118 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 26, NO. 1, FEBRUARY Boolean Gossip Networks

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1 118 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 26, NO. 1, FEBRUARY 2018 Booean Gossip Networs Bo Li, Junfeng Wu, Hongsheng Qi, Member, IEEE, Aexandre Proutiere, and Guodong Shi, Member, IEEE Abstract This paper proposes and investigates a Booean gossip mode as a simpified but non-trivia probabiistic Booean networ. With positive node interactions, in view of standard theories from Marov chains, we prove that the node states asymptoticay converge to an agreement at a binary random variabe, whose distribution is characterized for arge-scae networs by mean-fied approximation. Using combinatoria anaysis, we aso successfuy count the number of communication casses of the positive Booean networ expicity in terms of the topoogy of the underying interaction graph, where remaraby minor variation in oca structures can drasticay change the number of networ communication casses. With genera Booean interaction rues, emergence of absorbing networ Booean dynamics is shown to be determined by the networ structure with necessary and sufficient conditions estabished regarding when the Booean gossip process defines absorbing Marov chains. Particuary, it is shown that for the majority of the Booean interaction rues, except for nine out of the tota possibe nonempty sets of binary Booean functions, whether the induced chain is absorbing has nothing to do with the topoogy of the underying interaction graph, as ong as connectivity is assumed. These resuts iustrate the possibiities of reating dynamica properties of Booean networs to graphica properties of the underying interactions. Index Terms Booean networs, gossiping process, Marov chains, communication casses. I. INTRODUCTION A. Bacground AVARIETY of random networ dynamics with nodes taing ogica vaues arises from bioogica, socia, engineering, and artificia inteigence systems [1] [4]. In the Manuscript received October 25, 2016; revised May 2, 2017 and August 9, 2017; accepted October 8, 2017; approved by IEEE/ACM TRANSACTIONS ON NETWORKING Editor X. Lin. Date of pubication October 31, 2017; date of current version February 14, This wor was supported in part by the Nationa Natura Science Foundation of China under Grant , Grant , and Grant , in part by the Nationa Center for Mathematics and Interdiscipinary Sciences, Chinese Academy of Sciences, and in part by the Austraian Department of Education and Training through an Endeavour Feowship. (Corresponding author: Junfeng Wu.) B. Li is with the Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing , China (e-mai: ibo@amss.ac.cn). J. Wu is with the Coege of Contro Science and Engineering, Zhejiang University, Hangzhou , China (e-mai: jfwu@zju.edu.cn). H. Qi is with the Key Laboratory of Systems and Contro, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing , China, and aso with the University of Chinese Academy of Sciences, Beijing , China (e-mai: qihongsh@amss.ac.cn). A. Proutiere is with the Autonomic Compex Communication networs, Signas and Systems Linnaeus Centre, KTH Roya Institute of Technoogy, Stochom, Sweden (e-mai: aepro@th.se). G. Shi is with the Research Schoo of Engineering, The Austraian Nationa University, Canberra, ACT 0200, Austraia (e-mai: guodong.shi@anu.edu.au). Digita Object Identifier /TNET s, Kauffman introduced random Booean iteration rues over a networ [1] to describe proto-organisms as randomy aggregated nets of chemica reactions where the underying genes serve as a binary (ON-OFF) device. Inspired by neuron systems, the so-caed Hopfied networs [2] provided a way of reaizing coective computation inteigence, where nodes having binary vaues behave as artificia neurons by a weighted majority voting via random or deterministic updating. Rumors spreading over a socia networ [3] and virus scattering over a computer networ [4] can be modeed as epidemic processes with binary nodes states indicating whether a peer has received a rumor, or whether a computer has been infected by a type of virus. Booean dynamica networs, consisting of a finite set of nodes and a set of deterministic or random Booean interaction rues among the nodes, are natura and primary toos for the modeing of the above node dynamics with ogica vaues. The study of Booean networs received considerabe attention for aspects ranging from steady-state behaviors and input-output reations to imit cyce attractors and mode reduction, e.g., [5] [15]. It has been we understood that deterministic Booean rues are essentiay inear in the state space [5], [10], whie probabiistic Booean networs are merey standard Marov chains [6], [11] [15]. There however exist fundamenta chaenges in estabishing expicit and precise theoretica resuts due to computation compexity barriers [16] and the ac of anaytica toos. In this paper, we propose and study a randomized Booean gossip process, where Booean nodes pairwise meet over an underying graph in a random manner at each time step, and then the two interacting nodes update their states by random ogica rues in a prescribed set of Booean operations. B. The Mode We consider n nodes indexed by the set V={1,...,n}. The underying interaction structure of the networ is modeed by an undirected graph G = (V, E), where E is the edge set with each entry being an unordered pair of two distinct nodes in V. ThesetN i = {j : {i, j} E} represents the neighbourhood of node i. Throughout our paper we assume that the graph G=(V, E) is connected. Time is sotted at t =0, 1,... Node interactions foow a random gossip process [17], where independenty at each time t 0, a pair of nodes i and j with {i, j} E is randomy seected over the graph. Each node i hods a binary vaue from the set {0, 1} at each time t, denoted x i (t). Note that, there are a tota of 16 Booean functions with two arguments mapping from {0, 1} 2 to {0, 1}. Using hexadecima numbers, we index IEEE. Persona use is permitted, but repubication/redistribution requires IEEE permission. See for more information.

2 LI et a.: BOOLEAN GOSSIP NETWORKS 119 Fig. 1. The 16 Binary operators mapping from {0, 1} 2 to {0, 1}. Each diagram visuaizes a Booean mapping: the first coumn represents vaues of the first argument (in bac); the second coumn represents vaues of the second argument (in red); the third coumn (in bue) represents the outcome of the operation foowing the direction of the same type of ines. For exampe, the first diagram reads as 0 0 0=0, 0 0 1=0, 1 0 0=0, 1 0 1=0. these functions in the set (see Fig. 1) H := { 0,..., 9, A,..., F }, where 1 each specifies a binary Booean function in the way that a b is the vaue of the function with arguments (a, b). Let C be a subset of H specifying potentia node interaction rues aong the edges. Let q := C be the cardinaity of the set C. We index the eements in C by C1,..., Cq. Suppose the node pair {i, j} is seected at time t. Introduce p > 0 for 1, q, satisfying p =1. At time t,, nodes i and j jointy choose ( C, C ) C C with probabiity p. Under such a choice, the evoution of x m (t) is determined by x i (t +1)=x i (t) C x j (t), x j (t +1)=x j (t) C x i (t), (1) x m (t +1)=x m (t), m / {i, j}. C. Induced Marov Chain Let X t =(x 1 (t),...,x n (t)),t =0, 1,... be the random process driven by the gossip agorithm and the Booean rues (1). This random process X t,t 0 defines a 2 n -state Marov chain M G (C) =(S n,p), where S n = { [s 1...s n ]: s i {0, 1},i V } 1 These Booean functions have their respective names, for which we refer to [30]. is the state space, and P is the state transition matrix. Then the state transition matrix P is given by P = [ ] P [s1...s n][q 1...q n] R 2 n 2 n with its rows and coumns indexed by the eements in S n,i.e., ( ) P [s1...s n][q 1...q n] := P X t+1 =[q 1...q n ] X t =[s 1...s n ]. D. Reated Wor The proposed randomized Booean gossip mode apparenty cover the cassica gossip process [17] [20] as a specia case. The process (1) is aso a specia case of the probabiistic Booean networ mode [7], [8], where random Booean interactions are posed pairwise. Therefore conceptuay the mode (1) under consideration can certainy be paced into the studies of genera probabiistic Booean networs, e.g., [9], [12], [13]. Since the node interaction rues can be an arbitrary set of Booean functions, this Booean gossip mode is a usefu approximation or generaization to existing characterizations to gene reguation [1], socia opinion evoution [3], and virus spreading [4]. Gene Reguation: The evoution of gene expressions can be naturay described as a dynamica system where the two quantized eves, ON and OFF, are represented by ogic states 1 and 0, respectivey. Each gene normay woud ony interact with a sma number of neighbouring genes. 2 Therefore, the proposed Booean gossip networ mode at east serves as a good approximation for gene reguator networs, 2 Such number is two or three in Kauffman s origina proposa [1].

