IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. 4, NO. 2, JUNE 2017 369 Emergent Behaviors Over Signed Random Dynamical Networks: Relative-State-Flipping Model Guodong Shi, Member, IEEE, Alexandre Proutiere, Mikael Johansson, John S. Baras, Life Fellow, IEEE, and Karl Henrik Johansson, Fellow, IEEE

Abstract—We study asymptotic dynamical patterns that emerge Consensus problems aim to compute a weighted average of among a set of nodes interacting in a dynamically evolving signed the initial values held by a collection of nodes, in a distributed random network, where positive links carry out standard consen- manner. The DeGroot’s model [2], as a standard consensus sus, and negative links induce relative-state flipping. A sequence of deterministic signed graphs defines potential node interactions algorithm, described how opinions evolve in a network of that take place independently. Each node receives a positive rec- agents and showed that a simple deterministic opinion update ommendation consistent with the standard consensus algorithm based on the mutual trust and the differences in belief be- from its positive neighbors, and a negative recommendation de- tween interacting agents could lead to global convergence of fined by relative-state flipping from its negative neighbors. After the beliefs. Consensus dynamics have since then been widely receiving these recommendations, each node puts a deterministic weight to each recommendation, and then encodes these weighted adopted for describing opinion dynamics in social networks, recommendations in its state update through stochastic attentions for example, [6], [7], and [14]. In engineering sciences, a huge defined by two Bernoulli random variables. We establish a number amount of literature has studied these algorithms for distributed of conditions regarding almost sure convergence and divergence of averaging, formation forming, and load balancing between the node states. We also propose a condition for almost sure state collaborative agents under fixed or time-varying interaction net- clustering for essentially weakly balanced graphs, with the help of several martingale convergence lemmas. Some fundamental dif- works [15]–[22]. Randomized consensus seeking has also been ferences on the impact of the deterministic weights and stochastic widely studied, motivated by the random nature of interactions attentions to the node-state evolution are highlighted between the and updates in real complex networks [23]–[30]. current relative-state-flipping model and the state-flipping model This paper aims to study consensus dynamics with both col- considered previously. laborative and noncollaborative node interactions. A convenient Index Terms—Belief clustering, consensus dynamics, random framework for modeling different roles and relationships be- graphs, signed networks. tween agents is to use signed graphs introduced in the classical work by Heider in 1946 [10]. Each link is associated with I. INTRODUCTION a sign, either positive or negative, indicating collaborative or HE emergent behaviors, such as consensus, swarming, noncollaborative relationships. In [34], a model for consen- T clustering, and learning of the dynamics evolving over a sus over signed graphs was introduced for continuous-time large complex network of interconnected nodes have attracted a dynamics, where a node flips the sign of its true state to a significant amount of research attention in past decades [2]–[6]. negative (antagonistic) node during the interaction. The author In most cases, node interactions are collaborative, reflecting that of [34] showed that state polarization (clustering) of the signed their state updates obey the same rule which is spontaneous consensus model is closely related to the so-called structural or artificially designed, aiming for some particular collective balance in classical social signed graph theory [37]. In [35], task. This, however, might not always be true since nodes take the authors proposed a model for investigating the transition on different, or even opposing, roles, where examples arise in between agreement and disagreement when each link randomly biology [8], [9]; social science [10]–[12]; and engineering [13]. takes three types of interactions: 1) attraction; 2) repulsion; and 3) neglect, which was further generalized to a signed-graph Manuscript received December 5, 2014; revised October 4, 2015; accepted setting in [36]. December 5, 2015. Date of publication December 8, 2015; date of current We assume a sequence of deterministic signed graphs that version June 16, 2017. This work was supported in part by the Knut and Alice defines the interactions of the network. Random node in- Wallenberg Foundation, in part by the Swedish Research Council, in part by the KTH SRA TNG, and in part by AFOSR MURI under Grant FA9550-10-1- teractions take place under independent, but not necessarily 0573. Recommended by Associate Editor M. Franceschetti. identically distributed, random sampling of the environment. G. Shi is with the Research School of Engineering, College of Engineering Once interaction relations have been realized, each node re- and Computer Science, The Australian National University, Canberra, ACT 0200, Australia (e-mail: [email protected]). ceives a positive recommendation consistent with the standard A. Proutiere, M. Johansson, and K. H. Johansson are with ACCESS Linnaeus consensus algorithm from its positive neighbors. Nodes receive Centre, Royal Institute of Technology, 10044 Stockholm, Sweden (e-mail: negative recommendations from their negative neighbors. After [email protected]; [email protected]; [email protected]). J. S. Baras is with the Electrical and Computer Engineering Depart- receiving these recommendations, each node puts a (determin- ment, University of Maryland, College Park, MD 20742 USA (e-mail: baras@ istic) weight to each recommendation, and then encodes these umd.edu). weighted recommendations in its state update through stochas- Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. tic attentions defined by two Bernoulli random variables. In [1], Digital Object Identifier 10.1109/TCNS.2015.2506905 we studied almost sure convergence, divergence, and clustering

