Quasi-majority Functional Voting on Expander Graphs

Consider a distributed graph where each vertex holds one of two distinct opinions. In this paper, we are interested in synchronous voting processes where each vertex updates its opinion according to a predefined common local updating rule. For example, each vertex adopts the majority opinion among 1) itself and two randomly picked neighbors in best-of-two or 2) three randomly picked neighbors in best-of-three. Previous works intensively studied specific rules including best-of-two and best-of-three individually. In this paper, we generalize and extend previous works of best-of-two and best-of-three on expander graphs by proposing a new model, quasi-majority functional voting. This new model contains best-of-two and best-of-three as special cases. We show that, on expander graphs with sufficiently large initial bias, any quasi-majority functional voting reaches consensus within $O(\log n)$ steps with high probability. Moreover, we show that, for any initial opinion configuration, any quasi-majority functional voting on expander graphs with higher expansion (e.g., Erd\H{o}s-R\'enyi graph $G(n,p)$ with $p=\Omega(1/\sqrt{n})$) reaches consensus within $O(\log n)$ with high probability. Furthermore, we show that the consensus time is $O(\log n/\log k)$ of best-of-$(2k+1)$ for $k=o(n/\log n)$.


Introduction
Consider an undirected graph G = (V, E) where each vertex v ∈ V initially holds an opinion σ ∈ Σ from a finite set Σ.In synchronous voting process (or simply, voting process), in each round, every vertex communicates with its neighbors and then all vertices simultaneously update their opinions according to a predefined protocol.The aim of the protocol is to reach a consensus configuration, i.e., a configuration where all vertices have the same opinion.Voting process has been extensively studied in several areas including biology, network analysis, physics and distributed computing [10,33,31,23,27,2].For example, in distributed computing, voting process plays an important role in the consensus problem [23,27].
This paper is concerned with the consensus time of voting processes over binary opinions Σ = {0, 1}.Then voting processes have state space 2 V .A state of 2 V is called a configuration.The consensus time is the number of steps needed to reach a consensus configuration.

Previous works of specific updating rules
In pull voting, in each round, every vertex adopts the opinion of a randomly selected neighbor.This is one of the most basic voting process, which has been well explored in the past [34,28,14,18,8].In particular, the expected consensus time of this process has been extensively studied in the literature.For example, Hassin and Peleg [28] showed that the expected consensus time is O(n3 log n) for all non-bipartite graphs and all initial opinion configurations, where n is the number of vertices.From the result of Cooper, Elsässer, Ono, and Radzik [14], it is known that on the complete graph K n , the expected consensus time is O(n) for any initial opinion configuration.
In best-of-two (a.k.a.2-Choices), each vertex v samples two random neighbors (with replacement) and, if both hold the same opinion, v adopts the opinion.Otherwise, v keeps its own opinion.Doerr, Goldberg, Minder, Sauerwald, and Scheideler [21] showed that, on the complete graph K n , the consensus time of best-of-two is O(log n) with high probability 1 for an arbitrary initial opinion configuration.Since best-of-two is simple and is faster than pull voting on the complete graphs, this model gathers special attention in distributed computing and related area [26,15,16,17,19,20,39].There is a line of works that study best-of-two on expander graphs [15,16,17], which we discuss later.
In best-of-three (a.k.a.3-Majority), each vertex v randomly selects three random neighbors (with replacement).Then, v updates its opinion to match the majority among the three.It follows directly from Ghaffari and Lengler [26] that, on K n with any initial opinion configuration, the consensus time of best-of-three is O(log n) w.h.p.Kang and Rivera [29] considered the consensus time of best-of-three on graphs with large minimum degree starting from a random initial configuration.Shimizu and Shiraga [39] showed that, for any initial configurations, best-of-two and best-of-three reach consensus in O(log n) steps w.h.p. if the graph is an Erdős-Rényi graph G(n, p)2 of p = Ω (1).
Best-of-k (k ≥ 1) is a generalization of pull voting, best-of-two and best-of-three.In each round, every vertex v randomly selects k neighbors (with replacement) and then if at least k/2 + 1 of them have the same opinion, the vertex v adopts it.Note that the best-of-1 is equivalent to pull voting.Abdullah and Draief [1] studied a variant of best-of-k (k ≥ 5 is odd) on a specific class of sparse graphs that includes n-vertex random d-regular graphs 3 G n,d of d = o( √ log n) with a random initial configuration.To the best of our knowledge, best-of-k has not been studied explicitly so far.
In Majority (a.k.a.local majority), each vertex v updates its opinion to match the majority opinion among the neighbors.This simple model has been extensively studied in previous works [6,9,25,35,36,41].For example, Majority on certain families of graphs including the Erdős-Rényi random graph [6,41], random regular graphs [25] have been investigated.See [36] for further details.
Voting process on expander graphs.Expander graph gathers special attention in the context of Markov chains on graphs, yielding a wide range of theoretical applications.A graph G is λexpander if max{|λ 2 |, |λ n |} ≤ λ, where 1 = λ 1 ≥ λ 2 ≥ • • • ≥ λ n ≥ −1 are the eigenvalues of the transition matrix P of the simple random walk on G.For example, an Erdős-Rényi graph G(n, p) of p ≥ (1 + ) log n n for an arbitrary constant > 0 is O(1/ √ np)-expander w.h.p. [12].An n-vertex [13,40].Cooper et al. [14] showed that the expected consensus time of pull voting is O(n/(1 − λ)) on λ-expander regular graphs for any initial configuration.Compared to pull voting, the study of bestof-two on general graphs seems much harder.Most of the previous works concerning best-of-two on expander graphs put some assumptions on the initial configuration.Let A denote the set of vertices of opinion 0 and B = V \ A. Cooper, Elsässer, and Radzik [15] showed that, for any regular λ-expander graph, the consensus time is O(log n) w.h.p. if |A| − |B| = Ω(λn).This result was improved by Cooper, Elsässer, Radzik, Rivera, and Shiraga [16].Roughly speaking, they proved that, on λ-expander graphs, the consensus time is , where d(S) = v∈S deg(v) denotes the volume of S ⊆ V .To the best of our knowledge, the worst case consensus time of best-of-k on expander graphs has not been studied.