3 120 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 26, NO. 1, FEBRUARY 2018 where a pair of genes interact at any given time and the Booean function rues C describe random outcomes of the interactions. Socia Voting: Socia peers hod binary opinions for certain poitica or economica issues, where 1 represents a supportive opinion and 0 represents a non-supportive one. Peers meet with each other in pairs randomy and exchange their opinions. The two peers independenty decide their opinions after the meeting; the Booean function rues C describe how they might revise their opinions. Virus Spreading: Virus spreading across a computer networ can be modeed as a Booean networ, where 0 and 1 represent infected and heathy computers, respectivey [4]. The proposed Booean gossip process may characterize more possibiities for two computers during an interaction: two computers, infected or not, are both infected ( 1 ); two computers, infected or not, are both cured ( F ), etc. The graphica nature of the mode (1) maes it possibe to go beyond these existing wor [9], [12], [13] for more direct and expicit resuts. Additionay, majority Booean dynamics [27] and asynchronous broadcast gossiping [28] are reated to the mode (1) in the way that they describe Booean interactions between one node and a its neighbors at a given time instant, in contrast to the gossip interaction rue which happens between one node and one of its seected neighbors. networ topoogy as ong as the underying graph is connected; for the remaining nine sets of binary Booean functions, absorbing property of the induced chain is fuy determined by whether the underying graph G contains an odd cyce. The remainder of this paper is organized foows. Section II investigates positive Booean dynamics in terms of steady-state distribution and communication casses. Section III further studies genera Booean dynamics with a focus on how the interaction graph determines absorbing Marov chains aong the random Booean dynamics. Finay Section IV concudes the paper with a few remars. A notation tabe is provided in the Appendix B. II. POSITIVE BOOLEAN GOSSIPING In this section, we consider a specia case where the Booean interaction rues in the set C do not invove the negation. Note that conventionay represents Booean AND operation, whie represents Booean OR operation. We term such types of Booean interaction as positive Booean dynamics, and define C pst = {, } as the set of positive Booean functions. Let us denote C1 = and C2 =. E. Contributions and Paper Organization The proposed random Booean gossip mode is fuy determined by the underying graph G and the Booean interaction set C. Cassica (deterministic or probabiistic) Booean networs aso have graphica characterization [7] where a in appears if the state of the end nodes depend on each other in the Booean updating rues. To the best of our nowedge, few resuts have been obtained regarding how the structure of the interaction graph infuences detaied networ state evoution in the study of Booean networs. First of a, we study a specia networ where the Booean interaction rues in the set C do not invove the negation, which is termed positive Booean networs. Using standard theories from Marov chains, we show that the networ nodes asymptoticay converge to a consensus represented by a binary random variabe, whose distribution is studied for arge-scae networs in ight of mean-fied approximation methods. Moreover, by combinatoria anaysis the number of communication casses of positive Booean networs is fuy characterized with respect to the structure of the underying interaction graph G, where surprisingy oca cycic structures can drasticay change the number of communication casses of the entire networ. Next, we move to genera Booean interaction rues and study the reation between emergence of absorbing networ Booean dynamics and the networ structure. Necessary and sufficient conditions are provided for the induced Marov process M G (C) to be an absorbing chain. Interestingy, for the majority of the Booean interaction rues, except for nine of the possibe nonempty sets of binary Booean functions, whether the induced chain is absorbing does not rey on the A. State Convergence Reca that a state in a Marov chain is caed absorbing if it is impossibe to eave this state [26]. A Marov chain is caed absorbing if it contains at east one absorbing state and it is possibe to go from any state to at east one absorbing state in a finite number of steps. In an absorbing Marov chain, the non-absorbing states are caed transient. It is not hard to find that the Marov chain M G (C pst ) is an absorbing chain with [0...0] and [1...1] being the two absorbing states. Let I denote the -by- identity matrix for any integer. The state transition matrix P therefore wi have the form [ ] I2 0 P =, R Q where the I 2 boc corresponds to the two absorbing states [0...0] and [1...1], R is a (2 n 2) 2 matrix describing transition from the 2 n 2 transient states to the two absorbing states, and Q is a (2 n 2) (2 n 2) matrix describing the transition between the transient states. Note that foowing the definition of P, the rows of the matrix (I 2n 2 Q) 1 R are indexed by the entries in S n \ {[0...0], [1...1]}, and the coumns are indexed by [0...0] and [1...1]. Let [ (I 2n 2 Q) 1 R ] X be the X 0[1...1] 0-[1...1] entry of the matrix (I 2n 2 Q) 1 R. We can concude the foowing resut from standard theories for absorbing Marov chains (see [26, Th. 11.6, p. 420]). Proposition 1: Let X 0 = X(0) S n \{[0...0], [1...1]}. There exists a Bernoui random variabe x such that P ( im t x i (t) =x, for a i V ) =1.