2325-5870 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. 370 IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. 4, NO. 2, JUNE 2017 under the definition of Altafini [34] for negative interactions, for which we referred to as a state-flipping model. In this paper, we further investigate this random consensus model for signed networks under a relative-state-flipping set- ting, where instead of taking negative feedback of the relative state in standard consensus algorithms [2], [4], positive feed- back takes place along every interaction arc of a negative sign. This relative-state flipping formulation is consistent with the models in [35] and [36], and can be viewed as a natural opposite of the DeGroot’s type of node interactions. For the proposed relative-state-flipping model, we establish a number of con- Fig. 1. Signed network and its three positive clusters. The positive arcs are ditions regarding almost sure convergence and divergence of solid, and the negative arcs are dashed. Note that negative arcs are allowed the node states. We also propose a condition for almost sure within positive clusters. node state clustering for essentially weakly balanced graphs, with the help of several martingale convergence lemmas. Some definition of random signed networks as introduced in [1], fundamental differences on the impact of the deterministic where each link is associated with a sign indicating cooperative weights and stochastic attentions to the node-state evolution are or antagonistic relations. In this paper, we study relative-state- highlighted between the current relative-state-flipping model flipping dynamics along each negative arc, in contrast with the and the state-flipping model. state-flipping dynamics studied in [1]. The main difference of The remainder of this paper is organized as follows.Section II the information patterns between the two models will also be presents the network dynamics and the node update rules, and carefully explained. specifies the information-level difference between the relative- state-flipping and state-flipping models. Section III presents our A. Signed Random Dynamical Networks main results; the detailed proofs are given in Section IV. Finally, some concluding remarks are drawn in Section V. Consider a network with a set V = {1,...,n} of n nodes, ≥ {G V E }∞ with n 3. Time is slotted for t=0, 1,....Let t =( , t) 0 be a sequence of (deterministic) signed directed graphs over A. Graph Theory, Notations, and Terminologies node set V. We denote by σij(t) ∈{+, −} the sign of arc A simple directed graph (digraph) G =(V, E) consists of a (i, j) ∈Et. The positive and negative subgraphs containing the V E⊆V×V G G+ V E+ finite set of nodes and an arc set , where e = positive and negative arcs of t are denoted by t =( , t ) ∈E ∈V ∈V ∈ G− V E− (i, j) denotes an arc from node i to j with (i, i) and t =( , t ), respectively. We say that the sequence of E ∈V {G } for all i . We call node j reachable from node i if there is graphs t t≥0 is sign consistent if the sign of any arc (i, j) a directed path from i to j. In particular, every node is supposed does not evolve over time, that is, if for any s, t ≥ 0 to be reachable from itself. A node v from which every node in V is reachable is called a center node (root). A digraph G is (i, j) ∈Es and (i, j) ∈Et =⇒ σij(s)=σij (t). strongly connected if every two nodes are mutually reachable; G V E E ∞ E G has a spanning tree if it has a center node; G is weakly We also define ∗ =( , ∗) with ∗ = t=0 t as the total {G } connected if every two nodes are reachable from each other graph of the network. If t t≥0 is sign consistent, then the after removing all of the directions of the arcs in E. A subgraph sign of each arc E∗ never changes and in that case, G∗ =(V, E∗) of G =(V, E) is a graph on the same node set V, whose arc set is a well-defined signed graph. The notion of positive cluster in is a subset of E. The induced graph of Vi ⊆Von G, denoted as a signed directed graph is defined as follows. G G|Vi, is the graph (Vi, Ei) with Ei =(Vi ×Vi) ∩E. A weakly Definition 1: Let be a signed digraph with positive + connected component of G is a maximal weakly connected subgraph G . A subset V∗ of the set of nodes V is a positive induced graph of G. If each arc (i, j) ∈Eis associated uniquely cluster if V∗ constitutes a weakly connected component of G+ G V with a sign, either “+”or“−,” G is called a signed graph and the . A positive cluster partition of is a partition of into V Tp V ≥ V sign of (i, j) ∈Eis denoted as σij. The positive and negative = i=1 i for some Tp 1, where for all i =1,...,Tp, i subgraphs containing the positive and negative arcs of G are is a positive cluster. denoted as G+ =(V, E+) and G− =(V, E−), respectively. Note that G admitting a positive-cluster partition is a gen- Depending on the argument, |·| stands for the absolute eralization of the classical definition of a weakly structural value of a real number, the Euclidean norm of a vector, or balanced graph for which negative links are strictly forbidden the cardinality of a set. The σ-algebra of a random variable is inside each positive cluster [38]. Given any node i, there is a denoted as σ(·).WeuseP(·) to denote the probability and E{·} unique weakly connected component of G+ that contains the as the expectation of their arguments, respectively. node i (particularly, if all arcs pointing to and leaving from node i are negative, node i by itself is a weakly connected component of G+ from the definition). Therefore, it is clear II. RANDOM NETWORK MODEL AND NODE UPDATES G that for any signed graph , there is a unique positive cluster V Tp V G In this section, we present the considered random network partition = i=1 i of , where Tp is the number of positive model and specify individual node dynamics. We use the same clusters covering the entire set V of nodes. SHI et al.: EMERGENT BEHAVIORS OVER SIGNED RANDOM DYNAMICAL NETWORKS 371