Our model
In this paper, we propose a new class functional voting of voting process, which contains many known voting processes as a special case.Let A ⊆ V be the set of vertices of opinion 0 and A be the set in the next round.
A functional voting with respect to f is a synchronous voting process defined as We call the function f a betrayal function and the function an updating function.
Since f (0) = 0, consensus configurations are absorbing states.Hence the consensus time is well-defined 4 .The intuition behind the updating function H f is that, letting α = |A|/n and α = |A |/n, on a complete graph K n (with self-loop), the functional voting with respect to f satisfies . Functional voting contains many existing models as special cases.For example, pull voting, bestof-two, and best-of-three are functional votings with respect to x, x 2 and 3x 2 − 2x 3 , respectively.In general, best-of-k is a functional voting with respect to It is straightforward to check that Majority is a functional voting with respect to if a vertex adopts the random opinion when it meets the tie.
x pull voting best-of-three best-of-seven Figure 1: The update functions H f (x) of pull voting (solid line), best-of-three (dashed line) and best-of-seven (dotted line).One can easily observe that best-of-three and best-of-seven are quasimajority functional voting.Intuitively speaking, quasi-majority functional voting processes have updating functions H f with the property so-called "the rich get richer", which coincides with Definition 1.2.
Quasi-majority functional voting.In this paper, we focus on functional voting with respect to f satisfying the following property.
A voting process is a quasi-majority functional voting if it is a functional voting with respect to a quasi-majority function f .
For each constant k ≥ 2, best-of-k is quasi-majority functional voting but pull voting and Majority are not.Indeed, if It is straightforward to check that this function satisfies the conditions (3) to (5) if = 0 (pull-voting).See Figure 1 for depiction of the updating functions of pull voting, best-of-three and best-of-seven.

