Emergent Behaviors over Signed Random Dynamical Networks: Relative-State-Flipping Model
Guodong Shi, Alexandre Proutiere, Mikael Johansson, John. S. Baras, Karl H. Johansson
EEmergent Behaviors over Signed Random DynamicalNetworks: Relative-State-Flipping Model
Guodong Shi, Alexandre Proutiere, Mikael Johansson,John. S. Baras, and Karl H. Johansson ∗ Abstract
We study asymptotic dynamical patterns that emerge among a set of nodes interactingin a dynamically evolving signed random network, where positive links carry out standardconsensus and negative links induce relative-state flipping. A sequence of deterministic signedgraphs define potential node interactions that take place independently. Each node receivesa positive recommendation consistent with the standard consensus algorithm from its pos-itive neighbors, and a negative recommendation defined by relative-state flipping from itsnegative neighbors. After receiving these recommendations, each node puts a determinis-tic weight to each recommendation, and then encodes these weighted recommendations inits state update through stochastic attentions defined by two Bernoulli random variables.We establish a number of conditions regarding almost sure convergence and divergence ofthe node states. We also propose a condition for almost sure state clustering for essentiallyweakly balanced graphs, with the help of several martingale convergence lemmas. Some fun-damental differences on the impact of the deterministic weights and stochastic attentionsto the node state evolution are highlighted between the current relative-state-flipping modeland the state-flipping model considered in Altafini 2013 and Shi et al. 2014.
Keywords.
Random graphs, Signed networks, Consensus dynamics, Belief clustering ∗ G. Shi is with the College of Engineering and Computer Science, The Australian National University, CanberraACT 0200, Australia. A. Proutiere, M. Johansson, and K. H. Johansson are with ACCESS Linnaeus Centre, RoyalInstitute of Technology, Stockholm 10044, Sweden. J. S. Baras is with Electrical and Computer Engineering, Uni-versity of Maryland, College Park, MD 20742, USA. This work has been supported in part by the Knut and AliceWallenberg Foundation, the Swedish Research Council, KTH SRA TNG, and by AFOSR MURI grant FA9550-10-1-0573. e-mail: [email protected], [email protected], [email protected], [email protected], [email protected]. a r X i v : . [ c s . S I] D ec hi et al. Signed Random Dynamical Networks: Relative-State-Flipping Model
The emergent behaviors, such as consensus, swarming, clustering, and learning, of the dynamicsevolving over a large complex network of interconnected nodes have attracted a significantamount of research attention in the past decades [2, 3, 4, 5, 6]. In most cases node interactionsare collaborative, reflected by that their state updates obey the same rule which is spontaneousor artificially designed aiming for some particular collective task. This however might not alwaysbe true since nodes take on different, or even opposing, roles, where examples arise in biology[8, 9], social science [10, 11, 12], and engineering [13].Consensus problems aim to compute a weighted average of the initial values held by acollection of nodes, in a distributed manner. The DeGroot’s model [2], as a standard consensusalgorithm, described how opinions evolve in a network of agents, and showed that a simpledeterministic opinion update based on the mutual trust and the differences in belief betweeninteracting agents could lead to global convergence of the beliefs. Consensus dynamics have sincethen been widely adopted for describing opinion dynamics in social networks, e.g., [6, 7, 14]. Inengineering sciences, a huge amount of literature has studied these algorithms for distributedaveraging, formation forming and load balancing between collaborative agents under fixed ortime-varying interaction networks [15, 16, 17, 18, 19, 20, 21, 22]. Randomized consensus seekinghas also been widely studied, motivated by the random nature of interactions and updates inreal complex networks [23, 24, 25, 26, 27, 28, 29, 30].This paper aims to study consensus dynamics with both collaborative and non-collaborativenode interactions. A convenient framework for modeling different roles and relationships betweenagents is to use signed graphs introduced in the classical work by Heider in 1946 [10]. Each link isassociated with a sign, either positive or negative, indicating collaborative or non-collaborativerelationships. In [34], a model for consensus over signed graphs was introduced for continuous-time dynamics, where a node flips the sign of its true state to a negative (antagonistic) nodeduring the interaction. The author of [34] showed that state polarization (clustering) of thesigned consensus model is closely related to the so-called structural balance in classical socialsigned graph theory [37]. In [35], the authors proposed a model for investigating the transitionbetween agreement and disagreement when each link randomly takes three types of interactions:attraction, repulsion, and neglect, which was further generalized to a signed-graph setting in[36]. 2 hi et al.
Signed Random Dynamical Networks: Relative-State-Flipping Model
We assume a sequence of deterministic signed graphs that defines the interactions of the net-work. Random node interactions take place under independent, but not necessarily identicallydistributed, random sampling of the environment. Once interaction relations have been realized,each node receives a positive recommendation consistent with the standard consensus algorithmfrom its positive neighbors. Nodes receive negative recommendations from its negative neigh-bors. After receiving these recommendations, each node puts a (deterministic) weight to eachrecommendation, and then encodes these weighted recommendations in its state update throughstochastic attentions defined by two Bernoulli random variables. In [1], we studied almost sureconvergence, divergence, and clustering under the definition of Altafini [34] for negative interac-tions, for which we referred to as a state-flipping model.In this paper, we further investigate this random consensus model for signed networks undera relative-state-flipping setting, where instead of taking negative feedback of the relative statein standard consensus algorithms [2, 4], a positive feedback takes place along every interactionarc of a negative sign. This relative-state flipping formulation is consistent with the models in[35, 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 conditions regardingalmost sure convergence and divergence of the node states. We also propose a condition foralmost sure node state clustering for essentially weakly balanced graphs, with the help of severalmartingale convergence lemmas. Some fundamental differences on the impact of the deterministicweights and stochastic attentions to the node state evolution are highlighted between the currentrelative-state-flipping model and the state-flipping model.The remainder of the paper is organized as follows. Section 2 presents the network dynamicsand the node update rules, and specifies the information-level difference between the relative-state-flipping and state-flipping models. Section 3 presents our main results; the detailed proofsare given in Section 4. Finally some concluding remarks are drawn in Section 5.