Each node randomly interacts with her neighboring nodes neighbor relationships. In the state-flipping model, naturally in Gt at time t. We present a general model on the random it is the head node along each negative arc that possesses the node interactions at a given time t.Attimet, some pairs of knowledge of sign of that arc. In the relative-state-flipping nodes are randomly selected for interaction. We denote by model, on the other hand, it is the tail node that knows the Et ⊂Et, the random subset of arcs corresponding to interacting sign of each directed arc so that nodes know whether a specific node pairs at time t.Tobeprecise,Et is sampled from the neighbor is positive or negative to implement the state updates distribution μt defined over the set Ωt of all subsets of arcs that cause the repulsive influence from its negative neighbors. E { } { } in t. We assume that E0,E1,...form a sequence of indepen- Let Bt t≥0 and Dt t≥0 be two sequences of independent { } { } dent sets of arcs. Formally, we introduce the probability space Bernoulli random variables. We assume that Bt t≥0, Dt t≥0, F { } ≥ (Θ, , P) obtained by taking the product of the probability and Gt t≥0 define independent processes. For any t 0, S S E{ } E{ } { } spaces(Ωt, t,μt), where t is the discrete σ-algebra on Ωt: define bt = Bt and dt = Dt . The processes Bt t≥0 F S ≥ { } Θ= t≥0 Ωt; is the product of σ-algebras t, t 0; and and Dt t≥0 represent how much attention node i pays to the P is the product probability measure of μt,t≥ 0. We denote positive and negative recommendations, respectively. Node i by Gt =(V,Et), the random subgraph of Gt corresponding updates her state as follows: + − to the random set Et of arcs. The disjoint sets Et and Et + − denote the positive and negative arc set of Et, respectively. si(t +1)=si(t)+αBthi (t)+βDthi (t) (1) Finally, we split the random set of nodes interacting with node i at time t depending on the sign of the corresponding arc: for where α, β > 0 are two positive constants marking the weight + { node i, the set of positive neighbors is defined as Ni (t):= j : each node put on the positive and negative recommendations, ∈ +} (j, i) Et , whereas similarly, the set of negative neighbors is respectively. For the value of α, we impose the following − { ∈ −} Ni (t):= j :(j, i) Et . assumption. A1. i) There holds α ∈ (0, (n − 1)−1) and ii) there is a con- stant p∗ ∈ (0, 1) such that for all t ≥ 0 and i, j ∈V, P((i, j) ∈ B. Node Updates Et) ≥ p∗ if (i, j) ∈Et. Each node i holds a state si(t) ∈ R at t =0, 1,.... To update Assumption A1.i) is just to ensure that the node update her state at time t, node i considers recommendations received rule (1) falls to standard consensus algorithms with positive from her positive and negative neighbors: recommendations only, which has to be made a bit conservative 1) The positive recommendation node that i receives at since we aim to build the results on general random dynamical time t is networks. This bound for α can certainly be relaxed without affecting any of our results if the random dynamical network + − − hi (t):= (si(t) sj (t)) . takes certain particular forms, for example, it will only require ∈ + ∈ j Ni (t) α (0, 1) for the upcoming analysis if gossiping processes [24] are considered. Assumption A1.ii) is a regularity condition to 2) The negative recommendations node that i receives at the arc existence in different time slots, which is, in fact, quite time t is defined as general and holds for many existing random network models, for example, [23], [25], and [26]. Assumption A1 will be − − hi (t):= (si(t) sj(t)) . adopted throughout this paper without specific further mention. − j∈N (t) T i Let s(t)=(s1(t) ...sn(t)) be the random vector repre- senting the network state at time t. The main objective of In the above expressions, we use the convention that summing this paper is to analyze the behavior of the stochastic process over empty sets yields a recommendation equal to zero, for ex- {s(t)}t≥0. In the following text, we denote by P the probability + ample, when node i has no positive neighbors, then hi (t)=0. measure capturing all random components driving the evolution − + In view of the definition of hi (t) in contrast to hi (t), the model of s(t). is referred to as the relative-state-flipping model. In the remainder of this paper, we establish the asymp- Remark 1: In [1], we have considered another notion of totic properties of the network state evolution under the negative recommendations, namely, the state-flipping model relative-state-flipping model. As will be shown in the follow- − introduced in [34], defined as h (t):=− ∈ − (si(t)+ i j Ni (t) ing text, the state-flipping and relative-state-flipping models sj (t)). We remark that for the relative-state-flipping model, the share some common nature, for example, almost sure state network does not require a central global coordinate system and convergence/divergence, no-survivor property, etc. In the mean- nodes can interact based on the relative state only. As has been time, these common properties can be driven by fundamen- pointed out in [1], in the state-flipping model, the network nodes tally different parameters regarding network connectivity and are necessary to share a common knowledge of the origin of the recommendation weights and attentions. We introduce a few state space. assumptions on the random graph process and dynamical Remark 2: The two definitions of negative recommendations: environment. {G } G 1) the relative-state-flipping model considered in the current A2. t t≥0 is a sign consistent in admitting a total graph ∗. paper and 2) the state-flipping model studied in [1] and [34] A3. The events {(i, j) ∈ Et}, i, j ∈V, and t =0, 1,... are have different physical interpretations and make different as- independent and there is a constant p∗ ∈ (0, 1) such that for all ∗ sumptions on the knowledge that nodes possess about their t ≥ 0 and i, j ∈V, P((i, j) ∈ Et) ≤ p if (i, j) ∈Et. 372 IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. 4, NO. 2, JUNE 2017

G+ Assumption A2 ensures that although the arc sets of t and positive/negative arc contributes to the state convergence un- − G can be time varying, there is no arc that varies its sign over der constant attention {bt} and {dt}, as long as α + β ≤ t − time. This means that the nature of the relationship (positive or (n − 1) 1. We can easily build examples showing that it is a negative) between any two nodes is fixed while the underlying completely different story on this matter for the relative-state- interaction structure is changing. Assumption A3 is a technical flipping model considered in this paper. assumption, which again can be verified for a large class of Remark 4: The divergence statement in Theorem 1 is not random network models studied in the literature, for example, true for the state-flipping model [1], where almost sure state [23], [25], and [26]. divergence always requires sufficiently large β.

III. MAIN RESULTS B. Deviation Consensus In this section, we present the main results for the asymptotic We define almost sure deviation consensus as follows. behaviors of the random process defined by the considered Definition 2: Algorithm (1) achieves almost sure deviation relative-state-flipping model. consensus if

A. General Conditions P lim sup max |si(t) − sj(t)| =0 =1. t→∞ i,j∈V First of all, the following theorem provides general condi- tions for convergence and divergence. Note that almost sure deviation consensus means that the Theorem 1: Assume that for any t ≥ 0, Gt ≡G for some digraph G, and that each positive cluster of G admits a spanning distances among the node states converge to zero, but conver- tree in G+. For Algorithm (1) under the relative-state-flipping gence of each node state is not required. We need the following model, we have condition on the positive graphs, which assumes a directed ∞ ∞ P 1) If t=0 dt < , then (limt→∞ si(t) exits)=1for all spanning tree in the union graph of the environment over certain nodei ∈Vand all initial states s(0). bounded time intervals [19]. ∞ ∞ G ≥ 2) If t=0 dt = , has two positive clusters with no A4. There is an integer K 1 such that the union graph G+ V E+ negative links in each cluster, and there is a negative arc ([t, t + K]) = ( i, τ∈[t,t+K−1] τ ) has a spanning tree between any two nodes from different clusters, then there for all t ≥ 0. exist an infinite number of initial states s(0) such that Theorem 2: Assume that A4 holds. Denote K0 =(2n − 3)K and ρ∗ =min{α, 1 − (n − 1)α}. Define