4 LI et a.: BOOLEAN GOSSIP NETWORKS 121 Fig. 2. A four-node cyce graph. The imit x satisfies E{x } = [ (I 2 n 2 Q) 1 R ] X 0[1...1]. Fig. 3. Fu state transitions of the induced Marov chain by the positive Booean gossip process C pst = {, } over the four-node cyce graph as shown in Figure 2. States within the same communication cass are mared with the same coor. B. Communication Casses We continue to investigate the communication casses of M G (C pst ). Reca that a state [s 1...s n ] is said to be accessibe from state [q 1...q n ] if there is a nonnegative integer t such that P ( X t =[s 1...s n ] X 0 =[q 1...q n ] ) > 0. Itis termed that [s 1...s n ] communicates with state [q 1...q n ] if [s 1...s n ] and [q 1...q n ] are accessibe from each other [26]. This communication reationship forms an equivaence reation among the states in S n. The equivaence casses of this reation are caed communication casses of the chain M G (C pst ).The number of communication casses of M G (C pst ) is denoted as χ Cpst (G). The foowing theorem provides a fu characterization to χ Cpst (G). Theorem 1: There hod (i) χ Cpst (G) = 2n, ifg is a ine graph; (ii) χ Cpst (G) = m +3,ifG is a cyce graph with n =2m; χ Cpst (G) = m +2, if G is a cyce graph with n =2m +1; (iii) χ Cpst (G) = 5, ifg is neither a ine nor a cyce, and contains no odd cyce; (iv) χ Cpst (G) = 3, ifg is not a cyce graph but contains an odd cyce. Estabished by constructive proofs that can overcome the fundamenta computationa obstace in anayzing arge-scae Booean networs, Theorem 1 reveas how oca structures can drasticay change the number of communication casses as a goba property of networs. The detaied proof of Theorem 1 has been put in the Appendix. Beow we present a few exampes iustrating the statements of Theorem 1. Exampe 1: Let the underying graph G be the four-node cyce graph as dispayed in Figure 2. With the positive Booean rues C pst, the state transition map of the induced Marov chain is iustrated in Figure 3. Ceary the chain has 5 communication casses, consistent with Theorem 1. Exampe 2: Let the underying graph G be the four-node graph containing a three-node cyce subgraph as dispayed in Figure 4. With the positive Booean rues C pst,thestate transition map of the induced Marov chain is iustrated in Figure 5. In this case the chain has 3 communication casses, again verifying Theorem 1. Fig. 4. A four-node graph consisting of a three-node cyce subgraph. Fig. 5. Fu state transitions of the induced Marov chain by the positive Booean gossip process C pst = {, } over the four-node graph as shown in Figure 4. States within the same communication cass are mared with the same coor. C. Continuous-Time Approximation It has been cear from Proposition 1 that starting from X 0 S n \{[0...0], [1...1]}, the imit of the node states is fuy characterized by [ (I 2n 2 Q) 1 R ] X.However, 0[1...1] computing the exact vaue or even obtaining an approximation for the matrix (I 2 n 2 Q) 1 R is difficut for arge networs due to the exponentiay increasing dimension of the matrix. In this subsection, using mean-fied method [4], [25], we construct a continuous-time differentia equation to approximate the behavior of X(t) for arge scae networs (see [29] for a detaied survey on differentia equation approximations for Marov chains). To this end, we assume that the x i (0) are i.i.d Bernoui random variabes, and at time t the two seected

5 122 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 26, NO. 1, FEBRUARY 2018 nodes i and j update their states by independenty 3 seecting with some probabiity p > 0. 1) Compete Graph: Define n δ(t) = x i (t)/n i=1 as the proportion of nodes that tae vaue 1 at time t. Assume the underying networ forms a compete graph. Let the edges be seected uniformy at random at each time step. Denote δ(t) as the expected vaue of δ(t), i.e., δ(t) =E{δ(t)}. The density δ(t) evoves by the foowing rues: Let the two nodes in the seected pair {i, j} hod different vaues. When n is arge, and the graph is compete, this happens with an approximate probabiity 2δ(t) (1 δ(t)). Thevaueδ(t) wi increase by 1/n if the two seected nodes both use operations to update their vaues, an event with probabiity p 2.Thevaueδ(t) wi decrease by 1/n if the two seected nodes both appy operations, an event with probabiity (1 p ) 2. For a other cases, δ(t) is unchanged. As a resut, we concude that E{δ(t +1) δ(t) δ(t)} 1 n p 2 2δ(t)(1 δ(t)) 1 n (1 p ) 2 2δ(t)(1 δ(t)). (2) For a compete graph with n nodes, V{δ(t)} = E{δ 2 (t)} E 2 {δ(t)} can be considered very sma for arge n. Wefurther have δ(t +1) δ(t) 1 n p 2 2δ(t)(1 δ(t)) 1 n (1 p ) 2 2δ(t)(1 δ(t)). (3) Define s = t/n and δ(s) =δ(ns) =δ(t). Then, (3) can be written as δ(s +1/n) δ(s) 1 n p 2 2 δ(s)(1 δ(s)) 1 n (1 p ) 2 2 δ(s)(1 δ(s)) (4) We can therefore approximate (4) for arge n by the foowing differentia equation d ds δ(s) =p 2 2 δ(s)(1 δ(s)) (1 p ) 2 2 δ(s)(1 δ(s)), (5) whose soution reads anayticay as δ(0) δ(s) = (1 δ(0))e. (6) 2(1 2p )s + δ(0) Here δ(0) = δ(0) = δ 0 is the mean of the i.i.d Bernoui random variabes x i (0). Consequenty, we estabish the foowing approximate equation for δ(t): δ 0 δ(t) =. (7) (1 δ 0 )e 2(1 2p )t/n + δ 0 From (7), the foowing hods. 3 Precisey, this means p 11 = p 2,p 22 = (1 p ) 2,p 12 = p 21 = p (1 p ) in the dynamics (1). Fig. 6. A compete graph with 1000 nodes is considered. The soid ines are the approximate soution given by (7); the dashed ines are drawn according to the simuated reaization of the agorithm (1). The continuoustime approximations match the numerica reaizations rather precisey. Concusion: Assume G is a compete graph. For arge n, δ(t) approaches zero when p < 1/2, andδ(t) approaches one when p > 1/2, as time tends to infinity. To verify this concusion, we give some numerica resuts. Exampe 3: Consider a compete graph with n = 1000 nodes. Fix δ 0 =0.5, and we randomy distribute the vaues of nodes according to δ 0 =0.5. Forp =0.49 and 0.51, we et the nodes update their vaues randomy according to (1), respectivey. Each experiment is carried out over T = time steps, repeated for 2000 rounds. The average of the resuting 2000 sampe paths approximatey give the density of nodes with vaue one for every t. We compare the numerica simuation with the approximate soution given by (7). Figure 6 shows that (7) approximates the rea process (1) remaraby we. 2) Reguar Graph: A reguar graph is a graph where nodes have equa degrees. Suppose node i is seected to initiaize a gossip interaction at time t. Because i is uniformy seected from V, the probabiity that the seected node i is at state 1 is δ(t). IfG is a reguar graph with a random nature 4 and high node degrees where N i =Θ(n), the distribution of the random variabe j N i x j (t) N i wi tend to have a simiar distribution with j i x j(t), n 1 which is approximatey a Bernoui random variabe with mean δ(t). Therefore, δ(t) evoves foowing simiar rue as compete graphs, and the differentia equation (7) wi continue to be a good approximation for high-degree reguar graphs. Exampe 4: Consider a reguar graph of degree 500 with n = 1000 nodes. We seect p =0.49 and δ 0 =0.5. Again each experiment is carried out over T = time steps, 4 This is to say, the distribution of the ins shoud appear somehow independenty being cose to the concentration of random reguar graphs. The approximation can be quite inaccurate for graphs ie attices.

