Phase Transitions of Best-of-Two and Best-of-Three on Stochastic Block Models

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1. Phase Transitions of Best-of-Two and Best-of-Three on Stochastic Block Models Nobutaka Shimizu and Takeharu Shiraga (Chuo University) (The University of Tokyo) DISC19
2. /18 2 Sync. voting processes on a graph, Best-of-two (Bo2) and Best-of-three (Bo3). Each vertex supports one of two distinct opinions. ‣In each round, every node updates his opinion according to some rule. u • (Bo2): Sample two random neighbors and . Then, take the majority among opinions of and . x y x, y u • (Bo3): Sample three random neighbors and take their majority. ‣Markov Chain on . Each state is denoted by . ‣ = consensus time starting from (r.v.). 2V A ⊆ V T(A) A • c.f. In the well-known Voter process, adopts the opinion of a random neighbor. u u y x u y x
3. /18 Bo2 on with the half-and-half initial state. K500 3
4. /18 Known results (Bo2 & Bo3) 4 ‣ For Bo2 and Bo3 on , w.h.p for any [DGMSS11]. Kn T(A) = O(log n) A ‣ opinions setting: Bo2 and Bo3 on satisfies for any if is moderate [BCNPT18], [GL18], [BCNPST14]. k > 2 Kn T(A) = O(k log n) A k ‣ For Bo2 and Bo3 on expander graphs, under some initial bias assumption on [CER14], [CERRS15], [CRRS17]. T(A) = O(log n) A • core-periphery graph [CNNS18] • regular stochastic block model [CNS19] ‣ Bo2 on various graphs are investigated recently c.f. In Voter, it is known that [CEOR13]. E[T(A)] = O(n)
5. /18 Known results (Bo2 & Bo3) 5 ‣ w.h.p for any on [DGMSS11]. Much faster than Voter. T(A) = O(log n) A Kn ‣ If there are opinions, for any on if is moderate [BCNPT18], [GL18], [BCNPST14]. k > 2 T(A) = O(k log n) A Kn k ‣ Bo2 and Bo3 reaches consensus assumng an initial bias on expander graphs [CER14], [CERRS15], [CRRS17] O(log n) • core-periphery graph [CNNS18] • regular stochastic block model [CNS19] ‣ Bo2 on various graphs are investigated recently c.f. In Voter, it is known that [CEOR13]. E[T(A)] = O(n) How graph structure affect the Bo2 and Bo3 ?
6. /18 This Work ‣Bo2 and Bo3 on Stochastic Block Model (SBM) ‣SBM is a random graph consists of two vertex groups each of size , and two vertices form an edge with probability G(2n, p, q) n u, v G(300, 0.1, 0.01) 6 if and are in the same group. u v { p q o.w. ‣Generate and then set initial opinion state and observe the process. G(2n, p, q)
7. /18 7 Question: Where is the boundary?🤔? q p = 0.4 q p = 0.1
8. /18 Our result (phase transition) Consider Bo2 on such that and are constants. G(2n, p, q) p q Theorem 1 • If , w.h.p. for any . • If , w.h.p. for some . q/p > 5 − 2 T(A) = O(log n) A ⊆ V q/p < 5 − 2 T(A) ≥ exp(Ω(n)) A ⊆ V 8 Remarks: 1. Formally, we show that, w.h.p. exhibits a “nice” structure such that the statements above hold. G(2n, p, q) 2. Here, w.h.p. means “with probability for some const ”. 1 − n−c c > 0 3. Bo3 is more likely to reach consensus (since ). 1/7 < 5 − 2 Consider Bo3 on such that and are constants. G(2n, p, q) p q Theorem 2 • If , w.h.p. for any . • If , w.h.p. for some . q/p > 1/7 T(A) = O(log n) A ⊆ V q/p < 1/7 T(A) ≥ exp(Ω(n)) A ⊆ V
9. /18 Two components of our proof 9 1. Structual property of . G(2n, p, q) Tool: Janson’s ineq & Kim-Vu’s ineq 2. Bo2 and Bo3 on “nice” graphs Tools: Jacobi matrix, competitive dynamical system.
