Majority Determination in Binary-Valued Communication Networks

Majority determination is one of the fundamental problems in multiagent systems. It aims to cooperatively and distributedly determine the majority opinion of agents in a network, where the agents initially vote “in favor of” or “opposed” a proposal. An interesting aspect of this issue is to clarify the lowest resolution of communication required among the agents to determine the majority. In this study, we address this problem with binary-valued communication. To overcome the limitation of the finite capacity of communication channels, we exploit randomized communication, i.e., sending binary values (0 or 1), which are selected according to a probabilistic distribution. Based on this idea, we develop consensus-type algorithms that approximately solve the problem with an arbitrarily prescribed accuracy.


I. INTRODUCTION
Majority determination is one of the fundamental problems in multi-agent systems. The problem is quite simple: when the agents initially vote "in favor" or "opposed" for a proposal, how can they cooperatively and distributedly determine the majority opinion? Such cooperative decisionmaking is required for several applications, especially in distributed systems, e.g., system-level diagnosis, fault tolerance enhancement, database management, and fault-local mending [1]. Moreover, it typically appears in the decision-making algorithms of duplicated systems.
Majority determination is closely related to the so-called average consensus [2]. If the opinions in favor and opposed are encoded as 1 and 0, respectively, it is obvious that the majority is determined by comparing the average of the encoded opinions with the value 0.5. Therefore, a promising method for majority determination is to construct an average consensus algorithm for multi-agent systems. On the other hand, it should be noted that majority determination is a weaker objective than average consensus.
Here, we are interested in the lowest resolution of communication required among agents for majority determination. This demonstrates its potential for practical use.
Finally, we remark two things about our contribution. First, if each agent has a sufficiently large memory, it does not matter that the communication is binary-valued. This is because each agent can send any information as a sequence of binary numbers to an agent and the receiver agent can save the sequence in its large memory. In contrast, in our case with a limited amount of memory, binary-valued communication becomes a considerable limitation. Second, the idea of randomized communication has been employed in some earlier studies [5], [9]- [11]. However, to the best of our knowledge, a method for majority determination with binary-valued communication has never been presented. In this study, we have established a framework with binary-valued communication, which is used to reveal the lowest resolution of communication. In this sense, our result is distinguished from the others.
This paper is based on our preliminary version [18], published in a conference proceedings. This journal version contains a full explanation about our result and complete proofs, which are omitted in the conference version.
Notation: Let R, R + , and Z 0+ be a set of real numbers, a set of positive real numbers, and a set of nonnegative integers, respectively. We use 0 n×m and 1 n to represent an n × m zero matrix and an n-dimensional vector whose elements are all one, respectively. For a matrix M , round(M ) represents the matrix obtained by rounding off each of its elements to an integer. The matrix M is said to be irreducible if there exists no permutation matrix P such that P −1 M P is a block upper-triangular matrix. The cardinality of a finite set S is expressed by |S|. The probability of an event A to occur is represented by P [A]. The expectation and variance of a random variable x are denoted by E[x] and V [x], respectively. The conditional expectation of x with respect to a random variable y is expressed as E [x|y]. Further, the conditional expectation of x with respect to a filtration F is denoted by is the expectation of f (x) with respect to the random variable x. Finally, "w.p.p" stands for "with probability p" in this paper.

A. System Description
Consider a multi-agent system with n agents in which the dynamics of agent i ∈ {1, 2, . . . , n} is governed by where x i (t) ∈ R is the state, u i (t) ∈ Z 0+ is the input, and y i (t) ∈ {0, 1} is the binary-valued output, and f i : R×Z 0+ → R and g i : R × Z 0+ → {0, 1} are functions. Agent i receives the information of the sum of the outputs of the neighbors as follows: where N i ⊆ {1, 2, . . . , n} is the index set of the neighbors of agent i, i.e., the agents connected to agent i. Note that, in general, under (2), agent i cannot know the outputs of individual neighbors, which protects the privacy of the neighbors. The network structure of the above multi-agent system is represented by a directed graph G = (V, E) with the node set V := {1, 2, . . . , n} and the edge set We use ∆ ∈ {0, 1, . . . , n} to express the maximum in-degree of the network structure G, i.e., ∆ := max i∈V |N i |.