6 LI et a.: BOOLEAN GOSSIP NETWORKS 123 Fig. 7. A reguar graph with 1000 nodes is considered where node degree is 500 and p = The soid ine is the approximate soution given by (7) and the dashed ine is drawn according to numerica simuation. We see that (7) continues to be a good approximation of (1). repeated for 2000 rounds. The average of the resuting 2000 sampe paths aows us to obtain the approximate density of nodes with vaue 1 for a t. Figure 7 shows that (7) continues to provide an acceptabe approximation of the rea process (1). Remar: We have shown that Eq. (7) is a good mean-fied approximation for compete and reguar graphs. As indicated in [4], Erdős-Rényi random graph with average degree p(n 1) approximates we reguar graphs when n is arge. Hence, the equation (7) may aso provide ap reasonabe approximation for Erdős-Rényi random graph modes. However, most rea-word networs are neither reguar, nor behaving ie Erdős-Rényi random graphs. In future, it is worth exporing acceptabe approximations for more rea-word ie graphs, such as scae-free networs. III. GENERAL BOOLEAN DYNAMICS In this section, we discuss the evoution of (1) under genera Booean interaction set C 2 H,where2 H denotes the set containing a subsets of H. We are interested in how the induced chain M G (C) reies on the underying graph G and the set of Booean interaction rues C. Particuary, we woud ie to see when M G (C) defines an absorbing chain. Reca that absorbing states are the states that can never be eft once visited. Therefore, absorbing Marov chains behave fundamentay different with non-absorbing chains. We introduce two subsets of Booean mappings: B 1 = { C { A } 2 H : { A } C { 2, 3, A, B } } and B 2 = { C 2 H : { 2, B } C { 2, 3, A, B } }. We further et B := B 1 B2. Note that there are a tota of nine eements in B. Aswe show beow, Booean interaction rues in the set B ead to drasticay different infuences to the absorbing property of the induced chain, compared to the rues outside the set B. Fig. 8. Part of the state transitions of the induced Marov chain with C = { 2, 3 } for the underying graph in Figure 4. The chain is absorbing with seven absorbing states, which are dispayed in red. A. Main Resuts We first estabish a theorem reveaing the connection between the induced Marov chains of any two different underying graphs when connectivity is assumed. Theorem 2: Suppose C 2 H \ B. Then, for any two connected graphs G 1 and G 2 over the node set V, M G1 (C) is an absorbing Marov chain if and ony if M G2 (C) is an absorbing Marov chain. In view of Theorem 2 and the fact that G is a connected graph over a certain set V, whether M G (C) being an absorbing chain is fuy determined by the interaction rue set C when C does not beong to B. Next, we present the foowing theorem estabishing a necessary and sufficient condition for the induced chain to be absorbing when the Booean interaction rues come outside the set B. Theorem 3: Suppose C 2 H \ B. ThenM G (C) is an absorbing Marov chain if and ony if one of the foowing two conditions hods (i) C { 0, 1, 2, 3, 4, 5, 6, 7 }; (ii) C { 1, 3, 5, 7, 9, B, D, F }. When the interaction rues C indeed comes from the set B, the foowing theorem further gives a tight condition on the absorbing property of the induced chain. Specificay, if C is one of the nine function sets in B, the topoogy of G fuy determines whether the induced chain is absorbing. Theorem 4: Suppose C B. ThenM G (C) is an absorbing Marov chain if and ony if G does not contain an odd cyce. Note that, Theorem 2 can actuay be inferred from Theorem 3. Theorem 3 and Theorem 4 together present a comprehensive understanding of the absorbing property of the networ Booean evoution. Beow we present two exampes iustrating the usefuness of Theorems 3 and 4. Exampe 5: Consider again the graph G in Figure 4. With the set of Booean interaction rues being C = { 2, 3 }, part of pthe transition map of the induced Marov chain is iustrated in Figure 8. The chain is absorbing with seven absorbing states: [0000], [1010], [1001] [1000], [0100], [0010], and [0001]. This exampe is consistent with Theorem 3.(i). Exampe 6: Let the underying graph G be given in Figure 2. Let the set of Booean interaction rues be C = { 2, B }.The chain is absorbing as shown in Figure 9 with two absorbing

7 124 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 26, NO. 1, FEBRUARY 2018 Fig. 9. Part of the state transitions of the induced Marov chain with C = { 2, B } for the four-node cyce graph in Figure 2. There exist no outgoing transitions from [1001] and [0110], reveaing that they are absorbing states. Fig. 10. Part of the state transitions of the induced Marov chain with C = { 2, B } for the underying graph in Figure 4, which aready shows that the chain cannot be absorbing. states [1001] and [0110]. Note that, for carity, Figure 9 do not dispay a the state transitions. This exampe is consistent with Theorem 4 as the graph does not contain an odd cye. Exampe 7: Let the underying graph G be given in Figure 4, and et the set of Booean interaction rues continue to be C = { 2, B }. The chain is not absorbing as shown in Figure 10, further confirming the concusion drawn in Theorem 4 as the graph contains an odd cye. Note that, for carity, Figure 10 do not dispay a the state transitions. B. Key Lemma To simpify the discussion, we introduce some new notations. For any S =[s 1...s n ] S n, we denote [S0] as [s 1...s n 0] S n+1 and [S1] as [s 1...s n 1] S n+1.forany a {0, 1}, denote a =1 a. We categorize the states into the foowing five casses: C 1 (G) = {[s 1...s n ]:s i =0, 1 i n}, C 2 (G) = {[s 1...s n ]:s i =1, 1 i n}, C 3 (G) = {[s 1...s n ]:s i s j for any edge {i, j} of G}, C 4 (G) = {[s 1...s n ]: i, j,, s.t.{i, j} is an edge of G and 0=s i = s j s },and C 5 (G) = {[s 1...s n ]: i, j,, s.t.{i, j} is an edge of G and 1=s i = s j s }. We may simpy write C i instead of C i (G) whenever this simpification causes no confusion. In this subsection, we estabish a ey technica emma regarding whether a state in the C i can be an absorbing state in terms of the seection of C. Lemma 1: (i) The state in C 1 is an absorbing state if and ony if C { 0, 1, 2, 3, 4, 5, 6, 7 }. (ii) The state in C 2 is an absorbing state if and ony if C { 1, 3, 5, 7, 9, B, D, F }. (iii) A state in C 3 is an absorbing state if and ony if C { 2, 3, A, B }. (iv) A state in C 4 \ C 5 is an absorbing state if and ony if C { 2, 3 }. (v) A state in C 5 \ C 4 is an absorbing state if and ony if C { 3, B }. (vi) A state in C 5 C4 is an absorbing state if and ony if C { 3 }. Proof: (i) Note that [0...0] C 1 is a state at which any two nodes associated with a common edge must hod the same vaue 0. According to the agorithm (1), [0...0] is an absorbing state if and ony if for any i C there hods 0 i 0=0. Thus, [0...0] is an absorbing state if and ony if C { 0, 1, 2, 3, 4, 5, 6, 7 }. (ii) The proof is simiar to that in (i), whose detais are omitted. (iii) Let S C 3, at which two nodes sharing a in must hod different vaues. According to the structure of the agorithmp (1), S is an absorbing state if and ony if for any i C, 0 i 1=0and 1 i 0=1. That is, S is an absorbing state, if and ony if C { 2, 3, A, B }. (iv) It is cear that S C 4 \ C 5 is an absorbing state if and ony if for any i C, there hod 0 i 0=0, 0 i 1=0, and 1 i 0=1. In other words, S is an absorbing state if and ony if C { 2, 3 }. The proofs of the statements (v) and (vi) are simiar to that of (iv), which are, again omitted. C. Proof of Theorem 3 This subsection focuses on the proof of Theorem 3. (Necessity.) Assume M G (C) is an absorbing Marov chain. If both [0...0] C 1 and [1...1] C 2 are not absorbing, any state in C 4 or C 5 cannot be absorbing as we according to Lemma 1(i)-(ii)(iv)-(vi). This eaves the ony possibiity be that at east one of the states in C 3 is absorbing. Thus, C { 2, 3, A, B } from Lemma 1(iii). Next, we concude that C can ony be { A } by Lemma 1(i)-(ii) since C 2 H \ B. When C = { A }, according to Lemma 1, the ony possibe absorbing states are states in C 3. However, any state in C 3 cannot be accessed by any other states. This contradicts the assumption that M G (C) is an absorbing chain. Therefore, we can ony concude that either [0...0] C 1 or [1...1] C 2 is absorbing.