P | − | ∞ − lim max si(t) sj(t) = =1. (2) − (m+1)K0 1 t→∞ i,j∈V pn 1ρK0 X = ∗ ∗ (b (1 − d )) m 2 t t t=mK0 ⎛ ⎞ Note that by saying limt→∞ si(t) exists, we mean si(t) − (m+1)K0 1 converges to a finite limit. The first part of the above theorem in- K0 ⎝ ⎠ Ym =(1+2β(n − 1)) 1 − (1 − dt) . dicates that when the environment is frozen, and when positive t=mK0 clusters are properly connected, then irrespective of the mean { }∞ of the positive attentions bt 0 , the system states converge if ≤ − the attention each node puts in her negative neighbors decays Then, under the relative-state-flipping model, if 0 Xm ≤ ≥ ∞ − ∞ sufficiently fast over time. The second part of the theorem Ym 1 for all m 0 and m=0(Xm Ym)= , Algorithm (1) states that when this attention does not decay, divergence can achieves almost sure deviation consensus for all initial states. be observed. The following corollary is a direct consequence from Remark 3: Theorem 1 shows that well-structured positive Theorem 2, obtained by straightforward verification of the arcs and asymptotically decaying attention guarantee state con- parameter conditions in Theorem 2. ≡ ≡ vergence for a relative-state-flipping model. The essential rea- Corollary 1: Assume that A4 holds. Let bt b and dt d ∞ ∞ ∈ son is that when t=0 dt < , the first Borel-Cantelli lemma with b, d (0, 1). Then there exists d > 0 such that deviation [32, Theor. 2.3.1] ensures that along almost every sample path, consensus is achieved almost surely whenever d

{ } C. Almost Sure Divergence Lemma 1: Let vt t≥0 be a sequence of non-negative random variables with E{v } < ∞. Assume that for any t ≥ 0 We continue to provide conditions under which the maximal 0 gap between the states of two nodes grows large almost surely. E{vt+1|v0,...,vt}≤(1 − ξt)vt + θt We introduce a new technical condition on the negative graph, { } { } which assumes that the negative arcs span the entire network where ξt t≥0 and θt t≥0 are two (deterministic) sequences of ∀ ≥ ≤ ≤ ∞ ∞ leading to a weakly connected graph over certain bounded time non-negative numbers satisfying t 0, 0 ξt 1, t=0 ξt = , ∞ ∞ intervals. t=0 θt < , and limt→∞ θt/ξt =0. Then, limt→∞ vt =0a.s. ≥ A5. There is an integer K 1 such that the union graph We define h(t):=mini∈V si(t), H(t):=maxi∈V si(t), and G− V E− H − ([t, t + K]) = ( , τ∈[t,t+K−1] τ ) is weakly connected (t):=H(t) h(t), which will be used throughout the rest of for all t ≥ 0. this paper. The following lemma holds. ≡ ∞ ∞ Theorem 3: Assume that A3 and A5 hold. Let bt b and Lemma 2: Assume that t=0 dt < . Then, under the dt ≡ d for some constants b, d ∈ (0, 1). In addition, we require relative-state-flipping model, for all initial states, each h(t), α∈[0, (n−1)−1/2). Then, for Algorithm (1) under the relative- H(t), H(t) converges almost surely, respectively, to some state-flipping model, there is b > 0 such that whenever b

D. State Clustering P(Dt =1for finitely many t)=1.

Finally, we investigate the clustering of states of nodes In other words, we can define1 within each positive cluster. The following assumption ensures a minimum level of connectivity inside each positive cluster as TD := sup{Dt =1} t the environment changes. V Tp V A6. Assume that A2 holds and let = i=1 i be a which is finite with probability one. This is to say, the inequali- G positive-cluster partition of the total graph ∗. There is an ties in (5) hold almost surely for t ≥TD. + ≥ G |V integer K 1 such that the union graph ([t, t + K]) i = As a result, we conclude that V E+| ≥ ( i, τ∈[t,t+K−1] τ Vi ) has a spanning tree for all t 0. T ≤ ≤ ≤ ≤ ≤ T Theorem 4: Assume that A2 and A6 hold and let V = h( D) h(t) h(t +1) H(t +1) H(t) H( D) (4) Tp V be a positive-cluster partition of G∗. Define J(m)= i=1 i ≥T T T (m+1)K0−1 (m+1)K0−1 for all t D. It is obvious that h( D) and H( D) are random bt and W (m)= dt with K0 =(2n − t=mK0 ∞ t=mK0 ∞ variables taking finite values with probability one. The almost 3)K. Further assume that J(m)=∞, d < ∞, m=0 t=0 t sure convergence of h(t) and H(t) therefore follows directly and lim →∞ W (m)/J(m)=0. Then, under the relative-state- m from (4) which, in turn, implies the almost sure convergence flipping model, for any initial state s(0), Algorithm (1) achieves of H(t). a.s. state clustering in the sense that there are Tp real-valued ∗ ∗ We have now completed the proof.  random variables w ,...,w such that ∞ 1 Tp ∞ Lemma 3: Assume that t=0 dt < . Assume that for any † ∗ ≥ G ≡G G V P ∈V t 0, t for some digraph .Let be a positive cluster lim si(t)=wj ,i j ,j=1,...,Tp =1. t→∞ † † of G. Then, H (t):=maxi∈V† si(t) and h := mini∈V† si(t) converge almost surely, respectively, to some random variables Theorem 4 shows the possibility of state clustering for every that take finite value with probability one. positive cluster, whose proof is based on a martingale conver- Proof: Similarly, there holds gence lemma. H†(t +1)≤ H†(t); IV. PROOFS OF STATEMENTS h†(t +1)≥ h†(t); In this section, we establish the proofs of the various state- H†(t +1)≤H†(t) (5) ments presented in the previous section. if Dt =0. The desired conclusion follows from the same  A. Supporting Lemmas argument as the proof of Lemma 2.

