Modulus consensus in discrete-time signed networks and properties of special recurrent inequalities
MModulus consensus in discrete-time signed networks and properties ofspecial recurrent inequalities
Anton V. Proskurnikov and Ming Cao
Abstract — Recently the dynamics of signed networks, wherethe ties among the agents can be both positive (attractive) ornegative (repulsive) have attracted substantial attention of theresearch community. Examples of such networks are models ofopinion dynamics over signed graphs, recently introduced byAltafini (2012,2013) and extended to discrete-time case by Menget al. (2014). It has been shown that under mild connectivityassumptions these protocols provide the convergence of opinionsin absolute value, whereas their signs may differ. This “modulusconsensus” may correspond to the polarization of the opinions(or bipartite consensus , including the usual consensus as aspecial case), or their convergence to zero.In this paper, we demonstrate that the phenomenon ofmodulus consensus in the discrete-time Altafini model is amanifestation of a more general and profound fact, regardingthe solutions of a special recurrent inequality. Although sucha recurrent inequality does not provide the uniqueness of asolution, it can be shown that, under some natural assumptions,each of its bounded solutions has a limit and, moreover,converges to consensus. A similar property has previously beenestablished for special continuous-time differential inequalities(Proskurnikov, Cao, 2016). Besides analysis of signed networks,we link the consensus properties of recurrent inequalities to theconvergence analysis of distributed optimization algorithms andthe problems of Schur stability of substochastic matrices.
I. I
NTRODUCTION
In the recent years protocols for consensus and syn-chronization in multi-agent networks have been thoroughlystudied [1]–[4]. Much less studied are “irregular” behav-iors, exhibited by many real-world networks, such as e.g.desynchronization [5] and chaos [6]. An important stepin understanding these complex behaviors is to elaboratemathematical models for “partial” or cluster synchronization,or simply clustering [5], [7]–[9]. In social influence theory,this problem is known as the community cleavage problemor Abelson’s diversity puzzle [10], [11]: to disclose mech-anisms that hinder reaching consensus among social actorsand lead to splitting of their opinions into several clusters.One reason for clustering in multi-agent networks is thepresence of “negative” (repulsive, antagonistic) interactionsamong the agents [8]. Models of signed (or “coopetition”)
A.V. Proskurnikov is with the Delft Center for Systems and Con-trol (DCSC) at Delft University of Technology. He is also withITMO University and Institute for Problems of Mechanical Engineeringof Russian Academy of Sciences (IPME RAS), St. Petersburg, Rus-sia; [email protected]
M. Cao is with the Engineering and Technology Institute (ENTEG) atthe University of Groningen, The Netherlands; [email protected]
Financial support was provided by the ERC (ERC-StG-307207), NWO(vidi-438730), Russian Federation President’s Grant MD-6325.2016.8, andRFBR, grants 17-08-01728, 17-08-00715 and 17-08-01266. Theorem 1is obtained at IPME RAS under the sole support of Russian ScienceFoundation (RSF) grant 14-29-00142. networks with positive and negative couplings among thenodes describe a broad class of real-world systems, frommolecular ensembles [12] to continental supply chains [13].Positive and negative relations among social actors canexpress, respectively, trust (friendship) or distrust (hostility).Negative ties among the individuals may also result fromthe reactance or boomerang effects, first described in [14]:an individual may not only resist the persuasion process, buteven adopt an attitude that is contrary to the persuader’s one.A simple yet instructive model of continuous-time opin-ion dynamics over signed networks has been proposed byAltafini [15], [16] and extended to the discrete-time casein [17]. In the recent years, Altafini-type coordination pro-tocols over static and time-varying signed graphs have beenextensively studied, see e.g. [18]–[25]. It has been shownthat under mild connectivity assumptions these models ex-hibit consensus in absolute value, or modulus consensus : theagents’ opinions agree in modulus yet may differ in signs.The modulus consensus may correspond to the asymptoticstability of the network (the opinions of all individuals con-verge to zero), usual consensus (convergence of the opinionsto the same value, depending on the initial