B. Majority Determination Problem
In the multi-agent system, we assume that the agents initially vote opposed or in favor for some proposal, and the opinion of agent i is stored in its initial state x i (0) as a binary value (0 or 1). The values 0 and 1 represent opposed and in favor, respectively.
The sets of agents who voted opposed and in favor are denoted by I 0 ⊆ V and I 1 ⊆ V, respectively, i.e., I 0 := {i ∈ V|x i (0) = 0}, I 1 := {i ∈ V|x i (0) = 1}, and |I 0 | + |I 1 | = n. The sets I 0 and I 1 are called the opposition group and the supportive group, respectively.
for every i ∈ V. In this problem, (S1) specifies that the agents basically follow the same type of algorithm for scalability reasons. However, it may be noted that each agent is connected to a different number of neighbors, as mentioned in Section II-A. Consequently, it can design f i and g i depending on the number |N i | of the neighbors. Such a specification is typical in control problems in multi-agent systems, such as consensus control and coverage control. Meanwhile, (S2) is concerned with determining the majority opinion. If (S2) holds, the states of all the agents remain either less than or more than γ after a while. Thus the majority opinion can be determined by checking the relationship between the state of any agent and γ after a certain time, e.g., after a certain period when the state remains either less than or more than γ.

III. DISTRIBUTED ALGORITHM FOR MAJORITY DETERMINATION A. Algorithm Based on Randomized Communication
If a real-valued output is available for (1), Problem 1 can be easily solved by constructing an average consensus algorithm. In fact, the average consensus for x i (0) ∈ {0, 1} (i = 1, 2, . . . , n) indicates (3) for γ = 0.5. However, the output has a binary value in our problem. Thus, we propose an algorithm with randomized communication, which results in an approximation of the typical consensus algorithm.
The proposed algorithm is given as follows: where ε(t) ∈ [0, ∆] is the time-varying gain of this algorithm, u i (t) is the neighbors' information given in (2), and y i (t) is a random variable obtained from the Bernoulli distribution. Note that (S1) in Problem 1 holds for this algorithm. This algorithm is called the majority determination algorithm. This algorithm is in a similar form as that of the typical consensus algorithm. In fact, (2) is equivalent to which enables us to rewrite the state equation in (4) as However, it should be noted that y j (t) (j ∈ N i ) are binaryvalued and random.
In this algorithm, x i (t) must take a value from [0, 1] because the probability distribution of the random variable Lemma 1: Consider the multi-agent system with the majority determination algorithm in (4).
Proof: This lemma can be proved by mathematical induction. In particular, by noting that For an arbitrary i ∈ V, consider the state equation in (4). If Applying these facts to the state equation in (4), we obtain On the other hand, Thus, x i (t + 1) ∈ [0, 1], which completes the proof.

B. Example
Before showing the theoretical results for the majority determination algorithm in (4), we demonstrate the performance of the algorithm through simulations.
Consider a multi-agent system with n = 100 and network structure G, as shown in Fig. 1. For G, the maximum indegree ∆ is 42 and the average in-degree (1/100) This graph is randomly generated in such a way that its adjacent matrix is given by round(M + M ⊤ ) for a random matrix M ∈ R 100×100 whose diagonal elements are zero and off-diagonal elements are independently drawn from the uniform distribution on [0, 0.4]. Note that the matrix M + M ⊤ is symmetric and the graph is undirected.
For this system, we apply the majority determination algorithm in (4) with the gain Fig . 2 shows the result for the case where 59% of the agents are in favor and 41% oppose. It is clear that the states x i (t) (i = 1, 2, . . . , 100) converge to a value between 0.5 and 0.6. On the other hand, Fig. 3 shows the result for the case where 37% agents in favor and 63% oppose. Here, the states x i (t) (i = 1, 2, . . . , 100) converge to a value between 0.35 and 0.45.
These results suggest that (3) is satisfied for γ = 0.5, and the algorithm solves Problem 1 in the above cases.