The following is a martingale convergence lemma (see, for 1 From its definition, it is clear that TD is not a stopping time. Our argument, example, [33]). however, relies only on the fact that P(TD < ∞)=1. 374 IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. 4, NO. 2, JUNE 2017

Lemma 4: Assume that Dt =0for t=0,...,2(n−2)K−1. with K0 := inf{t ≥ K∗ : Bt =1}, which are stopping times ∈V − − { } { } Take i . Then, for any t=0,...,2(n 2)K 1, there holds for Bt t≥0. Now in view of the independence of Gt t≥0 and {Bt} ≥ , we know that {GK } ≥ is an independent process 1) If si(t) ≤ ζ0h(0) + (1 − ζ0)H(0) for some ζ0 ∈ (0, 1), t 0 m m 0 and each GK satisfies P((i, j) ∈ EK ) ≥ p∗ for all (i, j) ∈G then si(t +1)≤ λ∗ζ0h(0) + (1 − λ∗ζ0)H(0), where m m under Assumption A1. λ∗ =1− α(n − 1); Let V† be a positive cluster of G. By assumption, G|V† has 2) If si(t) ≤ ζ0h(0) + (1 − ζ0)H(0) for some ζ0 ∈ (0, 1), a spanning tree. Since α<1/(n − 1), the above discussion Bt =1, and (i, j) ∈ Gt, then sj(t +1)≤ αζ0h(0) + shows that at times Km,m=0, 1,..., the considered relative- (1 − αζ0)H(0). state-flipping model defines a standard consensus dynamics Proof: Note that the conditions that Dt =0 for t=0,..., on independent random graphs where each arc exists with a − − ∈ − −1 ≤ 2(n 2)K 1 and α (0, (n 1) ) yield H(t +1) H(t) probability of at least p∗ for any fixed time slot. Therefore, by ≥ − − and h(t +1) h(t) for all t =0,...,2(n 2)K 1. applying [39, Theor. 3.4] on randomized consensus dynamics ≤ − i) If Dt =0and si(t) ζ0h(0) + (1 ζ0)H(0) for some with arc-independent graphs, we conclude that the connectivity ∈ ζ0 (0, 1), then of V† ensures that + si(t +1)=si(t)+αBth (t) † i P lim H (Km)=0 =1 m→∞ ≤ si(t) − α (si(t) − sj (t)) † j∈N +(t) where H (t)=max∈V† s (t) − min ∈V† s (t). This immedi- i i i i i ≤ 1 − α N +(t) s (t)+α N +(t) H(t) ately gives us i i i ≤ − + − † 1 α Ni (t) (ζ0h(0) + (1 ζ0)H(0)) P lim H (t)=0 =1 →∞ + t + α Ni (t) H(0) ≤ λ∗ζ0h(0) + (1 − λ∗ζ0)H(0) (6) by the definition of the Km. Finally, according to Lemma 3, maxi∈V† si(t) and −1 in light of the fact that α ∈ (0, (n − 1) ), where λ∗ =1− mini∈V† si(t) converge almost surely. Define their limits † † α(n − 1). as, respectively, H∗ and h∗, which are finite with probability † ii) If si(t) ≤ ζ0h(0) + (1 − ζ0)H(0) for some ζ0 ∈ (0, 1), one. The fact that P(limt→∞ H (t)=0)=1 immediately † † Bt =1, and (i, j) ∈ Gt, there holds that leads to H∗ = h∗ almost surely. As a result, we conclude that − − † † sj(t +1)=sj (t) α (sj(t) sk(t)) P lim si(t)=H∗ = h∗ =1 t→∞ ∈ + k Nj (t) † − + for all i ∈V . = 1 α Nj (t) sj(t)+αsi(t) † Since V is chosen arbitrarily, we further know P(limt→∞ + α sk(t) si(t) exits)=1for all node i ∈V and all initial states s(0). ∈ + \{ } k Nj (t) i This proves the desired statement. V V G ≤ (1 − α)H(t)+α (ζ h(0) + (1 − ζ )H(0)) ii) Let 1 and 2 be the two positive clusters of .Let 0 0 s (0) = 0,i∈V and s (0) = C ,i∈V for some C > 0. ≤ − i 1 i 0 2 0 αζ0h(0) + (1 αζ0)H(0). (7) We define This proves the desired lemma.  f1(t):=maxsi(t); f2(t):=minsi(t). i∈V1 i∈V2

B. Proof of Theorem 1 Since either positive cluster contains positive links only and α ∈ (0, (n − 1)−1), there always holds that ∞ ∞ ∞ ∞ i) Let t=0 dt < . Then as long as t=0 bt < ,the first Borel-Cantelli Lemma guarantees that almost surely, each f1(t +1)≤ f1(t); f2(t +1)≥ f2(t). node revises its state for only a finite number of slots, which yields that the desired claim follows straightforwardly. In the Now that there is a negative arc between any two nodes from following text, we prove the desired conclusion based on the different clusters, it is straightforward to see that ∞ ∞ assumption that t=0 bt = . − ≥ − ∞ ∞ f2(t +1) f1(t +1) (1 + β)(f2(t) f1(t)) With t=0 dt < , from the first Borel-Cantelli Lemma ≥ f2(t) − f1(t)+C0 (8) K∗ := inf{k ≥ 0:Dt =0, ∀ t ≥ k} whenever Dt =1 and either (i∗,j∗) ∈ Et or (j∗,i∗) ∈ Et is a finite number almost surely. We note that K∗ is not a with i∗ =argmaxi∈V1 si(t) and j∗ =argmini∈V2 si(t).In { } { } stopping time for Dt t≥0, but a stopping time for Bt t≥0 light of Assumption A1, the second Borel-Cantelli Lemma by the independence of {Bt} ≥ and {Dt} ≥ . Hence, we can [32, Theor. 2.3.6] leads to that the event defined in (8) occurs t 0 t 0 ∞ ∞ recursively define infinitely often with probability one when t=0 dt = .The desired conclusion follows immediately. Km+1 := inf{t>Km : Bt =1},m=0, 1,... The proof is now complete.  SHI et al.: EMERGENT BEHAVIORS OVER SIGNED RANDOM DYNAMICAL NETWORKS 375