condition) and polarization , or “bipartite consensus”: the agents split intotwo groups, converging to the opposite opinions.In the recent works [26], [27] it has been shown that theeffect of modulus consensus in the continuous-time Altafinimodel is in fact a manifestation of a more profound result,concerned with the special class of differential inequalities ˙ x ( t ) ≤ − L ( t ) x ( t ) , (1)where L ( t ) stands for the Laplacian matrix of a time-varyingweighted graph. Although the inequality (1) is a seemingly“loose” constraint, any of its bounded solutions (under natu-ral connectivity assumptions) converges to a consensus equi-librium (this property is called consensus dichotomy ). Thisimplies, in particular, the modulus consensus in the Altafinimodel [26], [27] since the vector of the opinions’ absolutevalues obeys the inequality (1). In this paper, we extend thetheory of differential inequalities to the discrete-time case,where (1) is replaced by the recurrent inequality x ( k + 1) ≤ W ( k ) x ( k ) with { W ( k ) } k ≥ being a sequence of stochasticmatrices. We establish the consensus dichotomy criteria forthese inequalities, which imply the recent results on modulusconsensus in the discrete-time Altafini model [17], [23]. Wealso apply the recurrent inequalities to some problems ofmatrix theory and the analysis of distributed algorithms foroptimization and linear equations solving. a r X i v : . [ c s . S Y ] M a r I. P
ROBLEM S ETUP
We start with preliminaries and introducing some notation.
A. Preliminaries
First we introduce some notation. A vector x ∈ R n isnon-negative ( x ≥ ) if x i ≥ ∀ i . Given two vectors x, y ∈ R n , we write x ≥ y (respectively, x ≤ y ) if x − y ≥ (respectively, y − x ≥ ). The vector of ones is denoted by n = (1 , . . . , (cid:62) ∈ R n . Given a matrix A = ( a ij ) , we use | A | = ( | a ij | ) to denote the matrix of element-wise absolutevalues (the same rule applies to vectors). A matrix A = ( a ij ) is stochastic if its entries are non-negative and all rows sumto , i.e. (cid:80) j a ij = 1 ∀ i . We use ρ ( A ) to denote the spectralradius of a square matrix A . The standard Euclidean normof a vector x is denoted by (cid:107) x (cid:107) = √ x (cid:62) x .A non-negative matrix A = ( a ij ) i,j ∈ V can be associatedto a (directed) weighted graph G [ A ] = ( V, E [ A ] , A ) , whoseset of arcs is E [ A ] = { ( i, j ) : a ij (cid:54) = 0 } . B. Recurrent inequalities and consensus dichotomy.
In this paper, we are interested in the solutions of thefollowing discrete-time, or recurrent , inequality x ( k + 1) ≤ W ( k ) x ( k ) , k = 0 , , . . . (2)where x ( k ) ∈ R n is a sequence of vectors and W ( k ) ∈ R n × n stands for a sequence of stochastic matrices.Replacing the inequality in (2) by the equality, one obtainsthe well-known averaging, or consensus protocol [30]–[32] x ( k + 1) = W ( k ) x ( k ) , (3)dating back to the early works on social influence [33],[34], rational decision making [35] and distributed opti-mization [36]. The algorithm (3) may be interpreted as thedynamics of opinions formation in a network of n agents.At each step of the opinion iteration k agent i calculates theweighted average of its own opinion x i ( k ) and the others’opinions; this average is used as the new opinion of the i thagent x i ( k + 1) = (cid:80) j w ij ( k ) x j ( k ) . The graph G [ W ( k )] naturally represents the interaction topology of the networkat step k . Agent i is influenced by agent j if w ij ( k ) > ,otherwise the j th agent’s opinion x j ( k ) plays no role in theformation of the new agent i ’s opinion x i ( k + 1) .A similar interpretation can be given to the inequality (2).Unlike the algorithm (3), the opinion of agent i at eachstep of opinion formation is not uniquely determined bythe opinions from the previous step, but is only constrained by them x i ( k + 1) ≤ (cid:80) j w ij ( k ) x j ( k ) . The weight w ij ( k ) stands for the contribution of agent j ’s opinion x j ( k ) to thisconstraint, and in this sense it can also be treated as the“influence” weight. The inequality (2) does not provide thesolution’s uniqueness for a given x (0) , but only guaranteesthe existence of an upper bound for the solutions. We assume that the reader is familiar with the standard concepts ofgraph theory, regarding directed graphs and their connectivity properties,e.g. walks (or paths), cycles and strongly connected components [28], [29]. In the broad sense, “opinion” is just a scalar quantity of interest; it canstand for e.g. a physical parameters or an attitude to some event or issue.