8 LI et a.: BOOLEAN GOSSIP NETWORKS 125 If the state [0...0] is absorbing, we obtain C { 0, 1, 2, 3, 4, 5, 6, 7 } according to Lemma 1(i). Whie if [1...1] is absorbing, we have C { 1, 3, 5, 7, 9, B, D, F } from Lemma 1(ii). This proves the necessity statement. (Sufficiency.) We investigate a few cases. Let C { 0, 1, 2, 3, 4, 5, 6, 7 }. Then [0...0] is absorbing by Lemma 1(i). We divide the case into a few subcases: 1) If C { 0, 1, 4, 5 }, there is a positive probabiity that i { 0, 1, 4, 5 } is chosen. Because 1 i 0=0, any state other than [1...1] can transit to state [0...0] in some finite steps with a positive probabiity. If [1...1] is not absorbing, then it wi transit to some other state S with positive probabiity. Finay state S wi transit to the absorbing state [0...0] with positive probabiity. Therefore, no matter whether [1...1] is absorbing or not, M G (C) is an absorbing Marov chain. 2) Let C { 6 } and consider the update where 6 is aways seected. Then for any state in C 4 or C 3, two nodes with vaues 0 and 1 respectivey wi both hod vaue 1 after the interaction, i.e., the networ state enters C 5 or C 2. Furthermore, for any state in C 5 or C 2, two nodes both hoding vaue 1 wi both hod 0 after the interaction. Thus, for a states in C 2,...,C 5, the number of nodes hoding vaue 1 wi be stricty decreasing if 6 is aways present, unti the state transits to [0...0]. The chain M G (C) is an absorbing Marov chain since we aready now [0...0] is an absorbing state. 3) Assume C = { 2 } or C = { 2, 3 } and et 2 be chosen. Then any state in C 2 or C 5 wi transit to state in C 4 \ C 5 or C 1 in some finite steps. Thus, M G (C) is an absorbing Marov chain because a states in C 1, C 3, C 4 \C 5 are absorbing by Lemma 1. 4) If C = { 7 } or C = { 3, 7 }, we can use simiar discussion in 1) to concude that any state other than [0...0] can transit to state [1...1] in finite steps. The chain M G (C) is an absorbing Marov chain. 5) Let C = { 2, 7 } or C = { 2, 3, 7 }. The scenario is simiar to 2), where any state can transit to state [0...0] in finite steps. 6) If C = { 3 }, a states are absorbing. Of course M G (C) is an absorbing Marov chain. Assume C { 1, 3, 5, 7, 9, B, D, F }.The proof is simiar to the case above, whose detais are omitted. The proof of Theorem 3 is now compete. D. Proof of Theorem 4 In this subsection, we prove Theorem 4. If G contains an odd cyce, C 3 is empty. By Lemma 1, no state in C 1 C2 C4 C5 is absorbing. As S n = C 1 C2 C3 C4 C5, no state is absorbing. Thus, M G (C) is not absorbing. On the other hand, if G does not contain an odd cyce, C 3 is not empty consisting of two eements. We proceed to prove by induction on the number n of nodes that M G (C) is an absorbing Marov chain. For n =2, the concusion hods straightforwardy. Assume that M G (C) is absorbing for n =. There must be a spanning tree, denoted G T1, of G. We further find a subtree G T2 of G T1 with G T2 containing nodes of G T1. Without oss of generaity, et G T2 contain nodes 1,..., of G. By our induction assumption, M GT2 (C) is absorbing. Now any state in S +1 can be represented as [Su], where S S and u {0, 1}. AsM GT2 (C) is absorbing, there is a positive probabiity that in finite steps S transits to a state S in C 3 (G T2 ). Because G T2 is a subgraph of G T1, [Su] can transit to [S u] in finite steps in M GT1 (C). There wi be two cases. If [S u] C 3 (G T1 ), for G contains no odd cyce, [S u] C 3 (G). The proof is done. If [S u] / C 3 (G T1 ), there must be some node j associated with node +1 over graph G T1. Because C B, there is a positive probabiity that A or B is chosen. Note that 0 A 0=1, 0 B 0=1, 1 A 1=0 and 1 2 1=0. Thus, by (1), [S u] transits to [S u] with positive probabiity in M GT1 (C). Moreover, [S u] C 3 (G T1 ).ForG contains no odd cyce, [S u] C 3 (G) eads to the desired resut. The proof of Theorem 4 is competed. IV. CONCLUSIONS We proposed and investigated a Booean gossip mode, which may be usefu in describing socia opinion evoution as we as serves as a simpified probabiistic Booean networ. With positive node interactions, it was shown that the node states asymptoticay converge to a consensus represented by a binary random variabe, whose distribution was studied for arge-scae compete networs in ight of mean-fied approximation methods. By combinatoria anaysis the number of communication casses of the positive Booean networ was counted against the topoogy of the underying interaction graph. With genera Booean interaction rues, the emergence of absorbing networ Booean dynamics was expicity characterized by the networ structure. It turned out that oca structures in terms of existence of cyces can drasticay change fundamenta properties of the Booean networ. In future, it wi be interesting to oo into the possibiity of extending the graphica anaysis estabished in the current wor to mutistate Booean networs [14], [15] where each node may hod a state from a finite set with more than two vaues. APPENDIX A PROOF OF THEOREM 1 For each n, weusemod n (i) to denote the unique integer j satisfying 1 j n and i j mod (n). Reca that for any a {0, 1}, we denote a =1 a. We prove the statements of Theorem 1 in a few steps starting with a few fundamenta graphs.

9 126 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 26, NO. 1, FEBRUARY 2018 A. Line Graph In this subsection we prove Theorem 1.(i) stating that χ Cpst (G) = 2n when G is a ine graph. Without oss of generaity we assume the edges of G are {i, i +1} for i =1,...,n 1. The proof is outined as foows. We first introduce the notion of L-reduced state for each state in S n. Then, we prove that any two states communicate with each other if and ony if their L-reduced states are identica. Finay, we count the number of L-reduced states in the state space and therefore obtain the number of communication casses. Definition 1 (L-Reduced States): Let [s 1...s n ] S n. There exists a unique partition of s 1,...,s n into s 1 =...= s i1 = r 1, i 1 1; s i1+1 =...= s i2 = r 2, i 2 >i 1 ;... s id 2 +1 =...= s id 1 = r d 1, i d 1 >i d 2 ; s id 1 +1 =...= s n = r d such that r i r i+1 for a i =1,...,d 1. Then[r 1...r d ]:= L([s 1...s n ]) is termed the L-reduced states of [s 1...s n ]. Note that the vaues of any two consecutive eements in an L-reduced state are different. The foowing two emmas hod. Lemma 2: Suppose G is a ine graph. Then L([s 1...s n ]) is a subsequence of L([q 1...q n ]) if [s 1...s n ] is accessibe from [q 1...q n ]. More precisey, denoting L([s 1...s n ]) = [r 1...r d ], L([q 1...q n ]) = [h 1...h d ] there hods d d, and moreover, there exist 1 τ 1 <τ 2 <...