10. /18 10 Idea behind the proof (Bo3) The case of (i.e., ) is previously known. We expect that the same technique works for the case of contsants . p = q = 1 K2n p, q > 0 ‣Bo3 is a M.C. on in general graphs. But on in . ‣Let and be the set of opinion 1 and that in the next round. Then 2V {0,…, n} Kn A A′ |A′ | = ∑ v∈V 1[v∈A′ ] is the sum of independent random variables, which concentrates on its expectation. E[|A′ |] = ?
11. /19 α′ ≈ E[α′ ] = E[|A′ |] n = g(α) + O(n−1) . E[|A′ | |A] = ∑ v∈A Pr[v ∈ A′ ] = ∑ v∈V g ( degA (v) deg(v) ) . 11 ‣Evaluate . Indeed, letting , we have E[|A′ |] g(x) = ( 3 2) x2(1 − x) + ( 3 3) x3    Ћ g(α) degA (v) = 3 v ‣Let On , we have and thus α = |A|/n . Kn degA (v) deg(v) = |A| n + O(n−1) = α + O(n−1) ‣The orbit of is approximated by a discrete-time dynamical system on . α [0,1] deg(v) = 5
12. /18 Key lemma (concentration of for Bo3 on ) E[α′ |A] G(2n, p, p) ‣The lemma says “the behavior of can be approximated by a discrete-time dynamical system.” ‣We can evaluate , which leads to the symmetry breaking. ‣The result can be extended to . ‣The proof relies on concentration inequalities. α Var[α′ |A] G(2n, p, q) 12 Suppose that . Then satisfies the following w.h.p.: p = ω(log n/n) G(2n, p, p) ∀A ⊆ V, E[α′ |A] = g(α) +O ((np)−0.5), where g(x) := 3x2 − 2x3 . ‣This observation holds for . We prove it for !! Kn G(2n, p, q)
13. /18 On G(2n,p,q) ‣ Let . Ai := A ∩ Vi (i = 1,2) A1 A2 13 ‣ Fix the time step . ‣ : configuration at the current step . ‣ : that of the next step . t A t A′ t + 1
14. /18 2D dynamical system ‣Consider for each . E[α′ i ] = E[|A′ i |/n] i = 1,2 • This yields a 2D dynamical system on . [0,1]2 14 α′ 1 = 3 ( α1 + rα2 1 + r ) 2 − 2 ( α1 + rα2 1 + r ) 3 ± O((np)−1/2), α′ 2 = 3 ( rα1 + α2 1 + r ) 2 − 2 ( rα1 + α2 1 + r ) 3 ± O((np)−1/2) . ‣For Bo3, our key lemma implies where r = q/p .
15. /18 threshold r < 15 threshold r > Vertical axis = Horisontal axis = . α1 α2
16. /18 16 threshold (absorbing) r < threshold (saddle). r >
17. /18 Intuitive Explanation (p≠q) ‣If , two absorbing points appear: q/p < 1/7 It takes exponential time to escape from the ε- neighborhood of the absorbing points. 17 ‣We used competitive dynamical system theory to prove the orbit convergence for any initial point . (α1 , α2 ) ‣We do not explore the case where . r = 1/7 ‣If , these absorbing points disappear: q/p > 1/7 Escape from the fixed point in time. O(log n)
18. /18 • Approximate orbit by a dynamical system. Summary ‣ Technical tools: • Local behavior around fixed points. • Global behavior (orbit convergence) ‣ Bo2 and Bo3 on SBM • phase transition on q/p. • consensus time: log vs. exponential. 18 • Community detection is a potential application. • The case threshold is left as future task. • We obtained similar results for sparse SBM under initial bias assumption. q/p = Concentration inequalities to analyze random graphs. Essentially known in [Doerr et al. 2011]. Competitive dynamical system theory.