Proposition 1:
Any solution of (2) obeys the inequality x ( k ) ≤ M n , M ∆ = max i x i (0) . Proof:
Proposition 1 is proved via straightforwardinduction on k . By definition, x (0) ≤ M n ; if x ( k ) ≤ M n then x ( k + 1) ≤ W ( k ) x ( k ) ≤ M W ( k ) n = M n .Although many solutions of (2) are unbounded frombelow, under certain assumptions any its bounded solutionconverges to a consensus equilibrium c n , where c ∈ R .A similar property, called consensus dichotomy has beenestablished in [26], [27] for the differential inequalities (1). Definition 1:
The inequality (2) is said to be dichotomic if any of its bounded (from below) solutions has a limit x ∗ = lim k →∞ x ( k ) . It is called consensus dichotomic if theselimits are consensus equilibria x ∗ = c ∗ n , where c ∗ ∈ R .The main goal of this paper is to disclose criteria ofconsensus dichotomy in the recurrent inequalities (2). In Sec-tion IV we discuss applications of these criteria to models ofopinion dynamics and algorithms of distributed optimization.III. M AIN R ESULTS
The first step is to examine time-invariant inequalities (2).
A. A dichotomy criterion for the time-invariant case
In this subsection, we assume that W ( k ) ≡ W is aconstant matrix, whose graph G ∆ = G [ W ] has s ≥ stronglyconnected (or strong ) components G , . . . , G s ; in general,arcs between different components may exist (Fig. 1a). Astrong component is isolated if no arc enters or leaves it.All strong components are isolated (Fig. 1b) if and only ifevery arc of the graph belongs to a cycle [28, Theorem 3.2]. (a) (b) Fig. 1: Non-isolated (a) vs. isolated (b) strong components
Theorem 1:
The inequality (2) with the static matrix W ( k ) ≡ W is dichotomic if and only if all the strongcomponents G , . . . , G s of its graph G are isolated andaperiodic . The inequality is consensus dichotomic if andonly if G is strongly connected ( s = 1 ) and aperiodic, or,equivalently, the matrix W is primitive [29], [38].The proof of Theorem 1, as well as the remaining resultsof this section, is given in Appendix. The term dichotomy originates from ODE theory. A system is dichotomicif any of its solutions either grows unbounded or has a finite limit [37]. Recall that a graph is aperiodic if the greatest common divisor of itscycles’ lengths (that is also referred to as the graph’s period ) is equal to . emark 1: Let V j stand for the set of nodes of G j . Theo-rem 1 shows that the time-invariant dichotomic inequality (2)reduces to s independent inequalities of lower dimensions x ( m ) ( k + 1) ≤ W ( m ) x ( m ) ( k ) , m = 1 , . . . , s, (4)where x ( m ) ( k ) = ( x i ( k )) i ∈ V m , W ( m ) = ( w ij ) i,j ∈ V m andeach inequality (4) is consensus dichotomic. Remark 2:
The matrix is primitive if and only if [11],[29], [38] its powers W k are strictly positive for large k . B. Consensus dichotomy in the time-varying case
In this subsection, we extend the result of Theorem 1 tothe case of general time-varying inequality (2). Given ε > , let S ε denote the class of all stochastic matrices W =( w ij ) i,j ∈ V , satisfying the two conditions:1) w ii ≥ ε for any i ∈ V ;2) the graph G ε [ W ] = ( V, E ε [ W ]) is strongly connected,where E ε [ W ] ∆ = { ( i, j ) ∈ V × V : w ij ≥ ε } .In other words, removing from the graph G [ W ] all “light”arcs weighted by less than ε , the remaining subgraph G ε [ W ] is strongly connected and has self-loops at each node.For any integers k ≥ and m > k let Φ( m, k ) =( ϕ ij ( m, k )) ni,j =1 ∆ = W ( m − . . . W ( k ) stand for the evo-lutionary matrix of the equation (3); for convenience, wedenote Φ( k, k ) = I n . It is obvious that any solution of (2)satisfies also the family of inequalities x ( m ) ≤ Φ( m, k ) x ( k ) ∀ m ≥ k ≥ . The following theorem provides a consensus dichotomycriterion for the case of the time-varying matrix W ( k ) . Theorem 2:
The inequality (2) is consensus dichotomic if ε > exists that satisfies the following condition: for any k ≥ there exists m > k such that Φ( m, k ) ∈ S ε .Notice that for the static matrix W ( k ) ≡ W one has Φ( m, k ) = W m − k , so the condition from Theorem 2 meansthat W s ∈ S ε for some s . It can be easily shown thatin this case W s ( n − is a strictly positive matrix. On theother hand, if W d is strictly positive for some d , then W d ∈ S ε for sufficiently small ε > . In view of Remark 2and Theorem 1, in the static case W ( k ) ≡ W the sufficientcondition of consensus dichotomy from Theorem 2 is in factalso necessary , boiling down to the primitivity of W .The condition from Theorem 2 is implied by the twostandard assumptions on the sequence { W ( k ) } k ≥ . Assumption 1:
There exists δ > such that for any k ≥ w ii ( k ) ≥ δ for any i = 1 , . . . , n ;2) for any i, j such that i (cid:54) = j one has w ij ( k ) ∈ { }∪ [ δ ; 1] . Assumption 2: (Repeated joint strong connectivity) Thereexists an integer B ≥ such that the graph G [ W ( k ) + . . . + W ( k + B − is strongly connected for any k . Corollary 1:
Let Assumptions 1 and 2 hold. Then theinequality (2) is consensus dichotomic.
Proof:
We are going to show that the condition fromTheorem 2 holds for ε = δ B and m = k + B , i.e. Φ( k + B, k ) ∈ S δ B for any k . Indeed, ϕ ii ( m, k ) ≥ w ii ( m − . . . w ii ( k ) ≥ δ m − k ∀ i whenever m ≥ k due to Assump-tion 1. Supposing that ( i, j ) ∈ G [ W ( l )] , where k ≤ l < m ,one has Φ( m, k ) = Φ( m, l + 1) W ( l )Φ( l, k ) , and therefore ϕ ij ( m, k ) ≥ ϕ ii ( m, l + 1) w ij ( l ) ϕ jj ( l, k ) ≥ δ m − l − δδ l − k = δ m − k . Applying this to m = k + B , one easily notices that i is connected to j in the graph G δ B [Φ( k + B, k )] whenever w ij ( l ) > for some l = k, . . . , k + B − . Assumption 2implies now that Φ( k + B, k ) ∈ S δ B for any k .It should be noticed however that the condition of Theo-rem 2 may hold in many situations where Assumptions 1and 2 fail. Even in the static case W ( k ) ≡ W , thematrix W can be primitive yet have zero diagonal entries.The following corollary illustrates another situation whereboth Assumptions 1 and 2 may fail, whereas Theorem 2guarantees consensus dichotomy. Corollary 2:
Suppose that for any k one has W ( k ) ∈{ W }∪W , where W stands for the primitive matrix and W is a set of stochastic matrices, commuting with W : W W = W W ∀ W ∈ W . Let the set K = { k : W ( k ) = W } beinfinite. Then the inequality (2) is consensus dichotomic. Proof:
Let d be so large that W d is a positive matrix,whose minimal entry equals ε > . For any k , we can findsuch m > k that the sequence k, k + 1 , . . . , m − contains d elements from the set K . Since any W ( j ) commutes with W , Φ( m, k ) = T k W d , where T k is some stochastic matrix,and thus all entries of Φ( m, k ) are not less than ε .Many sequences { W ( k ) } , satisfying the conditions ofCorollary 2, fail to satisfy Assumptions 1 and 2. For instance,if W (cid:51) I n then the sequence { W ( k ) } can contain anarbitrary long subsequence of consecutive identity matrices,which violates Assumption 2. Both the matrix W andmatrices from W may have zero diagonal entries, which alsoviolates Assumption 1. The set W can also be non-compact,containing matrices with arbitrary small yet non-zero entries. C. The case of bidirectional interaction
It is known that in the case of bidirectional graphs w ij > ⇔ w ji > the conditions for consensus in the network (3)is reached under very modest connectivity assumptions.Under Assumption 1, consensus is reached if and only ifthe following relaxed version of Assumption 2 holds [31]. Assumption 3: (Infinite joint strong connectivity) Thegraph G ∞ = ( V, E ∞ ) is strongly connected, where E ∞ = (cid:40) ( i, j ) : ∞ (cid:88) k =1 w ij ( k ) = ∞ (cid:41) . The following result extends this consensus criterion tothe condition of consensus dichotomy in the inequality (2).