<τ d d such that r i = h τi for a i =1,...,d. Proof: By the definition of accessibiity, there is a nonnegative integer t such that P ( X t =[s 1...s n ] X0 =[q 1...q n ] ) > 0. First we assume t =1. According to the structure of (1), either [s 1...s n ]=[q 1...q n ],orthereisu {1,...,n} such that s u q u and s i = q i for a i u. The desired concusion obviousy hods if [s 1...s n ]=[q 1...q n ]. For the atter case, there is q v with v = u +1 or v = u 1 such that q u q v. Consequenty, the two states [s 1...s n ] and [q 1...q n ] differ with each other ony at s u and q u and satisfy s u q u, s u = s v, q u q v. Then it is easy to verify that L([s 1...s n ]) is a subsequence of L([q 1...q n ]) from the definition of L-reduced states. Now we proceed to et t = 2. There wi be a state [w 1...w n ] such that [s 1...s n ] is one step accessibe from [w 1...w n ], and [w 1...w n ] is one step accessibe from [q 1...q n ]. Utiizing the above understanding for the case with t = 1 we now L([s 1...s n ]) is a subsequence of L([w 1...w n ]) and L([w 1...w n ]) is a subsequence of L([q 1...q n ]), which in turn impy L([s 1...s n ]) is a subsequence of L([q 1...q n ]). Therefore the desired concusion hods for t =2. Apparenty the argument can be recursivey carried out and the resut hods for arbitrary integer t. Wehave now competed the proof of the emma. Lemma 3: Let G be a ine graph and consider S = [s 1...s n ],Q=[q 1...q n ] S n.thens and Q communicate with each other if and ony if they have identica L-reduced states. Proof: The necessity part of this emma foows directy from Lemma 2. In the foowing we focus ony on the sufficiency part. Let the identica L-reduced state of [s 1...s n ] and [q 1...q n ] be [r 1...r ]. We carry out an induction argument on for any n. Let =1.Then[0...0] n and [1...1] n are the two possibe states for [s 1...s n ] and [q 1...q n ]. The desired concusion hods straightforwardy. Now assume: Induction Hypothesis: The statement of the emma hods true for a and a n. We proceed to prove the statement for = +1 and n. Denote i 1 =max{h : r 1 = s i, 1 i h} and j 1 =max{h : r 1 = q i, 1 i h}. By symmetry we may assume i 1 j 1 and we use the foowing two observations: a) The state [q 1...q n ] communicates with the state [q 1...q i1 q i1+1...q j1 q j1+1...q n ] by the definition of j 1. b) The two states [q i1+1...q j1 q j1+1...q n ] and [s i1+1s i1+2...s n ] have the same L-reduced state [r 2...r ]. Therefore by our induction hypothesis, [q i1+1...q j 1 q j1+1...q n ] and [s i1+1...s n ] communicate with each other, which in turn yieds that [s 1...s n ] communicates with [q 1...q i1 q i1+1...q j1 q j1+1...q n ]. Combining a) and b) we immediatey now that [s 1...s n ] communicates with [q 1...q n ]. By the principe of mathematica induction we have competed the proof of the emma. We are now ready to count the number of communication casses for the ine graph, which equas to the number of L-reduced states according to Lemma 3. For each m = 1,...,n, there are two different L-reduced states with ength m, i.e., [r 1...r m ] with r 1 =0or r 1 =1. Consequenty, there are a tota of 2n different L-reduced states. This concudes the proof for Theorem 1.(i). B. Cyce Graph In this subsection, we prove the case with G being a cyce graph. Without oss of generaity, et G be the cyce graph with edges { i, Mod n (i +1) }, i =1,...,n. We introduce some usefu notations that wi be used subsequenty. For any, weuseσ to denote the permutation on set {1,...,} with σ (i) =Mod (i +1) for i =1,...,. We further define P σ as a mapping over S by P σ ([s 1...s ]) = [s σ (1)...s σ ()] for a [s 1...s ] S. Intuitivey, if we pace these nodes uniformy on a cyce and denote the vaue of each node on them, then the resut of P σ on a state is obtained by rotating a the vaues countercocwise. We aso define a mapping f [1, 2] over S n by that for any [t 1...t n ],

10 LI et a.: BOOLEAN GOSSIP NETWORKS 127 f [1, 2]( [t1...t n ] ) =[r 1...r n ] with r i = t i for i 2 and r i = t 1 for i = 2. Definition 2 (K -Reduced States): Let [s 1...s n ] S n with [r 1...r d ] = L([s 1...s n ]) being its L-reduced states. The K -reduced states of [s 1...s n ] S n, denoted K ([s 1...s n ]), is defined as foows: [r 1 ] if d =1; K ([s 1...s n ]) = [r 1...r d ] if d>1 and r d r 1 ; [r 1...r d 1 ] if d>1 and r d = r 1. Let K (S) be the number of digits in K (S) for S S n. According to the definition, the vaues of any two consecutive eements of K -reduced states are different. Moreover, if there are at east two entries of K -reduced states, the first entry is different from the ast one. The foowing emma can be estabished using a simiar anaysis as we used in Lemma 2. Lemma 4: Suppose G is a cyce graph, (i) K (S) is either 1 or an even integer; (ii) If d is one or an even integer, then there is S S n with K (S) = d. (iii) If S is accessibe from T,then K(S) K (T ). Lemma 5: Consider S, T S n. If 1 < K (S) = K (T ) <n,thens and T communicate with each other. Proof: Denote S = [s 1...s n ] and T = [t 1...t n ]. We prove this emma in a few steps. Step 1: We first prove that S communicates with Pσ n (S) for any integer if K (S) <n. Note that if K (S) =1, S must be [0...0] n or [1...1] n. The caim hods straightforwardy. Now we assume K (S) > 1. Since K (S) <n,theset I := {i : s i = s Modn(i+1), 1 i n} is nonempty. Moreover, because K (S) > 1, we can find j Isuch that s Modn(j+1) s Modn(j+2). By the structure of (1), the state f [Modn(j+2),Mod n(j+1)](s) is accessibe from S. By the definition of j, there hods f [j,modn(j+1)]f [Modn(j+2),Mod n(j+1)](s) =S. That is to say, the state S is accessibe from f [Modn(j+2),Mod n(j+1)](s). Therefore, S communicates with f [Modn(j+2),Mod n(j+1)](s). Appying this argument recursivey, we obtain that S communicates with f [Modn(j+n),Mod n(j+n 1)]...f [Modn(j+2),Mod n(j+1)](s), a state equa to P σn (S). It is then convenient to concude that S communicates with Pσ n (S) for any integer. Step 2: In this step, we prove that if S =[s 1...s n ] and T =[t 1...t n ] have identica K -reduced states, then S and T communicate with each other. Let K (S) =K (T )=[c 1...c d ]. If d =1or n, it is easy to see S = T. Now assume 1 <d<n. Because d>1, thesets{i : s i s 0 } and {i : t i t 0 } are not empty. Denote j 1 =max{i : s i s 0 },andj 2 =max{i : t i t 0 }. Without oss of generaity we assume j 1 > j 2. Apparenty T communicates with f [j2,j 2+1](T ). Furtherwe now that T communicates with T =[t 1...t n]:=f [j1 1,j 1]...f [j2+1,j 2+2]f [j2,j 2+1](T ). Moreover, we can concude that j 1 =max{i : t i t 0 }, and T and T have the same K -reduced state. So T and S have the same K -reduced state. By the definition of K - reduced state and the fact that j 1 =max{i : s i s 0 } = max{i : t i t 0}, we now that the L-reduced state of S is equa to the L-reduced state of T.Defineanewinegraph G, whose nodes are the nodes of G with edges being {i, i+1} for i =1,...n 1. According to Lemma 3, S aso communicates with T in M G(C pst ). Therefore, S communicates with T in M G (C pst ), because G is a subgraph of G. Thus, S and T communicate with each other. Step 3: This step wi compete the proof. Let d = K (S). IfK (S) =K (T ), wehavenownthats and T communicate with each other. We ony need to consider the case K (S) K (T ). Because K (S) = K (T ),theremust hod that K (T )=P σd (K (S)). Ford>1, theset{i : s i s 0 } is nonempty. Define j = min{i : s i s 0 }. According to Step 1, S communicates with Pσ j 1 n (S). By the definition of K -reduced states, we now that the K -reduced state of Pσ j 1 n (S) is P σd (K (S)), i.e., K (T ). Therefore, Pσ j 1 n (S) communicates with T, impying that S communicates with T. Now, we are ready to count the number of communication casses. According to Lemma 4, the digit number d of the K -reduced states of a the states in the same communication cass are identica. Moreover, d can be 1 or even numbers. If n =2m, there are three cases: (i) For d = 1, there are two communication casses {[0...0]} and {[1...1]}. (ii) For each d =2, 4,...,2m 2, according to Lemma 4 and Lemma 5, there is a unique communication cass whose eements have K -reduced with d digits. (iii) For d = 2m, the two states S 0 := [s 1...s 2m ] and T 0 := [ s 1... s 2m ] with s 2i 1 = 1 and s 2i = 0 for i =1,...,m, are the ony states whose K -reduced states are of ength 2m. Moreover, either S 0 or T 0 cannot be accessibe from any other state. That is to say, they form two communication casses. As a resut, there are a tota of m +3 communication casses. We have competed the proof for the case n =2m. The case with n =2m +1 can be simiary anayzed, whose detaied proof is omitted. This concudes the proof of Theorem 1(ii). C. Star Graph In this subsection, we prove that χ Cpst (G) = 5 if G is a star graph with n( 4) nodes. Note that a connected graph is caed a star graph if there is a node such that a the edges of the graph contain this node. This particuar node is caed the center node of the graph. The foowing proposition characterizes the communication casses for M G (C pst ) over a star graph G. Proposition 2: Let G be a star graph with n( 4) nodes. Then χ Cpst (G) = 5. Moreover, etting node 1 be the center node, the five casses are F 1 n = {[s 1...s n ]:s i =0, 1 i n}, F 2 n = {[s 1...s n ]:s i =1, 1 i n},