Graph Theory, Notations and Terminologies
A simple directed graph (digraph) G = ( V , E ) consists of a finite set V of nodes and an arc set E ⊆ V × V , where e = ( i, j ) ∈ E denotes an arc from node i ∈ V to j ∈ V with ( i, i ) / ∈ E for all i ∈ V . We call node j reachable from node i if there is a directed path from i to j . In particularevery node is supposed to be reachable from itself. A node v from which every node in V is3 hi et al. Signed Random Dynamical Networks: Relative-State-Flipping Model reachable is called a center node (root). A digraph G is strongly connected if every two nodesare mutually reachable; G has a spanning tree if it has a center node; G is weakly connected if aconnected undirected graph can be obtained by removing all the directions of the arcs in E . Asubgraph of G = ( V , E ), is a graph on the same node set V whose arc set is a subset of E . Theinduced graph of V i ⊆ V on G , denoted G|V i , is the graph ( V i , E i ) with E i = ( V i × V i ) ∩ E . Aweakly connected component of G is a maximal weakly connected induced graph of G . If eacharc ( i, j ) ∈ E is associated uniquely with a sign, either ’+’ or ’ − ’, G is called a signed graphand the sign of ( i, j ) ∈ E is denoted as σ ij . The positive and negative subgraphs containing thepositive and negative arcs of G , are denoted as G + = ( V , E + ) and G − = ( V , E − ), respectively.Depending on the argument, | · | stands for the absolute value of a real number, the Euclideannorm of a vector or the cardinality of a set. The σ -algebra of a random variable is denoted as σ ( · ).We use P ( · ) to denote the probability and E {·} the expectation of their arguments, respectively. In this section, we present the considered random network model and specify individual nodedynamics. We use the same definition of random signed networks as introduced in [1], whereeach link is associated with a sign indicating cooperative or antagonistic relations. In the currentwork we study relative-state-flipping dynamics along each negative arcs, in contrast with thestate-flipping dynamics studied in [1]. The main difference of the information patterns betweenthe two models will also be carefully explained.
Consider a network with a set V = { , . . . , n } of n nodes, with n ≥
3. Time is slotted for t = 0 , , . . . . Let (cid:8) G t = ( V , E t ) (cid:9) ∞ be a sequence of (deterministic) signed directed graphs overnode set V . We denote by σ ij ( t ) ∈ { + , −} the sign of arc ( i, j ) ∈ E t . The positive and negativesubgraphs containing the positive and negative arcs of G t , are denoted by G + t = ( V , E + t ) and G − t = ( V , E − t ), respectively. We say that the sequence of graphs {G t } t ≥ is sign consistent if thesign of any arc ( i, j ) does not evolve over time, i.e., if for any s, t ≥ i, j ) ∈ E s and ( i, j ) ∈ E t = ⇒ σ ij ( s ) = σ ij ( t ) . hi et al. Signed Random Dynamical Networks: Relative-State-Flipping Model
Figure 1: A signed network and its three positive clusters. The positive arcs are solid, and thenegative arcs are dashed. Note that negative arcs are allowed within positive clusters.We also define G ∗ = ( V , E ∗ ) with E ∗ = (cid:83) ∞ t =0 E t as the total graph of the network. If {G t } t ≥ issign consistent, then the sign of each arc E ∗ never changes and in that case, G ∗ = ( V , E ∗ ) is awell-defined signed graph. The notion of positive cluster in a signed directed graph is defined asfollows. Definition 1
Let G be a signed digraph with positive subgraph G + . A subset V ∗ of the set ofnodes V is a positive cluster if V ∗ constitutes a weakly connected component of G + . A positivecluster partition of G is a partition of V into V = (cid:83) T p i =1 V i for some T p ≥ , where for all i = 1 , . . . , T p , V i is a positive cluster. Note that G admitting a positive-cluster partition is a generalization of the classical definitionof weakly structural balanced graph for which negative links are strictly forbidden inside eachpositive cluster [38]. From the above definition, it is clear that for any signed graph G , there is aunique positive cluster partition V = (cid:83) T p i =1 V i of G , where T p is the number of positive clusterscovering the entire set V of nodes.Each node randomly interacts with her neighboring nodes in G t at time t . We present a generalmodel on the random node interactions at a given time t . At time t , some pairs of nodes arerandomly selected for interaction. We denote by E t ⊂ E t the random subset of arcs correspondingto interacting node pairs at time t . To be precise, E t is sampled from the distribution µ t definedover the set Ω t of all subsets of arcs in E t . We assume that E , E , . . . form a sequence ofindependent sets of arcs. Formally, we introduce the probability space (Θ , F , P) obtained bytaking the product of the probability spaces (Ω t , S t , µ t ), where S t is the discrete σ -algebra on5 hi et al. Signed Random Dynamical Networks: Relative-State-Flipping Model Ω t : Θ = (cid:81) t ≥ Ω t , F is the product of σ -algebras S t , t ≥
0, and P is the product probabilitymeasure of µ t , t ≥
0. We denote by G t = ( V , E t ) the random subgraph of G t correspondingto the random set E t of arcs. The disjoint sets E + t and E − t denote the positive and negativearc set of E t , respectively. Finally, we split the random set of nodes interacting with node i attime t depending on the sign of the corresponding arc: for node i , the set of positive neighborsis defined as N + i ( t ) := (cid:8) j : ( j, i ) ∈ E + t (cid:9) , whereas similarly, the set of negative neighbors is N − i ( t ) := (cid:8) j : ( j, i ) ∈ E − t (cid:9) . Each node i holds a state s i ( t ) ∈ R at t = 0 , , . . . . To update her state at time t , node i considers recommendations received from her positive and negative neighbors:(i) The positive recommendation node i receives at time t is h + i ( t ) := − (cid:88) j ∈ N + i ( t ) (cid:0) s i ( t ) − s j ( t ) (cid:1) ;(ii) The negative recommendations node i receives at time t is defined as: h − i ( t ) := (cid:88) j ∈ N − i ( t ) (cid:0) s i ( t ) − s j ( t ) (cid:1) . In the above expressions, we use the convention that summing over empty sets yields a recom-mendation equal to zero, e.g., when node i has no positive neighbors, then h + i ( t ) = 0. In view ofthe definition of h − i ( t ) in contrast to h + i ( t ), the model is referred to as the relative-state-flippingmodel. Remark 1
In [1], we have considered another notion of negative recommendations, namely thestate-flipping model introduced in [34], defined as h − i ( t ) := − (cid:80) j ∈ N − i ( t ) (cid:0) s i ( t )+ s j ( t ) (cid:1) . We remarkthat for the relative-state-flipping model, the network does not require a central global coordinatesystem and nodes can interact based on relative state only. As has been pointed in [1], in thestate-flipping model, the network nodes are necessary to share a common knowledge of the originof the state space. Remark 2
The two definitions of negative recommendations, the relative-state-flipping modelconsidered in the current paper, and the state-flipping model studied in [34, 1], have different hi et al. Signed Random Dynamical Networks: Relative-State-Flipping Modelphysical interpretations and make different assumptions on the knowledge that nodes possessabout their neighbor relationships. In the state-flipping model, naturally it is the head node alongeach negative arc, that possesses the knowledge of sign of that arc. In the relative-state-flippingmodel, on the other hand, it is the tail node knows the sign of each directed arc so that nodesknow if a specific neighbor is positive or negative to implement the state updates that cause therepulsive influence from its negative neighbors.