Theorem 3:
Suppose that Assumption 1 and 3 hold andfor any k one has w ij ( k ) > ⇔ w ji ( k ) > . Then theinequality (2) is consensus dichotomic.The relaxation of Assumption 1 in Theorem 3 remains anon-trivial open problem. To the best of the authors’ knowl-edge, the same applies to usual consensus algorithms (3):most of the existing results for consensus in discrete-timeswitching networks [2], [30]–[32] rely on Assumption 1 orat least require uniformly positive diagonal entries w ii ( k ) .V. E XAMPLES AND A PPLICATIONS
In this section we apply the criteria from Section III tothe analysis of several multi-agent coordination protocols.
A. Modulus consensus in the discrete-time Altafini model
We first consider the discrete-time Altafini model [17],[19] of opinion formation in a signed network. This modelis similar to the consensus protocol (3) and is given by ξ ( k + 1) = A ( k ) ξ ( k ) ∈ R n , or, equivalently ξ i ( k + 1) = n (cid:88) j =1 a ij ( k ) x j ( k ) . (5)Here the matrix ( a ij ( k )) satisfies the following assumption. Assumption 4:
For any k = 0 , , . . . , the matrix A ( k ) =( a ij ( k )) has non-negative diagonal entries a ii ( k ) ≥ , andthe modulus matrix | A ( k ) | = ( | a ij ( k ) | ) is stochastic.The non-diagonal entries a ij ( k ) in (5) may be bothpositive and negative. Considering the elements ξ i ( k ) as“opinions” of n agents, the positive value a ij ( k ) > canbe treated as trust or attraction among agents i and j . Inthis case, agent i shifts its opinion towards the opinion ofagent j . Similarly, the negative value a ij ( k ) < standsfor distrust or repulsion among the agents: the i th agent’sopinion is shifted away from the opinion of agent j . Thecentral question concerned with the model (5) is reachingconsensus in absolute value, or modulus consensus [17]. Definition 2:
We say that modulus consensus is estab-lished by the protocol (5) if the coincident limits exist lim k →∞ | ξ ( k ) | = . . . = lim k →∞ | ξ n ( k ) | for any ξ (0) ∈ R n . The absolute values x i ( k ) = | ξ i ( k ) | obey the inequalities x i ( k + 1) ≤ n (cid:88) j =1 | a ij ( k ) | x j ( k ) ∀ i, (6)and hence the vector x ( k ) = ( x ( k ) , . . . , x n ( k )) (cid:62) obeys (2)with W ( k ) = | A ( k ) | . If this recurrent inequality is consensusdichotomic, then modulus consensus in (5) is established.Theorems 2 and 3 yield in in the following criterion. Theorem 4:
Modulus consensus in (5) is established, if thesequence of matrices W ( k ) = | A ( k ) | satisfies the conditionsof Theorem 2 or Theorem 3.In particular, if Assumption 1 holds, then modulus con-sensus is ensured by the repeated strong connectivity (As-sumption 2), which can be relaxed to the infinite strongconnectivity (Assumption 3) if the network is bidirectional w ij ( k ) > ⇔ w ji ( k ) > . Theorem 4 includes thus theresults of Theorems 2.1 and 2.2 in [23]. As discussed inSection III, the condition from Theorem 2 holds in manysituations where Assumption 1 fails, e.g. W ( k ) ≡ W maybe a constant primitive matrix with zero diagonal entries.Unlike consensus algorithms (3), where the gains w ij ( k ) aredesign parameters, the social influence (or “social power”)of an individual over another one depends on many uncertainfactors [39], and the uniform positivity of the non-zero gains | a ij ( k ) | may become a restrictive assumption. In