C. Proof of Theorem 2 From the selection of vm1 ,...,vmn−1 , the above procedure eventually gives us the same bound for The proof relies on Lemma 1, cf., [29] for the analysis of nodes u ,...,u , and calculating the probability of randomized consensus. We proceed in steps. 1 n the required events in the above argument, we obtain Step 1) Consider 2n − 3 intervals [mK, (m +1)K − 1],m=0,...,2(n − 2). In this step, we select a few ρK0 ρK0 key intervals from them. P s (K )≤ ∗ h(0)+ 1− ∗ H(0),i∈V i 0 2 2 With Assumption A4, there is a center node vm ∈ − V in each G([mK, (m +1)K − 1]). Therefore, we K0 1 n−1 count 2n − 3 center nodes (repetitions are allowed, ≥ p∗ (bt(1 − dt)) . of course) in the 2n − 3 intervals. There must hold t=0 ≤ that either svm (0) (h(0) + H(0))/2 or svm (0) > (h(0) + H(0))/2 for each m =1,...,2n − 3.This This implies − − is to say, we can find n 1 center nodes (again, K K0 1 ρ∗ 0 − repetitions are allowed) out of the vm’s, denoted P H(K )≤ 1− H(0) ≥pn 1 (b (1−d )). 0 2 ∗ t t vm1 ,...,vmn−1 , such that either t=0 (11) h(0) H(0) s (0) ≤ + ,j=1,...,n− 1 (9) Using a simple coordinate change by yi(t)= vmj 2 2 −si(t) and repeating the above analysis to the yi(t), we immediately know that (11) continues to hold for or the case with (10). Step 3) In this step, we conclude the proof making use of h(0) H(0) − Lemma 1. svm (0) > + ,j=1,...,n 1 (10) j 2 2 From the definition of the algorithm, there always holds holds. Step 2) In this step, we establish a probabilistic bound for P (H(t +1)≤ (1 + 2β(n − 1)) H(t)) = 1 (12) H(K0). To this end, we first assume that (9) holds. − D =0 t =0,...,2(n − 2)K − 1 K0 1 Let t for .We P H H ≤ − − carry out the following recursive argument: ( (K0) > (0)) 1 (1 dt). (13) t=0

1) Withourselection,vm1 isacenternodeofthegraph G − { } { } { } ([τ1K, (τ1 +1)K 1]) for some τ1 =0,..., Since Bt t≥0, Dt t≥0, and Gt t≥0 define inde- 2(n − 2) with s (0) ≤ (h(0) + H(0))/2. pendent processes, we conclude from (11)–(13) that vm1 Applying Lemma 4.(i), we conclude that E {H |H } ((m+1)K0) (mK0) K K ρ∗ 0 ρ∗ 0 sv (K0) ≤ h(0) + 1 − H(0) ≤ − H m1 2 2 (1 Xm +Ym) (mK0). Then, from Lemma 1, we immediately conclude the where K0 and ρ∗ are defined in the statement of almost sure convergence of H(mK0) as m tends to Theorem 2. infinity. The desired almost sure convergence of H(t) 2) Since v is a center, there exist t ∈ [τ K, m1 1 1 holds by further noticing (12). This completes the (τ +1)K−1] and j∗ = v ∈Vsuch that (v , 1 m1 m1 proof.  ∈ j∗) Et1 with probability at least p∗.IfBt1 =1 ∈ and (vm1 ,j∗) Et1 , then we can apply Lemma 4 and then conclude D. Proof of Theorem 3 K0 K0 We first establish a technical lemma. ρ∗ ρ∗ ≤ − −1 sj∗ (K0) h(0) + 1 H(0). Lemma 5: Let α ∈ [0, (n − 1) /2). Then 2 2 i) P(H(t +1)≥ (1 − 2(n − 1)α)H(t)) = 1; ii) P(H(t +1)< H(t)) ≤ b. For convenience, we re-denote vm1 and j∗ as u1 and u2, respectively. Proof: i) Take i, j ∈V satisfying si(t)=h(t) and 3) We proceed for vm .Ifvm ∈{u1,u2}, applying 2 2 sj(t)=H(t). Similar to the proof of Lemma 4, we can estab- Lemma 4. (i) again and we can obtain the same lish that almost surely bound for sj∗ (K0). Otherwise, either vm2 = u1 or v = u allows us to find another node m2 2 si(t +1)≤ λ∗h(t)+(1− λ∗)H(t) (14) u with the bound for s (K ) obtained as in 3 u3 0 ≥ − Step 2). sj (t +1) λ∗H(t)+(1 λ∗)h(t) (15) 376 IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. 4, NO. 2, JUNE 2017

hold, where λ∗ =1− α(n − 1). Noting that (14) and (15) yield As a result, we can bound the probability of E∗ and conclude − P H ≥H L0 · − 1 H(t +1)≥|sj(t +1)− si(t +1)| (KL0 ) (0)(1 + β) (n 1) ≥| − |H 2λ∗ 1 (t) L ∗ n−2 0 ≥ − − KL0 =(1− 2(n − 1)α) H(t). (16) dp∗(1 p ) (1 b) . (19)

Claim A is proved. We can now apply the same argument as the proof of ii) If bt =0, only negative recommendations can be effective [1, Prop. 1]. With (19) and Lemma 5, there holds in the node-state update. The desired conclusion is obvious.  E { H − H } Now we define log (KL0 ) log (0) t L L0 := inf t ∈ Z :(1+β) ≥ 2(n − 1) ∗ n−2 0 ≥ − − KL0 dp∗(1 p ) (1 b) K = K (n2 − n)(L − 1) + 1 . L0 0 − · log (1 + β)L0 · (n − 1) 1 + b log (1 − 2(n − 1)α) 2 Consider the KL /K =(n − n)(L0 − 1) + 1 time intervals 0 L ∗ n−2 0 K [mK, (m +1)K − 1] ≥ dp∗(1 − p ) (1 − b) L0 log 2