Let { B t } t ≥ and { D t } t ≥ be two sequences of independent Bernoulli random variables. Weassume that { B t } t ≥ , { D t } t ≥ , and { G t } t ≥ define independent processes. For any t ≥
0, define b t = E { B t } and d t = E { D t } . The processes { B t } t ≥ and { D t } t ≥ represent how much attentionnode i pays to the positive and negative recommendations, respectively. Node i updates herstate as follows: s i ( t + 1) = s i ( t ) + αB t h + i ( t ) + βD t h − i ( t ) , (1)where α, β > s ( t ) = (cid:0) s ( t ) . . . s n ( t ) (cid:1) T be the random vector representing the network state at time t .The main objective of this paper is to analyze the behavior of the stochastic process { s ( t ) } t ≥ .In the following, we denote by P the probability measure capturing all random componentsdriving the evolution of s ( t ).In the remainder of the paper, we establish the asymptotic properties of the network stateevolution under relative-state-flipping model. As will be shown in the following, the state-flippingand relative-state-flipping models share some common nature, e.g., almost sure state conver-gence/divergence, no-survivor property, etc. In the mean time these common properties can bedriven by fundamentally different parameters regarding network connectivity and recommenda-tion weights and attentions. For the consistency of presentation we introduce the same set ofassumptions on the random graph process and the connectivity of the dynamical environmentas used in [1]. A1.
There is a constant p ∗ ∈ (0 ,
1) such that for all t ≥ i, j ∈ V , P (cid:0) ( i, j ) ∈ E t (cid:1) ≥ p ∗ if( i, j ) ∈ E t . A2.
There is an integer K ≥ G (cid:0) [ t, t + K − (cid:1) = (cid:0) V , (cid:83) τ ∈ [ t,t + K − E τ (cid:1) is strongly connected for all t ≥
0. 7 hi et al.
Signed Random Dynamical Networks: Relative-State-Flipping Model
A3. {G t } t ≥ is sign consistent admitting a total graph G ∗ . A4.
There is an integer K ≥ G + (cid:0) [ t, t + K ] (cid:1) = (cid:0) V , (cid:83) τ ∈ [ t,t + K − E + τ (cid:1) is strongly connected for all t ≥ A5.
There is an integer K ≥ G − (cid:0) [ t, t + K ] (cid:1) = (cid:0) V , (cid:83) τ ∈ [ t,t + K − E − τ (cid:1) is strongly connected for all t ≥ A6.
The events { ( i, j ) ∈ G t } , i, j ∈ V , t = 0 , , . . . are independent and there is a constant p ∗ ∈ (0 ,
1) such that for all t ≥ i, j ∈ V , P (cid:0) ( i, j ) ∈ G t (cid:1) ≤ p ∗ if ( i, j ) ∈ E t . In this section, we present the main results for the asymptotic behaviors of the random processdefined by the considered relative-state-flipping model.
First of all, the following theorem provides general conditions for convergence and divergence.
Theorem 1
Let A1 hold and α ∈ (0 , ( n − − ) and β > . Assume that for any t ≥ , G t ≡ G for some digraph G , and that each positive cluster of G admits a spanning tree in G + . ForAlgorithm (1) under the relative-state-flipping model, we have:(i) If (cid:80) ∞ t =0 d t < ∞ , then P (cid:0) lim t →∞ s i ( t ) exits (cid:1) = 1 for all node i ∈ V and all initial states s (0) ;(ii) If (cid:80) ∞ t =0 d t = ∞ , G has two positive clusters with no negative links in each cluster, andthere is a negative arc between any two nodes from different clusters, then there exist aninfinite number of initial states s (0) such that P (cid:0) lim t →∞ max i,j ∈V | s i ( t ) − s j ( t ) | = ∞ (cid:1) = 1 . (2)The first part of the above theorem indicates that when the environment is frozen, andwhen positive clusters are properly connected, then irrespective of the mean of the positive8 hi et al. Signed Random Dynamical Networks: Relative-State-Flipping Model attentions { b t } ∞ , the system states converge if the attention each node puts in her negativeneighbors decays sufficiently fast over time. The second part of the theorem states that whenthis attention does not decay, divergence can be observed. Remark 3
Theorem 1 shows that well-structured positive arcs and asymptotically decaying at-tention guarantee state convergence for relative-state-flipping model. The essential reason is thatwhen (cid:80) ∞ t =0 d t < ∞ , the first Borel-Cantelli lemma (cf. Theorem 2.3.1, [32]) ensures that alongalmost every sample path, negative interactions happen only for a finite number of time instants.The positive interactions continue to guide the network states to a finite limit under suitable con-nectivity. It is then clear that the same condition can also guarantee state-convergence for thestate-flipping model considered in [1].In fact, for the state convergence property of the state-flipping model, a much stronger con-clusion regarding state convergence was shown (Theorem 1 in [1]) indicating that each posi-tive/negative arc contributes to the state convergence under constant attention { b t } and { d t } , aslong as α + β ≤ ( n − − . We can easily build examples showing that it is a completely differentstory on this matter for the relative-state-flipping model considered in the current paper. Remark 4
The divergence statement in Theorem 1 is not true for the state-flipping model [1],where almost sure state divergence always requires sufficiently large β . Next, we provide a sufficient condition for almost sure deviation consensus as defined below.