general, the assumptions of Theorems 2 and 3 do notguarantee the exponential convergence rate to the equilib-rium, which is provided by Assumptions 1 and 2 [19], [25].In the case of exponential convergence, an additional crite-rion has been established in [19], [25] (see also Theorem 2.3in [23]), allowing to distinguish between “degenerate” mod-ulus consensus (asymptotic stability of the linear system (5))and polarization . In the latter case, the agents split intotwo “hostile camps” V ∪ V = V = { , . . . , n } , and theopinions of agents from V i converge to ( − i M , where M = M ( ξ (0)) (cid:54) = 0 for almost all ξ (0) . If V = ∅ or V = ∅ ,then polarization reduces to usual consensus of opinions. B. Substochastic matrices and the Friedkin-Johnsen model
A non-negative matrix A = ( a ij ) is called substochastic if (cid:80) nj =1 a ij = 1 ∀ i . We say that the i th row of A is a deficiency row of A if the latter inequality is strict (cid:80) j a ij < . Unlikea stochastic matrix, always having an eigenvalue at , asubstochastic square matrix is usually Schur stable ρ ( A ) < .Theorem 1 allows to give an elegant proof of the Schurstability criterion for substochastic matrices [40], [41]. Lemma 1:
Let G = G [ A ] be the graph of a substochasticsquare matrix A and I d = { i : (cid:80) j a ij < } is the subset ofits nodes, corresponding to the deficiency rows of A . If anynode j either belongs to the set I d , or I d is reachable fromit in G via some walk, then ρ ( A ) < . Proof:
Consider the matrix W = ( w ij ) , defined by w ij ∆ = a ij + 1 n (cid:32) − (cid:88) l a il (cid:33) ≥ a ij . Obviously, W = ( w ij ) is stochastic and w ij > a ij ≥ ∀ j when i ∈ I d . Hence in the graph G [ W ] each node i ∈ I d isconnected to any other node and to itself, and hence G [ W ] isaperiodic. The condition of Lemma 1 implies that G [ W ] isalso strongly connected. Choosing an arbitrary non-negativevector x ≥ , the vectors x ( k ) = A k x are non-negative forany k ≥ and satisfy the inequality (2) with W ( k ) ≡ W .Thanks to Theorem 1, x ( k ) → c , where c ≥ . It remainsto notice that is not an eigenvector of A since I d ( A ) (cid:54) = ∅ ,and hence c = 0 . Thus A k x → as k → ∞ for any x ≥ ,which implies the Schur stability of A since any vector x is a difference of two non-negative vectors.Notice that Lemma 1 implies the following well-knownproperty of substochastic irreducible matrices [38]: if G isstrongly connected then A is either stochastic or Schur stable.The condition from Lemma 1 is not only sufficient but alsonecessary for the Schur stability [41]. Lemma 1 implies thecondition of opinion convergence in the Friedkin-Johnsen model of opinion formation [10], [41], [42] x ( k ) = Λ W x ( k ) + ( I − Λ) u, u = x (0) . (7)Here W is a stochastic matrix of influence weights, and Λ isa diagonal matrix of the agents’ susceptibilities to the socialinfluence [42], ≤ λ ii ≤ . Without loss of generality, onemay suppose that λ ii = 0 ⇔ w ii = 1 ; in this case agent i is stubborn x i ( k ) ≡ x i (0) (often it is assumed [42] thatig. 