11 128 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 26, NO. 1, FEBRUARY 2018 F 3 n = {[s 1...s n ]:s 1 =0, s i =1, 2 i n}, F 4 n = {[s 1...s n ]:s 1 =1, s i =0, 2 i n}, F 5 n = {[s 1...s n ]: i, j, 2 i, j n, s i =0, s j =1}. Proof: Denote S n 1,...,S n 4 as the singeton state in F 1 n,...,f 4 n, respectivey. Moreover, any other state cannot be accessibe from S n 1 or S n 2, whie S n 3 or S n 4 cannot be accessibe from any other state. Thus, they do form communication casses, respectivey. We ony need to prove a the eements in F 5 n communicate with each other. We prove this by induction. First, et n =4.Thereare12 eements of F 5 4, isted as [0100], [0010], [0001], [0011], [0101], [0110], [1100], [1010], [1001], [1011], [1101], [1110]. It is easy to verify that they are in the same communication cass. Assume that for n = 4, a the eements in F 5 n communicate with each other. Now we prove the case for n = +1.LetG be a star graph with +1 nodes with node 1 being its center node. Let G be the subgraph of G with nodes 1,..., and a edges containing them in G. In fact, G is a star graph with nodes. By our induction assumption, a eements in F 5 communicate with each other in M G (C pst ). Because G is a subgraph of G, a eements in A := {[S0] S +1 : S F 5 } communicate with each other, and a eements in B := {[S1] S +1 : S F 5 } communicate with each other. Note that F 5 +1 = A B {[S 1 1], [S 2 0], [S 3 0], [S 4 1]}. Introduce U a = [ ], U b = [ ], U c = [ ] and U d = [ ]. They are eements of F 5. It is easy to verify that [U a 0] A is accessibe from [U a b 1] B. Moreover, [U 1] B is accessibe from [U b 0] A. Therefore, a eements in A B communicate with each other. It is straightforward to verify that [S 2 0] communicates with [S 3 c 0]. Aso,[U 0] A is accessibe from [S 3 0] and [S 2 b 0] is accessibe from [U 0] A. Thus, a eements in A {[S 2 0], [S 3 0]} communicate with each other. Moreover, [S 1 d 1] communicates with [S 4 1], [U 1] B is accessibe from [S 4 1], and [S 1 1] is accessibe from [U a 1] B. Therefore, a eements in B {[S 1 1], [S 4 1]} communicate with each other. Summarizing a these reations we now a eements in F 5 +1 = A B {[S 1 1], [S 2 0], [S 3 0], [S 4 1]} communicate with each other. This competes the proof of this proposition. D. Tree The foowing resut presents a characterization of the number of communication casses for tree graph that is not a ine. Proposition 3: Let G be a tree, having at east one node with degree greater than 2, i.e., G is not a ine graph. Then χ Cpst (G) = 5. The five communication casses can be described as foows: J 1 n = { [s 1...s n ]: s i =0, 1 i n }, J 2 n = { [s 1...s n ]: s i =1, 1 i n }, J 3 n = { [s 1...s n ]:s 1 =0, s i s j for any edge {i, j} of G }, J 4 n = { [s 1...s n ]: s 1 =1, s i s j for any edge {i, j} of G },and J 5 n = { [s 1 }...s n ]: i, j,, s.t.{i, j} is an edge of G and s i = s j s. Proof: It is straightforward to verify that any of J 1 n, J2 n, J 3 n, J4 n contains a unique eement, and forms a communication cass. We now prove J 5 n is a communication cass using an induction argument. For n = 4, G is a star graph which is proved in Proposition 2. Now assume that this proposition hods for n = 4. For any tree G with +1 nodes that is not a ine graph, there is a subgraph G with nodes which is sti a tree. Without oss of generaity, we denote the node not in G as node v = +1 V.Weusev 0 to denote the node with the highest degree in G (If there are more than one such nodes, we just choose one of them arbitrariy). There is a path (v 0,v 1,..., v h,v ) connecting node v 0 and node v in G, whereh 0 is an integer. By the induction assumption, the communication casses of M G (C) are J 1,...,J5 with each J defined by repacing n with in J n. Denote A = {[S0] S +1 : S J 5 } and B = {[S1] S +1 : S J 5 }. Note that J 5 +1 = A B {[S 1 1], [S 2 0], [S 3 0], [S 4 1]}. Because G is a subgraph of G, a eements in A communicate with each other, and a eements in B communicate with each other. Note that if G is a star graph with the v h being the center node, G wi be a star graph. This fas to the case discussed in Proposition 2. We assume G is not a star graph for the remainder of the proof. Introduce U a =[ v h 0...0], We have U a, U b J 5 is accessibe from [U a accessibe from [U b U b =[ v h 1...1]. a. It is easy to verify that [U 1] B 0] A. Moreover, [U b 0] A is 1] B. Therefore, a eements in A B communicate with each other. We further denote S 4 =[γ 1...γ ] and S 3 =[β 1...β ], and then U c := [γ 1...γ vh 1γ vh γ vh +1...γ ], := [β 1...β vh 1β vh β vh +1...β ]. It is straightforward to verify that [S 1 1] communicates with [U a 1] B, [S 2 0] communicates with [U b 0] A, [S 3 0] communicates with [U d 0] A, and[s 4 1] communicates with [U c 1] B. Thus, any two eements in J 5 +1 communicate with each other. U d E. Competion of the Proof The statements (i) and (ii) in Theorem 1 have been proved for the cases of ine and cyce graphs. We are now in

12 LI et a.: BOOLEAN GOSSIP NETWORKS 129 a pace to prove (iii) and (iv) based on our resuts for tree graphs. According to Proposition 3, for tree graphs without being a ine graph, there are five communication casses J 1 n, J 2 n, J3 n, J4 n and J5 n. Since any connected graph contains a spanning tree, the communication casses of M G (C pst ) for any connected graph G that is not a ine or cyce, can ony be unions of the J j n, j =1,...,5. Proof of Theorem 1(iii): Suppose G is neither a ine graph nor a cyce graph and it contains no odd cyce. There is a spanning tree of G, denoted G T. For M GT (C pst ), J 1 n, J 2 n, J3 n, J4 n and J5 n are communication casses. J1 n and J2 n are absorbing states in G. Because there is no odd cyce, J 3 n and J 4 n are the states that any pair of nodes associated with a common edge G share different vaues. That is to say, J 3 n and J 4 n cannot be accessibe from any other states in M G (C pst ). Thus, J 1 n, J 2 n, J 3 n, J 4 n and J 5 n are sti communication casses in M G (C pst ),i.e.