IV. PERFORMANCE ANALYSIS OF MAJORITY DETERMINATION ALGORITHMS
Here, we present theoretical results for validating the performance of the majority determination algorithm in (4).

A. Collective Dynamics and Properties
Let us first derive the collective dynamics of the multi-agent system. For the network structure G, the adjacency matrix, the degree matrix, and the graph Laplacian are denoted by A ∈ R n×n , D ∈ R n×n , and L ∈ R n×n , respectively, i.e., We introduce the following variables to represent the difference between x i (t) and y i (t): Using these variables and (2), the state equation of (4) can be rewritten as Combining the above equations for i = 1, 2, . . . , n, we eventually obtain where It is clear that (6) corresponds to the collective dynamics of the typical discretetime consensus algorithm, but with a time-varying gain ε(t) and an additional term ε(t)Aw(t).
By considering the probabilistic properties of w(t) (given in Appendix II), we can derive several results on the steadystate behavior of the multi-agent system with the majority determination algorithm in (4).
First, the following result is related to the consensus of x i (t) (i = 1, 2, . . . , n).
Lemma 2: For the multi-agent system with the majority determination algorithm in (4), suppose that the opinions x i (0) ∈ {0, 1} (i = 1, 2, . . . , n) are fixed. If (A1) the network structure G is strongly connected, (A2) the gain ε(t) is given by . This lemma guarantees that the states x i (t) (i = 1, 2, . . . , n) reach the consensus almost surely, i.e., for every (i, j) ∈ V×V. However, the result does not indicate that the states converge to a constant value. Now, we present the results for the average of the states x i (t) (i = 1, 2, . . . , n), i.e., Lemma 3: For the multi-agent system with the majority determination algorithm in (4), suppose that the opinions x i (0) ∈ {0, 1} (i = 1, 2, . . . , n) are given. If (A1') the network structure G is balanced, and (A2) holds, then hold for every t ∈ {1, 2, . . .}, where is the average of the squares of the ratio of in-degree to the maximum in-degree.
Proof: See Appendix IV. This lemma provides the expectation and variance of the average of the states. If the states x i (t) (i = 1, 2, . . . , n) reach the consensus (i.e., (7) holds for every (i, j) ∈ V × V), this result facilitates the characterization of the limit of the states 1, 2, . . . , n).
We comment on the tightness of the upper bound in (10). As easily seen in the proof in Appendix IV-B, the variance Then 0 ≤ E[(w i (τ )) 2 ] ≤ 1/4 for every i ∈ V and τ ∈ Z 0+ . The upper bound 1/4 will be given in Lemma 5 (iii) and it is a tight bound in the sense that E[(w i (τ ) (4) and (5)). Note that x i (τ ) = 0.5 often occurs, e.g., as illustrated in Figs. 2 and 3. Thus the upper bound in (10) is loose for the replacement of E[(w i (τ )) 2 ] by 1/4.