Definition 2
Algorithm (1) achieves almost sure deviation consensus if P (cid:0) lim sup t →∞ max i,j ∈V | s i ( t ) − s j ( t ) | = 0 (cid:1) = 1 . Note that almost sure deviation consensus means that the distances among the node statesconverge to zero, but convergence of each node state is not required. We need the followingassumption, which is a relaxed version of Assumption A4.
A7.
There is an integer K ≥ G + (cid:0) [ t, t + K ] (cid:1) = (cid:0) V i , (cid:83) τ ∈ [ t,t + K − E + τ (cid:1) has a spanning tree for all t ≥
0. 9 hi et al.
Signed Random Dynamical Networks: Relative-State-Flipping Model
Theorem 2
Assume that A1 and A7 hold and that α ∈ (0 , ( n − − ) . Denote K = (2 n − K and ρ ∗ = min { α, − ( n − α } . Define X m = p n − ∗ ρ K ∗ ( m +1) K − (cid:89) t = mK (cid:0) b t (1 − d t ) (cid:1) , and Y m = (cid:0) β ( n − (cid:1) K (cid:0) − ( m +1) K − (cid:89) t = mK (1 − d t ) (cid:1) . Then under the relative-state-flipping model, if ≤ X m − Y m ≤ for all m ≥ and (cid:80) ∞ m =0 ( X m − Y m ) = ∞ , Algorithm (1) achieves almost sure deviation consensus for all initial states. Remark 5
A direct consequence of Theorem 2 is that if b t ≡ b and d t ≡ d with b, d ∈ (0 , and β > , there exists d (cid:63) > such that whenever d < d (cid:63) , deviation consensus is achievedalmost surely. Observe that deviation consensus does not necessarily guarantee the convergenceof the state of each node. In fact, simple examples can be constructed with arbitrarily small β such that under the relative-state-flipping model, the state of each node grows arbitrarily largewhile deviation consensus still holds. This contrasts the result for the state-flipping model: thecondition α + β < ( n − − prevents the state of individual nodes to diverge (Theorem 1 in [1]). We continue to provide conditions under which the maximal gap between the states of two nodesgrows large almost surely, and establish a no-survivor property. We introduce a new connectivitycondition on the negative graph, which is a relaxed version of Assumption A5.
A8.
There is an integer K ≥ G − (cid:0) [ t, t + K ] (cid:1) = (cid:0) V , (cid:83) τ ∈ [ t,t + K − E − τ (cid:1) is weakly connected for all t ≥ Theorem 3
Assume that A1, A6, and A8 hold and that α ∈ [0 , ( n − − / . Let b t ≡ b and d t ≡ d for some constants b, d ∈ (0 , . Let β > and fix d . Then for Algorithm (1)under the relative-state-flipping model, there is b (cid:63) > such that whenever b < b (cid:63) , we have P (cid:0) lim t →∞ max i,j ∈V | s i ( t ) − s j ( t ) | = ∞ (cid:1) = 1 for almost all initial states (under the standardLebesgue measure). hi et al. Signed Random Dynamical Networks: Relative-State-Flipping Model
Remark 6
Theorem 3 indicates that in relative-state-flipping model, almost sure relative-statedivergence can be achieved as long as negative interactions happen sufficiently more often thanthe positive interactions. As explained in above remarks, for state-flipping model, state divergencenecessarily require sufficiently large weight on negative recommendations.
Finally, we investigate the clustering of states of nodes within each positive cluster.
A9.
Assume that A3 holds and let V = (cid:83) T p i =1 V i be a positive-cluster partition of the total graph G ∗ . There is an integer K ≥ G + (cid:0) [ t, t + K ] (cid:1)(cid:12)(cid:12) V i = (cid:0) V i , (cid:83) τ ∈ [ t,t + K − E + τ (cid:12)(cid:12) V i (cid:1) has a spanning tree for all t ≥ Theorem 4
Assume that A1, A3 and A9 hold and let V = (cid:83) T p i =1 V i be a positive-cluster partitionof G ∗ . Let α ∈ (0 , ( n − − ) . Define J ( m ) = (cid:81) ( m +1) K − t = mK b t and W ( m ) = (cid:80) ( m +1) K − t = mK d t with K = (2 n − K . Further assume that (cid:80) ∞ m =0 J ( m ) = ∞ , (cid:80) ∞ t =0 d t < ∞ , and lim m →∞ W ( m ) /J ( m ) =0 . Then under the relative-state-flipping model, for any initial state s (0) , Algorithm (1) achievesa.s. state clustering in the sense that there are T p real-valued random variables, w ∗ , . . . , w ∗ T p ,such that P (cid:16) lim t →∞ s i ( t ) = w ∗ j , i ∈ V j , j = 1 , . . . , T p (cid:17) = 1 . Theorem 4 shows the possibility of state clustering for every positive cluster, whose proof isbased on a martingale convergence lemma.
In this section, we establish the proofs of the various statements presented in the previoussection.
We list three martingale convergence lemmas (see e.g. [33]), and a result that will be instrumentalin the analysis of the system convergence under the relative-state-flipping model. 11 hi et al.