Our result
In this paper, we study the consensus time of quasi-majority functional voting on expander graphs 5 .Let T cons (A) denote the consensus time starting from the initial configuration A ⊆ V .For a graph G = (V, E), let π = (π(v)) v∈V denote the degree distribution defined as Note that v∈V π(v) = 1 holds.We denote by x p := denote the bias between A and V \ A.
Theorem 1.3 (Main theorem).Consider a quasi-majority functional voting with respect to f on an n-vertex λ-expander graph with degree distribution π.Then, the following holds: (i) Let C 1 > 0 be an arbitrary constant and ε : N → R be an arbitrary function satisfying The following result which we show in Section 5 indicates that the consensus time of Theorem 1.3(i) is optimal up to a constant factor.Theorem 1.4 (Lower bound).Under the same assumption of Theorem 1.3(i), T cons (A) = Ω(log n) w.h.p. for some A ⊆ V .Theorem 1.5 (Fast consensus for H f (0) = 0).Consider a quasi-majority functional voting with respect to f on an n-vertex λ-expander graph with degree distribution π.Let C > 0 be a constant depending only on f .Suppose that H f (0) = 0, λ ≤ C and π 2 ≤ C/ √ log n.Then, for any For example, for each constant k ≥ 2, best-of-k is quasi-majority with H f (0) = 0.
Remark 1.6.Roughly speaking, for p ≥ 2, π p measures the imbalance of the degrees.For any graphs, π p ≥ n −1+1/p and the equality holds if and only if the graph is regular.For star graphs, we have π p ≈ 1.
Results of best-of-k.Our results above do not explore Majority since it is not quasi-majority.A plausible approach is to consider best-of-k for k = k(n) = ω(1) since each vertex is likely to choose the majority opinion if the number of neighbor sampling increases.Also, note that the betrayal function f k of best-of-k given in (1)  (i) Suppose that p = Ω(n −1/2 ).Then (a) for any A ⊆ V , T cons (A) = O(log n) w.h.p.
(ii) Suppose that p ≥ (1 + ) log n n for an arbitrary constant > 0.Then, for any w.h.p., where C > 0 is a constant depending only on f .In Corollary 1.8(i), we stress that the worst-case consensus time on G(n, p) was known for p = Ω(1) [39].
Corollary 1.10.Consider a best-of-k on an n-vertex random d-regular graph G n,d for an arbitrary constant k ≥ 2.Then, G n,d w.h.p. satisfies the following: (ii) Suppose that d ≥ C and d ≤ n/2 for a constant C > 0 depending only on f .Then, Then, for any We can apply Theorems 1.3 and 1.5 if the ratio of the maximum and average degree is constant as follows.
Corollary 1.12.Consider a quasi-majority functional voting with respect to f on an n-vertex λexpander graph with degree distribution π.Suppose that d max ≤ C 1 d ave for an arbitrary constant C 1 > 0, where d max and d avr denote the maximum and average degree, respectively.Then, the following holds: (ii) Suppose that λ ≤ C 2 for some constant C 2 > 0 depending only on f .Then, for any (iii) In addition to the same assumption as (ii), suppose that H f (0) = 0.Then, it holds w.h.p. that Let C be an arbitrary constant.Consider best-of-(2k + 1) on an n-vertex λ-expander graph with degree distribution π such that λ ≤ Ck −1/2 n −1/4 , and d max ≤ Cd avr , where d max and d avr denote the maximum and average degree, respectively.Then, Corollaries 1.12 and 1.13 immediately follow from Theorems 1.3 to 1.5 and 1.7 since π 2 = O(n −1/2 ).Note that if the ratio of the maximum degree d max and average degree d avr is constant, We obtain Corollaries 1.8 to 1.11 from Corollaries 1.12 and 1.13.
Other quasi-majority functional voting.We can consider the ρ-lazy variant of a voting process, i.e., every vertex v individually tosses its private coin and operates the voting process with probability ρ, while v does nothing with probability 1 − ρ.Berenbrink, Giakkoupis, Kermarrec, and Mallmann-Trenn [8] studies 1/2-lazy pull voting.If the original voting process is a quasimajority functional voting with respect to f , then the corresponding ρ-lazy variant is quasi-majority functional voting with respect to ρf : x → ρf (x).Indeed, Corollary 1.14.Consider a ρ-lazy quasi-majority functional voting on G(n, p) for an arbitrary constant ρ ∈ (0, 1].Suppose that p = Ω(1/ √ n).Then, for any This implies the following interesting observation.In voting processes, the number of neighbor sampling queries per each vertex at each step affects the performance.In pull voting, each vertex communicates with one neighbor but it has a drawback on the slow consensus time.In best-oftwo, each vertex communicates with two random neighbors and its consensus time is much faster than that of pull voting.In ρ-lazy best-of-two, each vertex queries 2ρ vertices at each round in expectation, that is less queries than pull voting if ρ < 1/2.On the other hand, the consensus time is much faster than pull voting.
Additionally, we can deal with k-careful voting.In this model, each vertex v selects k random neighbors (with replacement), and if these sampled k opinions are the same one, v adopts it.Note that one-careful voting and two-careful voting are equivalent to pull voting and best-of-two, respectively.One can check easily that, for any constant k ≥ 2, this model is a quasi-majority functional voting with respect to f (

