Quasi-majority Functional Voting on Expander Graphs

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1. Quasi-Majority Functional Voting on Expander Graphs Nobutaka Shimizu and Takeharu Shiraga Chuo University Thu University of Tokyo
2. /23 2 • Each vertex has an opinion, 0 or 1 (configuration). • Application: model of opinion forming, consensus problem, etc. (i): All vertices simultaneously update their opinion according to a predefined rule. (ii): Repeat (i) until all vertices have the same opinion. 4ZODISPOPVT
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3. /23 3 • Examples: Pull Voting, Best-of-Two, Best-of-Three • (Pull Voting): Pick up a neighbor u.a.r. Adopt the opinion of random neighbor. • (Bo2): Pick up two random neighbors with replacement. Adopt the majority opinion among two random neighbors and the opinion of itself. • (Bo3): Pick up three random neighbors with replacement. Adopt the majority opinion among the three.
4. /23 Bo2 on starting with the half-and-half configuration. K500 4
5. /23 • := {vertex of opinion 0} at round . • . (consensus time) • We say that an event holds w.h.p. if for some const . At t ∈ ℕ ∪ {0} Tcons (A) := inf{t : At ∈ {∅, V}, A0 = A} Z Pr[Z] ≥ 1 − O(n−c) c > 0 5 'SBNFXPSL • Assumption: the underlying graph is connected and nonbipartite.
6. /23 • w.h.p. for expander graphs. ∀A ⊆ V, E[Tcons (A)] = O(n) • for any on any graph. E[Tcons (A)] = O(n3) A ⊆ V 6 • Well explored since 1950s in the literature of interactive particle system. • Pull Voting exhibits “linearity”. This provides a rich theory (e.g., duality) of Pull Voting. ,OPXO
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7. /23 7 • Expander : w.h.p. if has an initial bias. • [Cooper, Elsässer, Radzik;14] • [Cooper, R. Elsässer, T. Radzik, N. Rivera, and T. Shiraga;15] • [Cooper, Radzik, Rivera, Shiraga;17] Tcons (A) = O(log n) A • : w.h.p. for any [DGMSS11]. Kn Tcons (A) = O(log n) A • Introduced by [Doerr, Goldberg, Minder, Sauerwald, Scheideler;11]. Worst-case analysis is difficult if the underlying graph is not . Kn ,OPXO
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   #P • : w.h.p. for any if is const. G(n, p) Tcons (A) = O(log n) A p [Shimizu, Shiraga;19]
8. /23 8 • w.h.p. for any A on [Ghaffari, Lengler;18]. Tcons (A) = O(log n) Kn • Introduced by [Becchetti, Clemanti, Natale, Pasquale, Silvestri, Trevisan;14]. ,OPXO
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   #P • Expander : if has a bias [Cooper, Radzik, Rivera, Shiraga;17] Tcons (A) = O(log n) A • Dense graphs ( is random) [Kang, Rivera;19] A Bo2, Bo3, and other voting processes are studied via their own specific methods. • : w.h.p. for any if is const. G(n, p) Tcons (A) = O(log n) A p [Shimizu, Shiraga;19]
9. /23 9 • We propose a general class of voting processes that contains Bo2 and Bo3 as a special case. • We prove that, for the general voting process, on expander graphs. •We prove that, for Bok (a generalization of Bo2 and Bo3), on expander graphs. ∀A ⊆ V, Tcons (A) = O(log n) ∀A ⊆ V, Tcons (A) = O(log n/log k) 0VS
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10. /23 Definition 1 (Functional Voting) 10 For a function , a functional voting w.r.t. is a voting process such that We call betrayal function and update function. f : [0,1] → [0,1] f f A A′ ={vertices of opinion 0 currently} ={vertices of opinion 0 at the next round} Pr[v ∈ A′ ] = f ( degA (v) deg(v) ) Pr[v ∈ B′ ] = f ( degB (v) deg(v) ) JG
    v ∈ B B = V∖A B′ = V∖A′ JG
    v ∈ A Hf (x) := x(1 − f(1 − x)) + (1 − x)f(x) For , {vertices adjacent to } v ∈ V N(v) = v For , and S ⊆ V degS (v) = |N(v) ∩ S| deg(v) = degV (v)
11. /23 11 •Pull Voting : • •Bo2 : • •Bo3 : • f(x) = x f(x) = x2 f(x) = ( 3 2) x2(1 − x) + ( 3 3) x3 = 3x2 − 2x3 &YBNQMFT Pr[v ∈ A′ ] = f ( degA (v) deg(v) ) Pr[v ∈ B′ ] = f ( degB (v) deg(v) ) JG
    v ∈ B JG
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    Hf • Let and . • For fixed , is concentrated on . •Indeed, on (with self loop on each vertex), we have α = |A|/n α′ = |A′ |/n A α′ = 1 n ∑ v∈V 1v∈A′ E[α′ ] Kn E[α′ ] = |A| n + |B| n f(α) − |A| n f(1 − α) •The behavior of is α α → Hf (α) → Hf (Hf (α)) → ⋯ Hf (x) := x(1 − f(1 − x)) + (1 − x)f(x) = Hf (α)
13. /23 Definition 2 (Quasi-Majority Functional Voting) 13 A functional voting w.r.t. is quasi-majority if (I) is , (II) , (III) for all , (IV) , and (V) . f f C2 0 < f(1/2) < 1 Hf (x) < x 0 < x < 1/2 H′ f (1/2) > 1 H′ f (0) < 1 • holds in general. • So (III) implies for all . • Bo2 and Bo3 are quasi-majority, but Pull Voting is not. Hf (1 − x) = 1 − Hf (x) Hf (x) > x 1/2 < x < 1
    