2 for m=0, 1,...,(n −n)(L0 −1). In view of Assumption A5, + b log (1 − 2(n − 1)α) each graph G−([mK, (m +1)K − 1]) is weakly connected. As a result, there must be a pair of nodes, saying > 0 (20) − (um,vm), such that (um,vm) ∈G ([mK, (m +1)K − 1]) | − |≥H − and sum (mK) svm (mK) (mK)/(n 1). Note that when b 0. We can pro- (u ,v ) ∈G−([mK, (m +1)K − 1]) means that there is H m m ceed to define U(m)=log (mKL0 ) for m =0, 1,.... Recur- t ∈ [mK, (m +1)K − 1] (u ,v ) ∈G− t > { } m such that m m tm .If m sively applying the above arguments to the process Um ,we 2 mK, we now construct the following event obtain that U(m) has a strictly positive drift when b

(m+1)K0−1 Bt =0for all t ∈ [0,KL − 1]} . (18) 0 n−1 ≥ p∗ (bt(1 − dt)) . (21) 3 From the above argument, we know that the event E∗ implies t=mK0

− H ≥| − |≥H L0 · − 1 Moreover, from the effect of the negative recommendations on (KL ) si∗ (KL ) sj∗ (KL ) (0)(1+β) (n 1) . 0 0 0 nodes in V†, we can easily modify (12) and (13) to that for all t

P (Θ(t +1)> Θ(t)) ≤ dt (22) 2 Note that um, vm,andtm are all deterministic. 3 P ≤ − H Here, it is crucial to notice that H(t +1)≥H(t) always holds if Bt =0. (Θ(t +1) (1 + 2β(n 1)) (t)) = 1. (23) SHI et al.: EMERGENT BEHAVIORS OVER SIGNED RANDOM DYNAMICAL NETWORKS 377

With (21)–(23), we arrive at Finally, Θ(t) converging almost surely to zero means that Ψ(t) and ψ(t) must converge to the same limit (their con- E {Θ((m +1)K0) |Θ(m(K0))} vergence is established in the beginning of the proof), which must be the limit of the each node state in V†.Wehavenow ≤ − (1 Xm)Θ(mK0) completed the proof. 

− (m+1)K0 1 − H +(1+2β(n 1)) dt (t) (24) V. C ONCLUSION t=mK0 This paper continued the study of [35] and [36] by investigat- where Xm is defined in Theorem 2. ing a relative-state-flipping model for consensus dynamics over On the other hand, it holds from the structure of the algorithm signed random networks. A sequence of deterministic signed that H(t +1)≤H(t) if Dt =0, and that H(t +1)≤ (1 + graphs defines potential node interactions that happen indepen- 2β(n − 1))H(t) if Dt =1. We therefore deduce that dently but not necessarily i.i.d. The positive recommendations are consistent with the standard consensus algorithm; negative E {H(t +1)|H(t)}≤(1 + 2β(n − 1)dt) H(t). (25) recommendations are defined by relative-state flipping from its negative neighbors. Each node puts a (deterministic) weight to From (25), we know each recommendation, and then encodes these weighted rec- ∞ ommendations in its state update through stochastic attentions defined by two Bernoulli random variables. We have estab- E (H(t)) ≤H0 (1 + 2β(n − 1)dt) t=0 lished several fundamental conditions regarding almost sure convergence and divergence of the network states. A condition for all t ≥ 0. Taking the expectation from both sides of (24), we for almost sure state clustering was also proposed for weakly obtain balanced graphs, with the help of martingale convergence lemmas. Some fundamental differences were also highlighted E{Θ((m +1)K0)}≤(1 − Xm)E {Θ(mK0)} between the current relative-state-flipping model and the state- ∞ flipping model considered in [1] and [34]. + (1 + 2β(n − 1)) H0 (1 + 2β(n − 1)dt) W (m) (26) t=0 ACKNOWLEDGMENT (m+1)K0−1 where W (m)= t=mK dt. The authors thank the Associate Editor and the anonymous 0 ∞ ∞ Note that it is well known that t=0 dt < implies reviewers for their valuable comments, which helped us im- ∞ − ∞ − ∞ t=0(1 dt) > 0 and t=0(1 + 2β(n 1)dt) < . Conse- prove the technical clarity of the paper considerably. ∞ ∞ ∞ ∞ quently, m=0 J(m)= implies m=0 Xm = .Inview of Lemma 1, we have REFERENCES lim E {Θ((m +1)K0)} =0 (27) m→∞ [1] G. Shi, A. Proutiere, M. Johansson, J. S. Baras, and K. H. Johansson, “Emergent behaviors over signed random dynamical networks: State- flipping model,” IEEE Trans. Control Netw. Syst., vol. 2, no. 2, if lim →∞ W (m)/J(m)=0. Invoking Fatou’s lemma m pp. 142–153, Jun. 2015. [32, Theor. 1.6.5], we further conclude that [2] M. H. DeGroot, “Reaching a consensus,” J. Amer. Stat. Assoc., vol. 69, pp. 118–121, 1974. [3] T. Vicsek, A. Czirok, E. B. Jacob, I. Cohen, and O. Schochet, “Novel type E lim inf Θ((m +1)K0) ≤ lim E {Θ((m +1)K0)}=0 m→∞ t→∞ of phase transitions in a system of self-driven particles,” Phys. Rev. Lett., (28) vol. 75, pp. 1226–1229, 1995. [4] A. Jadbabaie, J. Lin, and A. S. Morse, “Coordination of groups of mobile which actually implies autonomous agents using nearest neighbor rules,” IEEE Trans. Autom. Control, vol. 48, no. 6, pp. 988–1001, Jun. 2003. [5] E. Bonabeau, M. Dorigo, and G. Theraulaz, Swarm Intelligence: From E lim Θ((m +1)K0) =0 (29) m→∞ Natural to Artificial System. New York, NY, USA: Oxford University Press, 1999. [6] B. Golub and M. O. Jackson, “Naive learning in social networks and the since Θ((m +1)K0) converges almost surely. Therefore, we wisdom of crowds,” Amer. Econ. J.: Microecon., vol. 2, pp. 112–149, have reached 2010. [7] P. M. DeMarzo, D. Vayanos, and J. Zwiebel, “Persuasion bias, social influence, and unidimensional opinions,” Quart. J. Econ., vol. 118, no. 3, P lim Θ((m +1)K0)=0 =1 (30) m→∞ pp. 909–968, 2003. [8] L. Edelstein-Keshet, Mathematical Models in Biology.NewYork,NY, which, in turn, leads to USA: McGraw-Hill, 1987. [9] N. Yosef et al., “Dynamic regulatory network controlling TH17 cell differentiation,” Nature, vol. 496, pp. 461–468, 2013. P lim Θ(t)=0 =1 (31) [10] F. Heider, “Attitudes and cognitive organization,” J Psychol., vol. 21, →∞ t pp. 107–112, 1946. [11] S. Galam, “Fragmentation versus stability in bimodal coalitions,” Phys. again, from the fact that Θ(t) converges almost surely. A., vol. 230, pp. 174–188, 1996. 378 IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. 4, NO. 2, JUNE 2017