Signed Random Dynamical Networks: Relative-State-Flipping Model
Lemma 1
Let { v t } t ≥ be a sequence of non-negative random variables with E { v } < ∞ . Assumethat for any t ≥ , E { v t +1 | v , . . . , v t } ≤ (1 + ξ t ) v t + θ t , where { ξ t } t ≥ and { θ t } t ≥ are two (deterministic) sequences of non-negative numbers satisfying (cid:80) ∞ t =0 ξ t < ∞ and (cid:80) ∞ t =0 θ t < ∞ . Then lim t →∞ v t = v a.s. for some random variable v ≥ . Lemma 2
Let { v t } t ≥ be a sequence of non-negative random variables with E { v } < ∞ . Assumethat for any t ≥ , E { v t +1 | v , . . . , v t } ≤ (1 − ξ t ) v t + θ t , where { ξ t } t ≥ and { θ t } t ≥ are two (deterministic) sequences of non-negative numbers satisfying ∀ t ≥ , ≤ ξ t ≤ , (cid:80) ∞ t =0 ξ t = ∞ , (cid:80) ∞ t =0 θ t < ∞ , and lim t →∞ θ t /ξ t = 0 . Then lim t →∞ v t = 0 a.s.. Lemma 3
Let { v t } t ≥ , { ξ t } t ≥ , { θ t } t ≥ be sequences of non-negative random variables. Assumethat for any t ≥ , E { v t +1 |F t } ≤ (1 + ξ t ) v t + θ t , where F t = σ ( v , . . . , v t ; ξ , . . . , ξ t ; θ , . . . , θ t ) . Suppose (cid:80) ∞ t =0 ξ t < ∞ and (cid:80) ∞ t =0 θ t < ∞ almostsurely. Then lim t →∞ v t = v a.s. for some random variable v ≥ . We define h ( t ) := min i ∈V s i ( t ), H ( t ) := max i ∈V s i ( t ), and H ( t ) := H ( t ) − h ( t ), which will beused throughout the rest of the paper. The following lemma holds. Lemma 4
Assume that α ∈ [0 , ( n − − ] and that (cid:80) ∞ t =0 d t < ∞ . Then under the relative-state-flipping model, for all initial states, each of h ( t ) , H ( t ) , H ( t ) converges almost surely.Proof. We build the proof in steps.Step 1. In this step, we prove the convergence of H ( t ). Since α ∈ [0 , ( n − − ], the proposedalgorithm simply does weighted averaging when D t = 0. It is therefore well known that H ( t +1) ≤ hi et al. Signed Random Dynamical Networks: Relative-State-Flipping Model H ( t ), h ( t + 1) ≥ h ( t ), and H ( t + 1) ≤ H ( t ) if D t = 0. On the other hand, when D t = 1, it holdsfrom the structure of the algorithm that H ( t + 1) ≤ (2 β ( n −
1) + 1) H ( t ). We deduce that: E (cid:8) H ( t + 1) |H ( t ) (cid:9) ≤ (cid:0) β ( n − d t (cid:1) H ( t ) , (3)which, in view of Lemma 1, implies that H ( t ) → H ∗ almost surely for some H ∗ ≥ H ( t ), we easily see from (3) that E (cid:8) H ( t + 1) | H ( t ) } ≤ H ( t ) + (cid:0) β ( n − d t (cid:1) H ( t ) . (4)Since we have proved that H ( t ) converges a.s. and (cid:80) t d t < ∞ , we deduce that (cid:80) t (cid:0) β ( n − d t (cid:1) H ( t ) < ∞ a.s.. Further, in light of the first Borel-Cantelli Lemma (cf. Theorem 2.3.1, [32]), (cid:80) t d t < ∞ ensures that P (cid:16) lim inf t →∞ H ( t ) > −∞ (cid:17) = 1because H ( t ) ≥ h ( t ) and (cid:8) h ( t + 1) < h ( t ) (cid:9) ⊆ (cid:8) D t = 1 (cid:9) for any t ≥
0. Thus, ¯ H ( t ) := H ( t ) − inf t ≥ H ( t ) is a well-defined nonnegative random variablefor any t ≥
0, and (4) implies E (cid:8) ¯ H ( t + 1) | ¯ H ( t ) } ≤ ¯ H ( t ) + (cid:0) β ( n − d t (cid:1) H ( t ) . (5)Hence, we can invoke Lemma 3 to conclude that ¯ H ( t ) converges to a nonnegative randomvariable almost surely as t grows to infinity, which immediately implies that H ( t ) convergesalmost surely.Step 3. The convergence of h ( t ) follows from a symmetric argument as the analysis to H ( t ). Wehave now completed the proof. (cid:4) Lemma 5
Assume that D t = 0 for t = 0 , . . . , n − K − . Let α ∈ (0 , ( n − − ) and i ∈ V .Then for any t = 0 , . . . , n − K − , there hold(i) If s i ( t ) ≤ ζ h (0) + (1 − ζ ) H (0) for some ζ ∈ (0 , , then s i ( t + 1) ≤ λ ∗ ζ h (0) + (1 − λ ∗ ζ ) H (0) , where λ ∗ = 1 − α ( n − ;(ii) If s i ( t ) ≤ ζ h (0) + (1 − ζ ) H (0) for some ζ ∈ (0 , , B t = 1 , and ( i, j ) ∈ G t , then s j ( t + 1) ≤ αζ h (0) + (1 − αζ ) H (0) . hi et al. Signed Random Dynamical Networks: Relative-State-Flipping ModelProof.