    
    
    
    
      
      
14. /23 14 Definition 3 (Expander Graph) A graph is -expander if , where are eigenvalues of the transition matrix of the simple random walk on . G λ max{|λ2 |, |λn |} ≤ λ 1 = λ1 ≥ λ2 ≥ ⋯ ≥ λn G • An Erdős–Rényi graph is -expander w.h.p. G(n, p) O(1/ np) Consider the stationary distribution of the simple random walk, i.e., π ∈ [0,1]V π(v) = deg(v) 2|E| . A graph is mildly regular if and . ∥π∥2 = O(n−1/2) ∥π∥3 = o(n−1/3) Definition 4 (Mildly Regular Graph) • For any regular graphs, . • For a star graph, and . ∥π∥2 = n−1, ∥π∥3 = n−2/3 ∥π∥2 ≈ 1 ∥π∥3 ≈ 1
15. /23 Theorem 2 (Sparse Expanders; Informal) 15 0VS
    3FTVMUT Any QMFV on a sparse expander graph reaches consensus in rounds if the initial configuration has a bias. O(log n) • Our results generalize and extend previous works of Bo2 and Bo3 on expander graphs! • Previous works says “For Bo2/Bo3, under an initial bias assumption”. Tcons (A) = O(log n) Theorem 1 (Main Result) For any quasi-majority functional voting on an -expander and mildly regular graph, w.h.p. for any . Moreover, for some O(n−1/4) Tcons (A) = O(log n) A ⊆ V Tcons (A) = Ω(log n) A ⊆ V .
16. /23 16 &YBNQMFT
    HSBQI Corollary 1 (Erdős–Rényi graph) An ER graph w.h.p. satisfies: G(n, n−1/2) • Consider Bo2/Bo3. For any , w.h.p. A ⊆ V Tcons (A) = O(log n) Corollary 2 (Random Regular Graph) A random -regular graph w.h.p. satisfies: Ω( n) • Consider Bo2/Bo3. For any , w.h.p. A ⊆ V Tcons (A) = O(log n) • Previous work : w.h.p. for any on . Tcons (A) = O(log n) A G(n, Ω(1)) [Shimizu, Shiraga;19] • Previous work : w.h.p. if initial bias. Tcons (A) = O(log n) ∃ [Cooper, Radzik, Rivera, Shiraga;17]
17. /23 17 &YBNQMFT
    NPEFM • -careful voting is a QMFV , where . k f(x) = xk • In -Careful Voting, a vertex samples random neighbors. If all of the neighbors have the same opinion, adopts it. k v k v • In -lazy Bo2, a vertex tosses a private coin with head probability . If head, performs Bo2. Otherwise, does nothing. ρ v ρ v v • -lazy Bo2 is a QMFV for constant , where . ρ ρ f(x) = ρx2 We can consider several models. If it is QMFV, we can obtain bounds of the consensus time from our result!!
18. /23 18 •Worst-case consensus time is known for . •We use this technique. Kn 1SPPG
    0VUMJOF •In general, a voting process is a Markov chain on . •On , the state space becomes . 2V Kn {0,…, n} •Our strategy: Even on expander graphs, the state space is roughly . {0,…, n}
19. /23 19 • Let and . α = |A|/n α′ = |A′ |/n 'VODUJPOBM
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    Kn •By the Chernoff bound, it holds w.h.p. that α′ = E[α′ ] ± 10 log n/n . α 0 1 Hf (α) = Hf (α) ± 10 log n/n .
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    Kn •If , then we are done since |α − 1/2| ≥ 100 log n/n •If , we use a useful result from the previous work [Doerr, Goldberg, Minder, Sauerwald, Scheideler;11]. |α − 1/2| < 100 log n/n α′ = Hf (α) ± 10 log n/n
    
    
    
    
    
      
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21. /23 Lemma 5 (informal; DGMSS11) 21 •For any , there exists such that holds with probability . • for some const if is sufficiently small. C δ |α′ − E[α′ ]| ≥ Cn−1/2 δ |E[α′ ] − 1/2| > (1 + ϵ)|α − 1/2| ϵ > 0 |α − 1/2| within rounds if both of the following conditions hold. |α − 1/2| > 100 log n/n O(log n) •The second condition easily follows since . •From CLT, has a fluctuation of size . We need to estimate . •On , it is easy to see that . H′ f (1/2) > 1 α′ Ω( Var[α′ ]) Var[α′ ] Kn Var[α′ ] = Θ(1/n)
22. /23 22 •We evaluate and on -expander graphs: E[α′ ] Var[α′ ] λ 0VS
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     A ⊆ V |A| − n/2 < 100 n log n Previous work did not evaluate . This is the reason for the lack of results of the worst-case consensus time. Var[α′ ] •The core is a variant of the Expander Mixing Lemma. 0 < f(1/2) < 1
23. /23 23 •We propose a general voting process that contains Bo2 and Bo3 as special cases. •We obtain the worst-case consensus time of this model on expander graphs. •The proof invokes several previous results. •The core of the proof is the evaluation of and . E[α′ ] Var[α′ ] $PODMVTJPO