Related work
In asynchronous voting process, in each round, a vertex is selected uniformly at random and only the selected vertex updates its opinion.Cooper and Rivera [18] introduced linear voting model.In this model, an opinion configuration is represented as a vector v ∈ Σ V and the vector v updates according to the rule v ← M v, where M is a random matrix sampled from some probability space.This model captures a wide variety model including asynchronous push/pull voting and synchronous pull voting.Note that best-of-two and best-of-three are not included in linear voting model.Schoenebeck and Yu [37] proposed an asynchronous variant of our functional voting.The authors of [37] proved that, if the function f is symmetric (i.e., f , smooth and has "majority-like" property (i.e., f (x) > x whenever 1/2 < x < 1), then the expected consensus time is O(n log n) w.h.p. on G(n, p) with p = Ω(1).This perspective has also been investigated in physics (see, e.g., [10]).Several researchers have studied best-of-two and best-of-three on complete graphs initially involving k ≥ 2 opinions [5,4,7,26].For example, the consensus time of best-of-three is Cooper, Radzik, Rivera, and Shiraga [17] considered best-of-two and best-of-three on regular expander graphs that hold more than two opinions.
Recently, Cruciani, Natale, and Scornavacca [20] studied best-of-two with a random initial configuration on a clustered regular graph.Shimizu and Shiraga [39] obtained phase-transition results of best-of-two and best-of-three on stochastic block models.

Preliminary and technical result 2.1 Formal definition
Let G = (V, E) be an undirected and connected graph.Let P ∈ [0, 1] V ×V be the matrix defined as where 1 Z denotes the indicator of an event Z.For v ∈ V and S ⊆ V , we write P (v, S) = s∈S P (v, s).Now, let us describe the formal definition of functional voting.For a given A ⊆ V , let (X v ) v∈V be independent binary random variables defined as where Note that this definition coincides with Definition 1.
For A ⊆ V , let T cons (A) denote the consensus time of the functional voting starting from the initial configuration A. Formally, T cons (A) is the stopping time defined as

Technical background
Consider best-of-two on a complete graph K n (with self loop on each vertex) with a current configuration A ⊆ V .Let α = |A|/n.We have P (v, A) = α for any v ∈ V and A ⊆ V .Then, for any from the Hoeffding bound.Therefore, the behavior of α can be written as the iteration of applying H f .
The most technical part is the symmetry breaking at α = 1/2.Note that H f (1/2) = 1/2 and thus, the argument above does not work in the case of |α − 1/2| = o( log n/n).To analyze this case, the authors of [21,11] proved the following technical lemma asserting that α w.h.p. escapes from the area in O(log n) rounds.
Intuitively speaking, the condition (ii) means that the bias |α − 1/2| is likely to be at least (1+ )|α−1/2| for some constant > 0. The condition (ii) is easy to check using the Hoeffding bound.The condition (i) means that α has a fluctuation of size Ω(1/ √ n) with a constant probability.We can check condition (i) using the Central Limit Theorem (the Berry-Esseen bound, see Lemma A.5).The Central Limit Theorem implies that the normalized random variable (α − E[α ])/ Var[α ] converges to the standard normal distribution as n → ∞.In other words, α has a fluctuation of size Θ( Var[α ]) with constant probability.Now, to verify the condition (i), we evaluate Var[α ].On K n , it is easy to show that Var[α ] = Θ(1/n), which implies the condition (i).
The authors of [16,17] considered best-of-two on expander graphs.They focused on the behavior of π(A) instead of α.Roughly speaking, they proved that . At the heart of the proof, they showed the following result.Lemma 2.2 (Special case of Lemma 3 of [17]).Consider a λ-expander graph with degree distribution π.Then, for any S ⊆ V , Then, from the Hoeffding bound, we have . Thus, if the initial bias |π(A) − 1/2| is Ω(max{λ 2 , log n/n}), we can show that the consensus time is O(log n).
Unfortunately, we can not apply the same technique to estimate Var[π(A )] on expander graphs, and due to this reason, it seems difficult to estimate the worst-case consensus time on expander graphs.Actually, any previous works put assumptions on the initial bias due to the same reason.It should be noted that Lemma 2.1 is well-known in the literature.For example, Cruciani et al. [20] used Lemma 2.1 from random initial configurations.
The technique of estimating E[π(A )] by Cooper et al. [16,17] is specialized in best-of-two.Thus, it is not straightforward to prove the estimation of E[π(A )] for voting processes other than best-of-two.