Note that the conditions that D t = 0 for t = 0 , . . . , n − K − α ∈ (0 , ( n − − )yield H ( t + 1) ≤ H ( t ) and h ( t + 1) ≥ h ( t ) for all t = 0 , . . . , n − K − D t = 0 and s i ( t ) ≤ ζ h (0) + (1 − ζ ) H (0) for some ζ ∈ (0 , s i ( t + 1) = s i ( t ) + αB t h + i ( t ) ≤ s i ( t ) − α (cid:88) j ∈ N + i ( t ) (cid:0) s i ( t ) − s j ( t ) (cid:1) ≤ (1 − α | N + i ( t ) | ) s i ( t ) + α | N + i ( t ) | H ( t ) ≤ (1 − α | N + i ( t ) | ) (cid:0) ζ h (0) + (1 − ζ ) H (0) (cid:1) + α | N + i ( t ) | H (0) ≤ λ ∗ ζ h (0) + (1 − λ ∗ ζ ) H (0) (6)in light of the fact that α ∈ (0 , ( n − − ), where λ ∗ = 1 − α ( n − s i ( t ) ≤ ζ h (0) + (1 − ζ ) H (0) for some ζ ∈ (0 , B t = 1, and ( i, j ) ∈ G t , there holds that s j ( t + 1) = s j ( t ) − α (cid:88) k ∈ N + j ( t ) (cid:0) s j ( t ) − s k ( t ) (cid:1) = (1 − α | N + j ( t ) | ) s j ( t ) + αs i ( t )+ α (cid:88) k ∈ N + j ( t ) \{ i } s k ( t ) ≤ (1 − α ) H ( t ) + α (cid:0) ζ h (0) + (1 − ζ ) H (0) (cid:1) ≤ αζ h (0) + (1 − αζ ) H (0) . (7)This proves the desired lemma. (cid:4) (i). Let (cid:80) ∞ t =0 d t < ∞ . Then as long as (cid:80) ∞ t =0 b t < ∞ , the first Borel-Cantelli Lemma guaranteesthat almost surely, each node revises its state for only a finite number of slots, which yields thedesired claim follows straightforwardly. In the following, we prove the desired conclusion basedon the assumption that (cid:80) ∞ t =0 b t = ∞ .With (cid:80) ∞ t =0 d t < ∞ , from the first Borel-Cantelli Lemma, K ∗ := inf { k ≥ D t = 0 , ∀ t ≥ k } hi et al. Signed Random Dynamical Networks: Relative-State-Flipping Model is a finite number almost surely. We note that K ∗ is not a stopping time for { D t } t ≥ , but astopping time for { B t } t ≥ by the independence of { B t } t ≥ and { D t } t ≥ . Hence, we can recursivelydefine K m +1 := inf { t > K m : B t = 1 } , m = 0 , , . . . with K := inf { t ≥ K ∗ : B t = 1 } , which are are stopping times for { B t } t ≥ . Now in view of theindependence of { G t } t ≥ and { B t } t ≥ , we know that { G K m } m ≥ is an independent process andeach G K m satisfies P (cid:0) ( i, j ) ∈ E K m (cid:1) ≥ p ∗ for all ( i, j ) ∈ G under Assumption A1.Let V † be a positive cluster of G . By assumption, V † has a spanning tree. Since α < / ( n − K m , m = 0 , , . . . , the considered relative-state-flippingmodel defines a standard consensus dynamics on independent random graphs where each arcexists with probability at least p ∗ for any fixed time slot. Therefore, applying Theorem 3.4in [39] on randomized consensus dynamics with arc-independent graphs, we conclude that theconnectivity of V † ensures that P (cid:0) lim m →∞ H † ( K m ) = 0 (cid:1) = 1 , where H † ( t ) = max i ∈V † s i ( t ) − min i ∈V † s i ( t ). This immediately gives us P (cid:0) lim t →∞ H † ( t ) = 0 (cid:1) = 1by the definition of the K m .Finally, applying Lemma 4 to the subgraph generated by node set V † , both max i ∈V † s i ( t ) andmin i ∈V † s i ( t ) almost surely converge, and define their limits as, respectively, H †∗ and h †∗ . Thus,there holds that P (cid:0) lim t →∞ max i ∈V † s i ( t ) = H †∗ (cid:1) = 1and that P (cid:0) lim t →∞ min i ∈V † s i ( t ) = h †∗ (cid:1) = 1 . The fact that P (cid:0) lim t →∞ H † ( t ) = 0 (cid:1) = 1 immediately leads to H †∗ = h †∗ almost surely. As aresult, we conclude that P (cid:0) lim t →∞ s i ( t ) = H †∗ = h †∗ (cid:1) = 1for all i ∈ V † . This proves the desired statement. 15 hi et al. Signed Random Dynamical Networks: Relative-State-Flipping Model (ii) Let V and V be the two positive-clusters of G . Let s i (0) = 0 , i ∈ V and s i (0) = C , i ∈ V for some C >
0. We define f ( t ) := max i ∈V s i ( t ); f ( t ) := min i ∈V s i ( t ) . Since the either of the positive cluster contains positive links only and α ∈ (0 , ( n − − ), therealways holds that f ( t + 1) ≤ f ( t ); f ( t + 1) ≥ f ( t ) . Now that there is a negative arc between any two nodes from different clusters, it is straightfor-ward to see that f ( t + 1) − f ( t + 1) ≥ (1 + β ) (cid:0) f ( t ) − f ( t ) (cid:1) ≥ f ( t ) − f ( t ) + C (8)whenever D t = 1 and either ( i ∗ , j ∗ ) ∈ E t or ( j ∗ , i ∗ ) ∈ E t with i ∗ = arg max i ∈V s i ( t ) and j ∗ = arg min i ∈V s i ( t ). In light of Assumption A1, the second Borel-Cantelli Lemma (cf., Theorem2.3.6, [32]) leads to that the event defined in (8) happens infinitely often with probability onewhen (cid:80) ∞ t =0 d t = ∞ . The desired conclusion follows immediately.The proof is now complete. (cid:4) The proof relies on Lemma 2, cf., [29] for the analysis of randomized consensus.Consider 2 n − mK, ( m + 1) K − , m = 0 , . . . , n − v m ∈ V in each of G ([ mK, ( m + 1) K − n − v m ’s and denote them as v m , . . . , v m n − , thatsatisfy either or s v mj (0) > ( h (0) + H (0)) /