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John S. Baras (LF’14) received the Diploma Karl Henrik Johansson (F’13) received the M.Sc. in electrical and mechanical engineering (Hons.) and Ph.D. degrees in electrical engineering from from the National Technical University of Athens, Lund University, Lund, Sweden. Athens, Greece, in 1970 and the M.S. and Currently, he is Director of the ACCESS Linnaeus Ph.D. degrees in applied mathematics from Harvard Centre and Professor at the School of Electrical University, Cambridge, MA, USA, in 1971 and 1973, Engineering, KTH Royal Institute of Technology, respectively. Stockholm, Sweden. He is a Wallenberg Scholar Since 1973, he has been with the Department of and has held a Senior Researcher Position with Electrical and Computer Engineering, University the Swedish Research Council. He also heads the of Maryland at College Park, College Park, MD, Stockholm Strategic Research Area ICT The Next USA, where he is currently Professor; member Generation. He has held visiting positions at UC of the Applied Mathematics, Statistics and Scientific Computation Program Berkeley, Berkeley, CA, USA; California Institute of Technology, Pasadena, Faculty; and Affiliate Professor in the Fischell Department of Bioengineering CA, USA; Nanyang Technological University, Nanyang, China; and the Insti- and the Department of Mechanical Engineering. From 1985 to 1991, he was tute of Advanced Studies, Hong Kong University of Science and Technology, the Founding Director of the Institute for Systems Research (ISR) (one of Hong Kong, China. He has been on the editorial boards of several journals, the first six National Science Foundation Engineering Research Centers). including Automatica, IEEE TRANSACTIONS ON AUTOMATIC CONTROL,and In 1990, he was appointed to the Lockheed Martin Chair in Systems Engi- IET Control Theory and Applications. He is currently a Senior Editor of IEEE neering. Since 1991, he has been the Director of the Maryland Center for Hybrid TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS and Associate Editor Networks (HYNET), which he co-founded. He has held visiting research of European Journal of Control. He has been Guest Editor for a special issue of scholar positions with Stanford University, Stanford, CA, USA; Massachusetts IEEE TRANSACTIONS ON AUTOMATIC CONTROL on cyberphysical systems Institute of Technology, Cambridge, MA, USA; Harvard, Cambridge; the and one for IEEE Control Systems Magazine on cyberphysical security. His Institute National de Reserche en Informatique et en Automatique (INRIA); research interests are in networked control systems, cyberphysical systems, and the University of California at Berkeley, Berkeley, CA, USA; Linkoping applications in transportation, energy, and automation systems. University; and the Royal Institute of Technology (KTH), Stockholm, Sweden. Dr. Johansson has served on the executive committees of several European Dr. Baras’ awards are: the 1980 George S. Axelby Award of the IEEE Control research projects in the area of networked embedded systems. He received Systems Society; the 1978, 1983, and 1993 Alan Berman Research Publication the Best Paper Award of the IEEE International Conference on Mobile Awards from the NRL; the 1991, 1994, and 2008 Outstanding Invention of the Ad-hoc and Sensor Systems in 2009 and the Best Theory Paper Award of Year Awards from the University of Maryland; the 1998 Mancur Olson Research the World Congress on Intelligent Control and Automation in 2014. In 2009, Achievement Award from the University of Maryland, College Park; the 2002 he was awarded Wallenberg Scholar, as one of the first ten scholars from and 2008 Best Paper Awards at the 23rd and 26th Army Science Conferences; all sciences, by the Knut and Alice Wallenberg Foundation. He was awarded the 2004 Best Paper Award at the Wireless Security Conference WISE04; the Future Research Leader from the Swedish Foundation for Strategic Research 2007 IEEE Communications Society Leonard G. Abraham Prize in the Field in 2005. He received the triennial Young Author Prize from IFAC in 1996 of Communication Systems; the 2008 IEEE Globecom Best Paper Award for and the Peccei Award from the International Institute of System Analysis, wireless networks; and the 2009 Maryland Innovator of the Year Award. In Austria, in 1993. He received the Young Researcher Awards from Scania in 2012, he was awarded the Principal Investigator with Greatest Impact and for 1996 and from Ericsson in 1998 and 1999. He was the General Chair of the the Largest Selling Product with Hughes Network Systems for HughesNet, over ACM/IEEE Cyber-Physical Systems Week 2010, Stockholm, and IPC Chair the last 25 years of operation of the Maryland Industrial Partnerships Program. of many conferences. He has been a member of the IEEE Control Systems These awards recognized his pioneering invention, prototyping, demonstration, Society Board of Governors and the Chair of the IFAC Technical Committee and help with commercialization of Internet protocols and services over satel- on Networked Systems. lites in 1994, which created a new industry serving tens of millions worldwide. In 2014, he was awarded the 2014 Tage Erlander Guest Professorship by the Swedish Research Council, and a three-year (2014–2017) Hans Fischer Senior Fellowship from the Institute for Advanced Study of the Technical University of Munich. He has been the initial architect and continuing innovator of the pioneering MS on Systems Engineering program of the ISR. His research interests include control, communication, and computing systems. He holds eight patents and has four more pending. He is a Fellow of SIAM and a Foreign Member of the Royal Swedish Academy of Engineering Sciences (IVA).