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.