Our technical contribution
For simplicity, in this part, we focus on a quasi-majority functional voting with respect to a symmetric function f (i.e., f (1 − x) = 1 − f (x) for every x ∈ [0, 1]) on a λ-expander graph with degree distribution π.For example, f (x) = 3x 2 − 2x 3 of best-of-three is a symmetric function.Note that f = H f if f is symmetric.Similar results mentioned in this subsection holds for non-symmetric f (see Section 3.3).For a C 2 function h : R → R, let be some constants 6 depending only on h.The following technical result enables us to estimate E[π(A )] and Var[π(A )] of functional voting.Lemma 2.3.Consider a functional voting with respect to a symmetric C 2 function f on a λexpander graph with degree distribution π.Let g(x) Note that, if f is symmetric, the corresponding functional voting satisfies that Pr[v ∈ A ] = f (P (v, A)) for any v ∈ V .Thus we have To evaluate E[π(A )] and Var[π(A )] above, we prove the following key lemma that is a generalization of Lemma 2.2 and implies Lemma 2.3.Lemma 2.4 (Special case of Lemmas 3.2 and 3.3).Consider a λ-expander graph with degree distribution π.Then, for any S ⊆ V and any C 2 function h : R → R,

Proof sketch of Theorem 1.3
We present proof sketch of Theorem 1.3(i).From the assumption of Theorem 1.
for some constant 2 > 0. Note that we invoke the property that H f (x) < x whenever 0 < x < 1/2.
• If π(A) ≤ C 3 for sufficiently small constant C 3 , we use the Markov inequality to show π(A t ) = O(n −3 ) w.h.p. for some t = O(log n).Since π(A) ≥ 1/n 2 whenever A = ∅, this implies that the consensus time is O(log n) w.h.p.Note that, since for some constant

Technical tools for reversible Markov chains
To begin with, we briefly summarize the notation of Markov chain, which we will use in this section 7 .Let V be a set of size n.A transition matrix P over V is a matrix P ∈ [0, 1] V ×V satisfying v∈V P (u, v) = 1 for any u ∈ V .Let π ∈ [0, 1] V denote the stationary distribution of P , i.e., a probability distribution satisfying πP = π.A transition matrix P is reversible if π(u)P (u, v) = π(v)P (v, u) for any u, v ∈ V .It is easy to check that the matrix ( 4) is a reversible transition matrix and its stationary distribution is (3).Let λ 1 ≥ • • • ≥ λ n denote the eigenvalues of P .If P is reversible, it is known that λ i ∈ R for all i.Let λ = max{|λ 2 |, |λ n |} be the second largest eigenvalue in absolute value 8 .For a function h : R → R and subsets S, T ⊆ V , consider the quantity Q h (S, T ) defined as The special case of h(x) = x, that is, Q(S, T ) := v∈S π(v)P (v, T ), is well known as edge measure [30] or ergodic flow [3,32].Note that, for any reversible P and subsets S, T ⊆ V , Q(S, T ) = Q(T, S) holds.The following result is well known as a version of the expander mixing lemma.Lemma 3.1 (See, e.g., p.163 of [30]).Suppose P is reversible.Then, for any S, T ⊆ V , We show the following lemma which gives a useful estimation of Q h (S, T ).Lemma 3.2.Suppose P is reversible.Then, for any S, T ⊆ V and any C 2 function h : R → R, Proof of Lemma 3.2.From Taylor's theorem, it holds for any x, y ∈ [0, 1] that Hence Note that the last inequality follows from Corollary A.2.