B. Performance of Majority Determination Algorithm
Based on Lemmas 2 and 3, the following result is obtained. Theorem 1: For the multi-agent system with the majority determination algorithm in (4), suppose that the opinions 1, 2, . . . , n) are fixed. Moreover, suppose that σ ∈ (0, 1] and θ ∈ (0, 1) are arbitrarily given. If (A1), (A1'), and (A2) are true, and (A3) either the opposition group I 0 or the supportive group I 1 outnumbers the other in the sense that (A4) the number n of agents is sufficiently large such that then there exists a time T ∈ Z 0+ such that for every i ∈ V.
Proof: See Section IV-C. Theorem 1 provides an approximate solution to Problem 1 in the sense that the states of all the agents remain less than or more than 0.5 after some time T in a probabilistic sense. In fact, sup t≥T x i (t) < 0.5 implies that lim sup t→∞ x i (t) < 0.5 and its converse is also true. Therefore, (14) means that (3) holds with probability greater than or equal to θ. Thus, if θ is sufficiently large, there is a sufficiently high probability that the majority opinion of x i (0) (i = 1, 2, . . . , n) is determined. Hence, the parameter θ corresponds to the level of confidence of the solution. Meanwhile, the parameter σ corresponds to the resolution of distinguishing two groups: if σ is smaller, the algorithm can be applied to the case in which the two groups have more identical sizes. For this result, three remarks are given. First, the parameters σ and θ, which specify the accuracy of this algorithm, can be arbitrarily selected. However, as shown in (A4), if the gain sequence ε(t) (t = 0, 1, . . . ,) is fixed (i.e., ∞ t=1 (c(t)) 2 is fixed in (13)), the applicable size (i.e., n) of the multi-agent system becomes more limited as σ → 0 and θ → 1. Table I shows the lower bounds of σ satisfying (13) for δ = 0.75 2 and ∞ t=1 (c(t)) 2 = 0.9. If (A4) does not hold for given parameters n, δ, σ, θ, and c(t), an alternative is to reselect the design parameter c(t) so as to satisfy (13). In fact, c(t) in the form of satisfies the three conditions in (A2) and ∞ t=1 (c(t)) 2 < 2c 1 subject to 0 < c 1 < c 0 . Thus, by appropriately selecting c 1 , we can satisfy (13) for the given parameters. In this sense, the proposed method approximately solves the problem with an arbitrarily prescribed accuracy.
Second, the convergence rate of x i (t), which is useful for the (rough) estimation of the length of T , is given as follows. If c(t) is given by (15) and x(t) converges to the set span(1 n ) w.p.1. as shown in Lemma 2, the convergence rate is This is the straightforward consequence from the convergence rate analysis of classical stochastic approximation (Robbins-Monro algorithms) for finding roots (see, e.g., [20], [21]) and the fact that (6) is a Robbins-Monro algorithm as shown in Appendix III. It should be remarked that the convergence rate does not depend on the network structure unlike the typical consensus algorithm whose convergence rate depends on the network structure (i.e., the eigenvalues of graph Laplacian). This property may be useful in the sense that the convergence rate is known in advance even when we do not have the exact information on the network structure.
Finally, in Figs. 2 and 3, it seems that the states do not reach the consensus, although the consensus is guaranteed by Lemma 2. However, it is not the case. This is because the convergence rate is O(1/ √ t) as mentioned above and it is not so fast. In fact, we can observe the consensus after a long time in both the cases; for example, in the case of Fig. 2, we obtain max (i,j)∈{1,2,...,n} |x i (300) − x j (300)| ≃ 0.047 and max (i,j)∈{1,2,...,n} |x i (3000) − x j (3000)| ≃ 0.011.