2, for all j = 1 , . . . , n −
1. The two cases are symmetricand without loss of generality, we consider the first case only.Now we assume that D t = 0 for t = 0 , . . . , n − K −
1. We carry out the following recursiveargument:1) By our selection v m is a center node of the graph G ([ τ K, ( τ + 1) K − τ = 0 , . . . , n −
2) with s v m (0) ≤ ( h (0) + H (0)) /
2. Applying Lemma 5.(i) we concludethat s v m (cid:0) K (cid:1) ≤ ρ K ∗ h (0) + (cid:0) − ρ K ∗ (cid:1) H (0) , hi et al. Signed Random Dynamical Networks: Relative-State-Flipping Model where K and ρ ∗ are defined in the statement of Theorem 2.2) Since v m is a center, there exist t ∈ [ τ K, ( τ + 1) K −
1] and j ∗ (cid:54) = v m ∈ V such that( v m , j ∗ ) ∈ E t with probability at least p ∗ . If B t = 1 and ( v m , j ∗ ) ∈ E t , then we canapply Lemma 5 and then conclude s j ∗ (cid:0) K (cid:1) ≤ ρ K ∗ h (0) + (cid:0) − ρ K ∗ (cid:1) H (0) . For convenience we re-denote v m and j ∗ as u and u , respectively.3) We proceed for v m . If v m / ∈ { u , u } , applying Lemma 5.(i) again and we can obtain thesame bound for s j ∗ (cid:0) K (cid:1) . Otherwise either v m = u or v m = u allows us to find anothernode u with the bound for s u (cid:0) K (cid:1) obtained as step 2).From the selection of v m , . . . , v m n − , the above procedure eventually gives us the same bound fornodes u , . . . , u N , and calculating the probability of the required events in the above argumentwe obtain P (cid:16) s i (cid:0) K (cid:1) ≤ ρ K ∗ h (0) + (cid:0) − ρ K ∗ (cid:1) H (0) , i ∈ V (cid:17) ≥ p n − ∗ K − (cid:89) t =0 (cid:0) b t (1 − d t ) (cid:1) . This implies P (cid:16) H (cid:0) K (cid:1) ≤ (cid:0) − ρ K ∗ (cid:1) H (0) (cid:17) ≥ p n − ∗ K − (cid:89) t =0 (cid:0) b t (1 − d t ) (cid:1) . (9)On the other hand, from the definition of the algorithm there always hold P (cid:0) H (cid:0) t + 1 (cid:1) ≤ (cid:0) β ( n − (cid:1) H (0) (cid:1) = 1 (10)and P (cid:0) H (cid:0) K (cid:1) > H (0) (cid:1) ≤ − K − (cid:89) t =0 (1 − d t ) . (11)Since { B t } t ≥ , { D t } t ≥ , and { G t } t ≥ define independent processes, we conclude from (9),(10), and (11) that E (cid:8) H (cid:0) ( m + 1) K (cid:1)(cid:12)(cid:12) H (cid:0) mK (cid:1)(cid:9) ≤ (cid:0) − X m + Y m (cid:1) H (cid:0) mK (cid:1) . The desired theorem then follows directly from Lemma 2 and (10). (cid:4) hi et al. Signed Random Dynamical Networks: Relative-State-Flipping Model
In light of α ∈ [0 , ( n − − / Claim A. P (cid:16) H ( t + 1) ≥ (cid:0) − n − α (cid:1) H ( t ) (cid:17) = 1. Claim B. P (cid:0) H ( t + 1) < H ( t ) (cid:1) ≤ b .Take i, j ∈ V satisfying s i ( t ) = h ( t ) and s j ( t ) = H ( t ). Similarly as the proof of Lemma 5, wecan establish that almost surely, s i ( t + 1) ≤ λ ∗ h ( t ) + (1 − λ ∗ ) H ( t ) (12)and s j ( t + 1) ≥ λ ∗ H ( t ) + (1 − λ ∗ ) h ( t ) (13)hold, where λ ∗ = 1 − α ( n − H ( t + 1) ≥ | s j ( t + 1) − s i ( t + 1) |≥ | λ ∗ − |H ( t )= (cid:0) − n − α (cid:1) H ( t ) , (14)Claim A is proved.Furthermore, if b t = 0, only negative recommendations can be effective in the node stateupdate. This implies Claim B.Now we define L := inf { t ∈ Z : (1 + β ) t ≥ n − } . Consider time intervals [ mK, ( m +1) K −
1] for m = 0 , , . . . , ( n − n )( L − K L = K (( n − n )( L −
1) + 1). UnderAssumption A8 and based on the fact that there are at most n ( n −
1) arcs, there are twonodes i ∗ , j ∗ ∈ V and L instants 0 ≤ τ < τ < · · · < τ L < K L such that ( i ∗ , j ∗ ) ∈ G − τ k and | s i ∗ ( τ k ) − s j ∗ ( τ k ) | ≥ H ( τ k ) / ( n −
1) for all τ k .Consider the following event:E ∗ := (cid:110) D τ k = 1 , i ∗ = N − j ∗ ( τ k ) for all τ k ; B t = 0 for all t ∈ [0 , K L − (cid:111) . (15)The event E ∗ implies H ( K L ) ≥ | s i ∗ ( K L ) − s j ∗ ( K L ) | ≥ H (0)(1 + β ) L · ( n − − . hi et al. Signed Random Dynamical Networks: Relative-State-Flipping Model
As a result, we can bound the probability of E ∗ and conclude P (cid:16) H ( K L ) ≥ H (0)(1 + β ) L · ( n − − (cid:17) ≥ (cid:0) dp ∗ (1 − p ∗ ) n − (cid:1) L (1 − b ) K L . (16)We can now apply the same argument as the proof of Proposition 1 in [1]. With (16), ClaimA, and Claim B, there holds E (cid:110) log H ( K L ) − log H (0) (cid:111) ≥ (cid:0) dp ∗ (1 − p ∗ ) n − (cid:1) L (1 − b ) K L · log (cid:16) (1 + β ) L · ( n − − (cid:17) + b log (cid:0) − n − α (cid:1) ≥ (cid:0) dp ∗ (1 − p ∗ ) n − (cid:1) L (1 − b ) K L log 2+ b log (cid:0) − n − α (cid:1) > b < b (cid:63) for some sufficiently small b (cid:63) >