Next, consider
for a function h : R → R and S, T ⊆ V .For notational convenience, for S ⊆ V , let π 2 (S) := v∈S π(v) 2 .We show the following lemma that evaluates R h (S, T ).Lemma 3.3.Suppose that P is reversible.Then, for any S, T ⊆ V and any C 2 function h : R → R, Proof.We first observe that holds for any x, y ∈ [0, 1] from Taylor's theorem.Hence, Then, applying the Cauchy-Schwarz inequality and Corollary A.2, and we obtain the claim.
Remark 3.4.The results of this paper can be extended to voting processes where the sampling probability is determined by a reversible transition matrix P .This includes voting processes on edge-weighted graphs G = (V, E, w), where w : E → R denotes an edge weight function.Consider the transition matrix P defined as follows: P (u, v) = w({u, v})/ x:{u,x}∈E w({u, x}) for {u, v} ∈ E and P (u, v) = 0 for {u, v} / ∈ E. A weighted functional voting with respect to f is determined by For simplicity, in this paper, we do not explore the weighted variant and focus on the usual setting where P is the matrix (4) and its stationary distribution π is (3).

Proof of Lemma 2.4
For the first inequality, by substituting V to S of Lemma 3.2, we obtain Note that Q(V, T ) = Q(T, V ) = π(T ) from the reversibility of P .Similarly, we obtain the second inequality by substituting V to S of Lemma 3.3.

Non-symmetric functions
This section is devoted to evaluate E[π(A )] and Var[π(A )] for non-symmetric f .To be more specifically, we prove the following.
Lemma 3.5.Consider a functional voting with respect to a C 2 function f on a λ-expander graph.
Then, for all A ⊆ V , Lemma 3.6.Consider a functional voting with respect to a C 2 function f on a λ-expander graph.Let g(x) Recall that we use Proof of Lemma 3.5.From Definition 1.1, ( 6) and ( 9), we have For notational convenience, for S, T ⊆ V , let The equality follows from the reversibility of P (see Section 3).From Lemma 3.2, we have Then, combining ( 11) and ( 12), we have and we obtain the claim.Note that the last inequality follows from Taylor's theorem (8) and Lemma 3.1.
4 Proofs of Theorems 1.3 and 1.5 Consider a quasi-majority functional voting with respect to f on an n-vertex λ-expander graph with degree distribution π.Let A 0 , A 1 , . . ., be the sequence given by the functional voting with initial configuration A 0 ⊆ V .Theorems 1.3 and 1.5 follow from the following lemma.(I) Let C 1 > 0 be an arbitrary constant and ε : N → R be an arbitrary function satisfying Proof of Theorem 1.3(ii).Since π 2 ≥ 1/ √ n, we have |δ(A 0 )| = Ω( log n/n).This implies that Phase (II) takes at most O(log n).Thus, we obtain the claim since we can merge Phases (II) to (IV) by taking appropriate constants c 2 , c 3 in Phase (III).
Proof of Theorem 1.3(i).Under the assumption of Theorem 1.3(i), for any positive constant C, a positive constant C exists such that . Thus, we can combine Phase (I) and Theorem 1.3(ii), and we obtain the claim.

