My slide for ICALP20

1. Quasi-Majority Functional Voting on
Expander Graphs
Nobutaka Shimizu and Takeharu Shiraga
Chuo UniversityThu University of Tokyo

2. /232
• 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.
(Synchronous) Voting Process

3. /233
• 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
Framework
• 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.
Known Results (Pull Voting)

7. /237
• 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
Known Results (Bo2)
• : w.h.p. for any if is const.G(n, p) Tcons(A) = O(log n) A p
[Shimizu, Shiraga;19]

8. /238
• 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].
Known Results (Bo3)
• 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. /239
• 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)
Our Contribution
w.h.p.
(the detail is omitted in this talk)

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) )
if v ∈ B
B = V∖A
B′ = V∖A′
if v ∈ A
Hf(x) := x(1 − f(1 − x)) + (1 − x)f(x)
For , {vertices adjacent to }v ∈ V N(v) = v
For , andS ⊆ V degS(v) = |N(v) ∩ S| deg(v) = degV(v)

11. /2311
•Pull Voting :
•
•Bo2 :
•
•Bo3 :
•
f(x) = x
f(x) = x2
f(x) =
(
3
2)
x2
(1 − x) +
(
3
3)
x3
= 3x2
− 2x3
Examples
Pr[v ∈ A′] = f
(
degA(v)
deg(v) )
Pr[v ∈ B′] = f
(
degB(v)
deg(v) )
if v ∈ B
if v ∈ A

12. /2312
Intuition behind 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 isα
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. /2314
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
Our Results
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. /2316
Examples (graph)
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. /2317
Examples (model)
• -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. /2318
•Worst-case consensus time is known for .
•We use this technique.
Kn
Proof Outline
•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. /2319
• Let and .α = |A|/n α′ = |A′|/n
Functional Voting on Kn
•By the Chernoff bound, it holds w.h.p. that
α′ = E[α′] ± 10 log n/n .
α0 1Hf(α)
= Hf(α) ± 10 log n/n .

20. /2320
Functional Voting on 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 holds w.h.p.
forHf(x) > x x ∈ (1/2,1)

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. /2322
•We evaluate and on -expander graphs:E[α′] Var[α′] λ
Our Ingredient
E[α′] = Hf(α) ± O(λ2
)
Var[α′] =
f(1/2)(1 − f(1/2)) + o(1)
n
for any with
.
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. /2323
•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′]
Conclusion