V. APPLICATION TO ANOMALY DETECTION BY SENSOR NETWORK
An application of majority determination is anomaly detection by a sensor network. In this section, we demonstrate our framework through this application.
Consider a sensor network composed of n sensor nodes, which aims at detecting anomaly of a target system. In this network, binary-valued communication is available among nodes. At a certain moment, each node measures the state of the system, which is either normal or anomaly; however, the measurements are not accurate, e.g., in the sense that the measurement is false with a probability. In this case, it is reasonable to exploit the majority of the measurements as the output of the sensor network. This is exactly our case. Now, let us illustrate how the above anomaly detection is performed by the majority determination algorithm in (4). Consider a sensor network with n = 50 and network structure G whose adjacency matrix is given by The network structure G is illustrated in Fig. 4, where node 1 bidirectionally communicate with all other nodes, node i + 1 is connected to node i (i = 2, 3, . . . , 49), and node 50 is connected to node 1. Note that G is directed but balanced, ∆ = 49, and δ ≃ 0.0216. The measurement of sensor node i is stored in its initial state x i (0) as a binary value. The values 0 and 1 represent normal and anomaly, respectively. It is assumed that we have the prior information σ = 0.65 for the collective measurements. For determining the majority in the sense of (14) with the confidence level θ = 0.95, we apply the majority determination algorithm in (4) with the gain In this case, the right hand side of (13) is less than 47.11, which implies that (13) holds. Fig. 5 shows the result for the case where 10% of the measurements are normal and 90% anomaly. It is observed that the states x i (t) (i = 1, 2, . . . , 50) are more than 0.5, from which the sensor network outputs anomaly. Note that the consensus among nodes is achieved as guaranteed by Lemma 2 but the consensus value is not always equal to the average of x i (0) (i = 1, 2, . . . , n) (see (9) and (10)).
In this way, our framework is useful for cooperative decision making in sensor networks. solved the problem with an arbitrarily prescribed accuracy. We also clarified the relationship among the number of agents, the distribution of opinions, and the accuracy of the method. Our results revealed that the lowest resolution required for majority determination is two-level. In future, we hope to extend our framework to the case with malicious agents. Moreover, it may be interesting to handle time-varying opinions.

A. Cooperativity and Irreducibility of Vector-valued Functions
We introduce the notions of cooperativity and irreducibility for vector-valued functions, according to Section 4 in [22].
Definition 1: Consider a function h : R n → R n . (i) The function h is said to be cooperative if it is continuously differentiable and the off-diagonal elements of the Jacobian matrix ∂h(x)/∂x are nonnegative for every x ∈ R n . (ii) The function h is said to be irreducible if the Jacobian matrix ∂h(x)/∂x is irreducible for every x ∈ R n .

B. Robbins-Monro Algorithm and Convergence
The Robbins-Monro algorithm is given as where x(t) ∈ R n is the state, a(t) ∈ R is the time-varying gain, e(t) ∈ R n is a random vector, and h : R n → R n is a function.
The following result is a straightforward consequence of Theorem 2 in [23], Theorems A and 4.4, and Corollary 4.6 in [22].
Lemma 4: Consider the Robbins-Monro algorithm in (20). If the following conditions hold, then the state x(t) converges to the set of zeros of the function h.

APPENDIX II PROBABILISTIC PROPERTIES OF VARIABLES w i (t)
It is important to clarify the probabilistic properties of the variables w i (t) (i = 1, 2, . . . , n) in (5). The following lemma provides the expectations of w i (t) and their products.
Proof: (i) Considering that w i (t) only depends on x i (t) according to (4), it is straightforward to calculate E[w i (t)] using (4) and (5). In fact, (ii) We first consider the former case, i.e., i = j and t 1 = t 2 .

APPENDIX III PROOF OF LEMMA 2
Here, we prove Lemma 2. The collective dynamics in (6) corresponds to the Robbins-Monro algorithm in (20) (Appendix I) by considering −Lx(t), Aw(t), and ε(t) as h(x(t)), e(t), and a(t), respectively. Thus, we show that the conditions (B1)-(B4) hold for h(x(t)) := −Lx(t), e(t) := Aw(t), and a(t) := ε(t). (B1) Since the function −Lx is linear with respect to x, −Lx is Lipschitz. Next, the off-diagonal elements of any graph Laplacian are nonpositive because L = D − A, D is diagonal, and the elements of A are nonnegative [19]. Thus, the off-diagonal elements of the Jacobian of −Lx, i.e., −L, are nonnegative for every x ∈ R n , which proves that the Jacobian of −Lx is cooperative. Finally, it is well established that the matrix −L is irreducible under (A1). Consequently, (B1) holds for h(x(t)) := −Lx(t). (B2) Condition (A2) implies (B2).