Proof of Lemma 4.1
For notational convenience, let

Phase (I):
We use the following lemma to show Lemma 4.1(I).
Lemma 4.2 (Lemma 4.5 of [11]).Consider a Markov chain (X t ) ∞ t=1 with finite state space Ω and a function Ψ : Ω → {0, . . ., n}.Let C 3 be arbitrary constant and m = C 3 √ n log n.Suppose that Ω, Ψ and m satisfies the following conditions: (i) For any positive constant h, there exists a positive constant (ii) Three positive constants γ, C 2 and h exist such that, for any Then, Ψ(X t ) ≥ m holds w.h.p. for some t = O(log n).
Let us first prove the following lemma concerning the growth rate of |δ|, which we will use in the proofs of (I) and (II) of Lemma 4.1.
Lemma 4.3.Consider a quasi-majority functional voting with respect to f on an n-vertex λexpander graph with degree distribution π.
be positive constants depending only on f .Suppose that λ ≤ h (f ) 2K(f ) .Then, for any A ⊆ V satisfying Proof.Combining Lemma 3.5 and Taylor's theorem, we have Note that Hence, it holds that We observe that, for any κ > 0, from Corollary A.4.Note that δ = v∈V π(v)(2X v −1) for independent indicator random variables (X v ) v∈V (see (5) for the definition of X v ).Thus, and we obtain the claim.
Proof of Lemma 4.1(I).We check the conditions (i) and (ii) of Lemma 4.
) is a positive constant depending only on f .
Proof of the claim.From Lemma 3.6 and assumptions, we have for any x ∈ R, where Φ(x) = 1 √ 2π x −∞ e −y 2 /2 dy.Thus, for any constant h > 0, there exists some constant C > 0 such that which verifies the condition (i).
Thus, we can apply Lemma 4.3 and positive constants γ, C exist such that, for any h Note that π 2 2 = Θ(1/n) from the assumption.This verifies the condition (ii).Thus, we can apply Lemma 4.2 and we obtain the claim.

Phase (II
Proof of Lemma 4.

Phase (IV):
We show the following lemma which is useful for proving (IV) and (V) of Lemma 4.1.
Thus, for any α 0 ≤ α * , we have This implies that, and we obtain the claim.
5 Proof of Theorem 1.4 This section is devoted to prove Theorem 1.4.In particular, we show the following theorem.
6 Proof of Theorem 1.7 We show Theorem 1.7.The proof is almost same as the one given in Section 4 but we need some special care.We assume k = ω(1) and thus k is sufficiently large.Consider best-of-(2k + 1) on an n-vertex λ-expander graph with degree distribution π.Suppose that the graph satisfies the conditions of Theorem 1.7.Let A 0 , A 1 , . . ., be the sequence given by the best-of-(2k + 1) with initial configuration A 0 ⊆ V .For notational convenience, let α := π(A), α := π(A ), α t := π(A t ), The dynamics of best-of-(2k + 1) are divided into four phases.More specifically, we prove the following key result that corresponds to Lemma 4.1.Lemma 6.1.Consider best-of-(2k + 1) on an n-vertex λ-expander graph with degree distribution π.Suppose that the graph satisfies the conditions of Theorem 1.7.Then, the following holds: Proof of Theorem 1.7 using Lemma 6.1.Theorem 1.7 is straightforward from Lemma 6.1.For any initial configuration A 0 ⊆ V , A 0 satisfies one of (I) to (IV).If A 0 satisfies (IV), the consensus time is O(log n/ log k).Otherwise, from Lemma 6.1, for some t = O(log n/ log k), A t satisfies |δ(A t )| > 0.9 and then apply Lemma 6.1(IV).
The rest of this section is devoted to prove Lemma 6.1.We begin with preparing useful facts concerning with best-of-(2k + 1).Let f 2k+1 be the betrayal function of best-of-(2k + 1).Then, we have for sufficiently large k.Here, we used Var where Thus, from the Hoeffding bound (Lemma A.3), it holds w.h.p. that On the other hand, it is routine to check the following facts.
We begin with proving the following result that corresponds to Lemma 4.3.
6.4 Phase (IV): 0.9 < |δ| ≤ 1 We may assume π(A 0 ) ≤ 0.1 without loss of generality.We claim that π(A t ) < 1 n 2 for some t = O(log n/ log k), which implies In the second part of the last inequality, we assume that k ≥ 2; hence, it holds that r a ≤ r a for 0 and thus, by the Markov inequality, Finally, we consider τ ( C 5 log n/ log k ).Let W 0 , W 1 , . . .be binary random variables defined as In the fifth inequality, we used the union bound over the choice for Ŵt .Note that 1 − Ŵt = 1 with probability C 1 √ k .

Conclusion
In this paper we propose functional voting as a generalization of several known voting processes.We show that the consensus time is O(log n) for any quasi-majority functional voting on O(n −1/2 )expander graphs with balanced degree distributions.This result extends previous works concerning voting processes on expander graphs.Possible future direction of this work includes 1.Does O(log n) worst-case consensus time holds for quasi-majority functional voting on graphs with less expansion (i.e., λ = ω(n −1/2 ))?
2. Is there some relationship between best-of-k and Majority?Proof.For each i ∈ [n], let Z i .