − 1
)/|L(i)|
}P` ← P` − γ, ` ∈ L(i)
k ← k + 1
It can be implemented in a distributed fashion [Boyd et al ’06].
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Examples
ER-graph [Babaee, Draief’13+]
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Examples
Preferential attachment [Babaee, Draief’13+]
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Summary
Algorithms for solving Majority consensusPerformance: memory, error, time to convergenceTime to convergence related to spectral properties of ratematrixSpeedingup convergence via convex optimisation
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Majority consensus: algorithms Examples Optimization of Majority Consensus Conclusion
Future Work
Lower-bounds of convergence time- O. Ayaso, D. Shah and M. Dahleh, Information TheoreticBounds for Distributed Computation over Networks ofPoint-to-Point Channels, IEEE IT, 2010.- M. Abdullah, M. Draief, Consensus on the Initial GlobalMajority by Local Majority Polling for a Class of Sparse Graphs,Arxiv1209.5025, 2013.
Trade-off between memory, error, time to convergence.Distributed spectral computations- David Kempe, Frank McSherry: A Decentralized Algorithm forSpectral Analaysis, Journal of Computer and System Sciences,2008.- S. Korada, A. Montanari, and S. Oh, Gossip PCA, Sigmetrics2011.