χ Cpst (G) = 5. Proof of Theorem 1(iv): Now suppose G contains an odd cyce. Again, there is a spanning tree G T of G. J 1 n, J2 n, J3 n, J 4 n and J5 n are communication casses in M G T (C). Aso,J 1 n and J 2 n are absorbing states in M G (C pst ). For states in J 3 n and J 4 n, there is an edge e beonging to the odd cyce such that the pair nodes of this edge tae different vaues. Now, by choosing another spanning tree G T containing the edge e, we can prove that eements in J 3 n, J4 n and J5 n communicate with each other in M G (C pst ). In turn, χ Cpst (G) = 3. APPENDIX B TABLE OF NOTATIONS V The node set {1,...,n} E The edge set with each entry being an unordered pair of two distinct nodes in V G The graph (V, E) N i The neighbourhood of node i, {j : {i, j} E} x i (t) The binary vaue node i hods at time t A binary Booean function H The set of a binary Booean functions C Potentia node interaction rues aong the edges, subset of H X t The vector (x 1 (t),...,x n (t)) [s 1...s n ] Any state for s i {0, 1} S n State space { [s 1...s n ]: s i {0, 1},i V } The cardinaity of a set or the number of digits of a state P Transition matrix between states whose eements are represented by P [s1...s n][q 1...q n] M G (C) The Marov chain (S n,p) defined by random process X t,t 0 Booean AND operation Booean OR operation C pst The set {, } P( ) Probabiity of an event E{ } Expectation of a random variabe V{ } Variation of a random variabe χ Cpst (G) The number of communication casses of M G (C pst ) δ(t) Proportion of nodes that tae vaue 1 at time t δ(t) The expected vaue of δ(t) B 1 The set {C { A }:{ A } C { 2, 3, A, B }} B 2 B The set {C 2 H :{ 2, B } C { 2, 3, A, B }} The set B 1 B2 ACKNOWLEDGMENT The authors than the Associate Editor and the anonymous reviewers for their constructive comments, which heped us improve the presentation of the wor consideraby. 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13 130 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 26, NO. 1, FEBRUARY 2018 [20] D. Shah, Gossip agorithms, Found. Trends Netw., vo. 3, no. 1, pp , [21] S. A. Kauffman, The Origins of Order: Sef-Organization and Seection in Evoution. New Yor, NY, USA: Oxford Univ. Press, [22] D. G. Green, T. G. Leishman, and S. Sadedin, The emergence of socia consensus in Booean networs, in Proc. IEEE Symp. Artif. Life, Apr. 2007, pp [23] H. Kitano, Foundations of Systems Bioogy. Cambridge, MA, USA: MIT Press, [24] L. Tournier and M. Chaves, Uncovering operationa interactions in genetic networs using asynchronous Booean dynamics, J. Theor. Bio., vo. 260, no. 2, pp , [25] J. O. Kephart and S. R. White, Directed-graph epidemioogica modes of computer viruses, in Proc. IEEE Comput. Soc. Symp. Res. Secur. Privacy, May 1991, pp [26] C. M. Grinstead and J. L. Sne, Introduction to Probabiity, 2nd ed. Providence, RI, USA: American Mathematica Society, [27] Y. Kanoria and A. Montanari, Majority dynamics on trees and the dynamic cavity method, Ann. App. Probab., vo. 21, no. 5, pp , [28] H. Amini, M. Draief, and M. Learge, Fooding in weighted sparse random graphs, SIAM J. Discrete Math., vo. 27, no. 1, pp. 1 26, [29] R. W. R. Daring and J. R. Norris, Differentia equation approximations for Marov chains, Probab. Surv., vo. 5, no. 1, pp , [30] R. E. Simpson, Introductory Eectronics for Scientists and Engineers, 2nd ed. Boston, MA, USA: Ayn & Bacon, Bo Li received the B.Sc. degree in economics and the B.Sc. degree in mathematics and appied mathematics from Wuhan University, Wuhan, China, in 2005, and the Ph.D. degree in fundamenta mathematics from the Academy of Mathematics and Systems Science (AMSS), Chinese Academy of Sciences (CAS), Beijing, China, in From 2010 to 2012, he was a Post-Doctora Researcher with the Key Laboratory of Systems and Contro, AMSS, CAS. Since 2012, he has been an Assistant Professor with the Key Laboratory of Mathematics Mechanization, AMSS, CAS. He hed a visiting position from 2015 to 2016 with the Research Schoo of Engineering, The Austraian Nationa University, Canberra, ACT, Austraia. His research interests incude mathematica bioogy, bioinformatics, and networed contro systems. Junfeng Wu received the B.Eng. degree from the Department of Automatic Contro, Zhejiang University, Hangzhou, China, in 2009, and the Ph.D. degree in eectrica and computer engineering from The Hong Kong University of Science and Technoogy, Hong Kong, in In 2013, he was a Research Associate with the Department of Eectronic and Computer Engineering, The Hong Kong University of Science and Technoogy. From 2014 to 2017, he was a Post-Doctora Researcher with the Autonomic Compex Communication networs, Signas and Systems Linnaeus Center, Schoo of Eectrica Engineering, KTH Roya Institute of Technoogy, Stochom, Sweden. He is currenty with the Coege of Contro Science and Engineering, Zhejiang University, Hangzhou. His research interests incude networed contro systems, state estimation, and wireess sensor networs, mutiagent systems. He received the Guan Zhao- Zhi Best Paper Award at the 34th Chinese Contro Conference in He was seected to the nationa 1000-Youth Taent Program of China in Hongsheng Qi (M 10) received the Ph.D. degree in systems theory from the Academy of Mathematics and Systems Science, Chinese Academy of Sciences, in From 2008 to 2010, he was a Post-Doctora Feow with the Key Laboratory of Systems Contro, Chinese Academy of Sciences, where he currenty is an Associate Professor with the Academy of Mathematics and Systems Science. His research interests incude ogica dynamic systems, game theory and contro, and so on. He was a co-recipient of the Automatica Theory/Methodoogy Best Paper Prize in 2011 and a second recipient of the second Nationa Natura Science Award of China in Aexandre Proutiere received the degree in mathematics from Écoe Normae Supérieure, Paris, France, the degree in engineering from Téécom ParisTech, Paris, and the Ph.D. degree in appied mathematics from Écoe Poytechnique, Paaiseau, France, in He is an Engineer with the Corps of Mines. In 2000, he joined France Teecom Research and Deveopment as a Research Engineer. From 2007 to 2011, he was a Researcher with Microsoft Research, Cambridge, U.K. He is currenty a Professor with the Department of Automatic Contro, KTH The Roya Institute of Technoogy, Stochom, Sweden. He was a recipient in 2009 of the ACM Sigmetrics Rising Star Award, and received the best paper awards at the ACM Sigmetrics Conference in 2004 and 2010, and at the ACM Mobihoc Conference in He was an Associate Editor of the IEEE/ACM TRANSACTIONS ON NETWORKING and an Editor of the IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, and is currenty an EditoroftheIEEETRANSACTIONSON INFORMATIONTHEORY and Queuing Systems. Guodong Shi (M 15) received the B.Sc. degree in mathematics and appied mathematics from the Schoo of Mathematics, Shandong University, Jinan, China, in 2005, and the Ph.D. degree in systems theory from the Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China, in From 2010 to 2014, he was a Post-Doctora Researcher with the Autonomic Compex Communication networs, Signas and Systems Linnaeus Centre, KTH Roya Institute of Technoogy, Stochom, Sweden. Since 2014, he has been with the Research Schoo of Engineering, The Austraian Nationa University, Canberra, ACT, Austraia, where he is currenty a Senior Lecturer and Future Engineering Research Leadership Feow. He hed a visiting position at the University of New South Waes, Canberra, in 2014, and is currenty an Honorary Associate with the Austraian Center for Fied Robotics, Schoo of Aerospace, Mechanica and Mechatronic Engineering, The University of Sydney, Austraia. His current research interests incude distributed contro systems, quantum networing and decisions, and socia opinion dynamics.