0. We can proceed to define U ( m ) = log H ( mK L )for m = 0 , , . . . . Recursively applying the above arguments to the process { U m } we obtain that U ( m ) has a strictly positive drift when b < b (cid:63) , which implies that lim inf m →∞ U ( m ) = ∞ holdsalmost surely.This completes the proof. (cid:4) Let us focus on a given positive cluster V † of G . We use the following notationsΨ( t ) = max i ∈V † s i ( t ) , ψ ( t ) = min i ∈V † s i ( t ) , Θ( t ) = Ψ( t ) − ψ ( t ) . Applying Lemma 4 on the positive cluster V † , we conclude that each of Θ( t ), Ψ( t ), and ψ ( t )converge to a finite limit almost surely if (cid:80) ∞ t =0 d t < ∞ .In light of Assumption A9, applying the same argument we used in order to establish (9) ofTheorem 2 on the cluster V † , we similarly have P (cid:16) Θ (cid:0) ( m + 1) K (cid:1) ≤ (cid:0) − ρ K ∗ (cid:1) Θ( mK ) (cid:17) ≥ p n − ∗ ( m +1) K − (cid:89) t = mK (cid:0) b t (1 − d t ) (cid:1) . (18)19 hi et al. Signed Random Dynamical Networks: Relative-State-Flipping Model
Moreover, from the effect of the negative recommendations on nodes in V † , we can easily modify(10) and (11) to that for all t , P (cid:0) Θ (cid:0) t + 1 (cid:1) > Θ( t ) (cid:1) ≤ d t (19)and P (cid:16) Θ (cid:0) t + 1 (cid:1) ≤ (1 + 2 β ( n − H ( t ) (cid:17) = 1 . (20)With (18), (19) and (20), we arrive at E (cid:8) Θ (cid:0) ( m + 1) K (cid:1)(cid:12)(cid:12) Θ (cid:0) m ( K ) (cid:1)(cid:9) ≤ (cid:0) − X m (cid:1) Θ (cid:0) mK (cid:1) + (1 + 2 β ( n − ( m +1) K − (cid:88) t = mK d t H ( t ) , (21)where X m is defined in Theorem 2.On the other hand, from (3) we know that E ( H ( t )) ≤ H ∞ (cid:89) t =0 (cid:0) β ( n − d t (cid:1) for all t ≥
0. Taking the expectation from the both sides of (21), we obtain: E (cid:8) Θ (cid:0) ( m + 1) K (cid:1)(cid:9) ≤ (cid:0) − X m (cid:1) E (cid:8) Θ (cid:0) mK (cid:1)(cid:9) + (cid:104) (1 + 2 β ( n − H ∞ (cid:89) t =0 (cid:0) β ( n − d t (cid:1)(cid:105) W ( m ) , (22)where W ( m ) = (cid:80) ( m +1) K − t = mK d t .Note that it is well known that (cid:80) ∞ t =0 d t < ∞ implies (cid:81) ∞ t =0 (1 − d t ) > (cid:81) ∞ t =0 (cid:0) β ( n − d t (cid:1) < ∞ . Consequently, (cid:80) ∞ m =0 J ( m ) = ∞ implies (cid:80) ∞ m =0 X m = ∞ . In view of Lemma 2, wehave lim m →∞ E (cid:8) Θ (cid:0) ( m + 1) K (cid:1)(cid:9) = 0 (23)if lim m →∞ W ( m ) /J ( m ) = 0. Invoking Fatou’s lemma (e.g., Theorem 1.6.5, [32]), we furtherconclude that E (cid:8) lim inf m →∞ Θ (cid:0) ( m + 1) K (cid:1)(cid:9) ≤ lim t →∞ E (cid:8) Θ (cid:0) ( m + 1) K (cid:1)(cid:9) = 0 , (24)20 hi et al. Signed Random Dynamical Networks: Relative-State-Flipping Model which actually implies E (cid:8) lim m →∞ Θ (cid:0) ( m + 1) K (cid:1)(cid:9) = 0 (25)since Θ (cid:0) ( m + 1) K (cid:1) converges almost surely. Therefore, we have reached P (cid:0) lim m →∞ Θ (cid:0) ( m + 1) K (cid:1) = 0 (cid:1) = 1 , (26)which in turn leads to P (cid:0) lim t →∞ Θ( t ) = 0 (cid:1) = 1 , (27)again, from the fact that Θ( t ) converges almost surely.Finally, Θ( t ) converging almost surely to zero means that Ψ( t ) and ψ ( t ) must converge tothe same limit (their convergence is established in the beginning of the proof), which must bethe limit of the each node state in V † . We have now completed the proof. (cid:4) This paper continued the study of [35, 36] investigating a relative-state-flipping model for con-sensus dynamics over signed random networks. A sequence of deterministic signed graphs definepotential node interactions that happen independently but not necessarily i.i.d. The positive rec-ommendations are consistent with the standard consensus algorithm; negative recommendationsare defined by relative-state flipping from its negative neighbors. Each node puts a (determinis-tic) weight to each recommendation, and then encodes these weighted recommendations in itsstate update through stochastic attentions defined by two Bernoulli random variables. We haveestablished several fundamental conditions regarding almost sure convergence and divergence ofthe network states. A condition for almost sure state clustering was also proposed for weakly bal-anced graphs, with the help of martingale convergence lemmas. Some fundamental differenceswere also highlighted between the current relative-state-flipping model and the state-flippingmodel considered in [1, 34]. 21 hi et al.
Signed Random Dynamical Networks: Relative-State-Flipping Model
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