2: The projection onto a closed convex set λ ii = 1 − w ii ). Another extremal case is λ ii = 1 , whichmeans that agent i “forgets” its initial opinion u i = x i (0) anditerates the usual procedure of opinion averaging x i ( k +1) = (cid:80) j w ij x j ( k ) . If < λ ii < , then agent i is “partiallystubborn” or prejudiced [11], [43]: such an agent adopts theothers’ opinions, however it is “attached” to its initial opinion x i (0) and factors it into every opinion iteration.If the substochastic matrix Λ W is Schur stable, then theopinion vector x ( k ) in (7) converges to the equilibrium x ( k ) −−−−→ k →∞ ( I − Λ W ) − ( I − Λ) u. (8)By noticing that the graphs G [Λ W ] and G [ W ] differ only bythe structure of self-loops (recall that λ ii > unless w ii = 1 and w ij = 0 ∀ j (cid:54) = i ), Lemma 1 implies the following. Corollary 3: [41] The opinions (8) converge if from eachagent i with λ ii = 1 there exists a walk in G [ W ] to someagent j with λ jj < , that is, each agent is either prejudicedor influenced (directly or indirectly) by a prejudiced agent.Using Theorems 2 and 3, some stability criteria for thetime-varying extension [43] of the Friedkin-Johnsen modelcan be obtained that are beyond the scope of this paper. C. Constrained consensus
In this subsection, we consider another application ofthe recurrent inequalities case, related to the problem of constrained or “optimal” consensus that is closely relatedto distributed convex optimization [44]–[46] and distributedalgorithms, solving linear equations [47]–[49].For any closed convex set Ω ⊂ R d and x ∈ R d the projection operator P Ω : x ∈ R d (cid:55)→ P Ω ( x ) ∈ Ω canbe defined, mapping a point to the closest element of Ω ,i.e. (cid:107) x − P Ω ( x ) (cid:107) = min y ∈ Ω (cid:107) x − y (cid:107) . This implies that (cid:93) ( y − P Ω ( x ) , x − P Ω ( x )) ≥ π/ (Fig. 2) and (cid:107) x − y (cid:107) ≥ (cid:107) x − P Ω ( x ) (cid:107) + (cid:107) y − P Ω ( x ) (cid:107) ∀ y ∈ Ω . (9)The distance d Ω ( x ) ∆ = (cid:107) x − P Ω ( x ) (cid:107) is a convex function.Consider a group of n discrete-time agents with the statevectors ξ i ( k ) ∈ R d . Each agent is associated with a closedconvex set Ξ i ⊆ R d (e.g., the set of minima of some convexfunction). The agents’ cooperative goal is to find some point ξ ∗ ∈ Ξ ∆ = Ξ ∩ . . . ∩ Ξ n . To solve this problem, variousmodifications of the protocol (3) have been proposed. We consider the following three algorithms ξ i ( k + 1) = P Ξ i (cid:104)(cid:88) nj =1 w ij ( k ) ξ j ( k ) (cid:105) , (10) ξ i ( k + 1) = P Ξ i (cid:104)(cid:88) nj =1 w ij ( k ) P Ξ j ( ξ j ( k )) (cid:105) , (11) ξ i ( k + 1) = w ii ( k ) P Ξ i ( ξ i ( k )) + (cid:88) j (cid:54) = i w ij ( k ) ξ j ( k ) . (12)Here W ( k ) = ( w ij ( k )) stands for the sequence of stochasticmatrices. The protocol (10) has been proposed in the in-fluential paper [44] (see also [46]), dealing with distributedoptimization problems. The special cases of protocols (11)and (12) naturally arise in distributed algorithms, solvinglinear equations, see respectively [47], [48] and [49]; arandomized version of (12) has been also examined in [45]. Theorem 5:
Let the set Ξ i be closed and convex, andassume that Ξ = Ξ ∩ . . . ∩ Ξ n (cid:54) = ∅ . Suppose that thematrices W ( k ) satisfy Assumptions 1 and 2. Then each ofthe protocols (10)-(12) establishes constrained consensus: lim k →∞ x ( k ) = . . . = lim k →∞ x n ( k ) ∈ Ξ . (13) Proof:
Due to the page limit, we give only an outlineof the proof. By assumption, there exists some ξ ∈ Ξ .Denote P i ( · ) ∆ = P Ξ i ( · ) , d i ( · ) ∆ = d Ξ i ( · ) and let η i ( k ) ∆ = (cid:80) j w ij ( k ) ξ j ( k ) . Under Assumptions 1 and 2, to prove theconstrained consensus (13) it suffices to show [46] that e i ( k ) ∆ = ξ i ( k + 1) − η i ( k ) −−−−→ k →∞ , d i ( ξ i ( k )) −−−−→ k →∞ . (14)Applying (9) to Ω = Ξ i , x = ξ , y = ξ ∈ Ξ i , one gets (cid:107) ξ − ξ (cid:107) ≥ (cid:107) P i ( ξ ) − ξ (cid:107) + d i ( ξ ) ∀ ξ ∈ R d , (15)and therefore (cid:107) ξ − ξ (cid:107) ≥ (cid:107) P i ( ξ ) − ξ (cid:107) . Each protocol (10)-(12) thus implies the recurrent inequality (2), where x i ( k ) ∆ = (cid:107) ξ i ( k ) − ξ (cid:107) ∀ i . For instance, the equation (10) entails that ≤ x i ( k +1) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n (cid:88) j =1 w ij ( k ) ξ j ( k ) − ξ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ n (cid:88) j =1 w ij ( k ) x j ( k ) . Corollary 1 implies the existence of the common limit x ∗ =lim k →∞ x i ( k ) ≥ . We are now going to prove (14) for theprotocol (10). The second statement in (14) is obvious since d i ( ξ i ( k + 1)) ≡ . Substituting ξ = η i ( k ) into (15), (cid:107) e i ( k ) (cid:107) (10) = d i ( η i ( k )) (15) ≤ (cid:107) η i ( k ) − ξ (cid:107) − x i ( k + 1) ≤≤ (cid:88) j w ij ( k ) x j ( k ) − x i ( k + 1) −−−−→ k →∞ . (16)To prove (14) for the protocol (12), notice that x i ( k + 1) (10) ≤ w ii ( k ) (cid:107) P i ( ξ i ( k )) − ξ (cid:107) + (cid:88) j (cid:54) = i w ij x j ( k ) (15) ≤ w ii ( k ) (cid:112) x i ( k ) − d i ( ξ i ( k )) + (cid:88) j (cid:54) = i w ij x j ( k ) . (17)Recalling that w ii ( k ) ≥ δ and x i ( k ) → x ∗ ∀ i , it can be shownthat d i ( ξ i ( k )) → and hence (cid:107) e i ( k ) (cid:107) = w ii ( k ) d i ( ξ i ( k )) → . The property (14) for the protocol (11) is proved similarly,combining the arguments from (16) and (17).. C ONCLUSIONS
In this paper, we have examined a class of recurrentinequalities (2), inspired by the analysis of “modulus consen-sus” in signed networks. Under natural connectivity assump-tions the inequality is shown to be consensus dichotomic ,that is, any of its solution is either unbounded or convergesto consensus. Besides signed networks, we illustrate theapplications of this profound property to some problems ofmatrix theory and distributed optimization algorithms.R
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PPENDIX P ROOFS OF T HEOREMS x ( k ) ∈ R n of (2), let j ( k ) , . . . , j n ( k ) be the permutation of the indices { , . . . , n } , sorting theelements of x ( k ) in the ascending order. In other words,the numbers y i ( k ) ∆ = x j ( k ) ( k ) satisfy the inequalities min i x i ( k ) = y ( k ) ≤ y ( k ) ≤ . . . ≤ y n ( k ) = max i x i ( k ) . We also introduce the sets J i ( k ) ∆ = { j ( k ) , . . . , j i ( k ) } and J ci ( k ) ∆ = { , . . . , n } \ J i ( k ) (where ≤ i ≤ n ). Recall that Φ( m, k ) = ( ϕ ij ( m s , k s )) = W ( m − . . . W ( k ) stands forthe evolutionary matrix of the linear equation (3).We start with the following simple proposition. Proposition 2:
For any p ≥ , q > p and j, l = 1 , . . . , n the inequality holds as follows x l ( q ) ≤ y n ( p ) − ϕ lj ( q, p ) ( y n ( p ) − x j ( p )) . In particular, for the case where q = p + 1 one has x l ( p + 1) ≤ y n ( p ) − w lj ( p ) ( y n ( p ) − x j ( p )) . (18) Proof:
The proof is immediate from the inequalities x l ( q ) ≤ n (cid:88) s =1 ϕ ls ( q, p ) x s ( p ) = ϕ lj ( q, p ) x j ( p )++ (cid:88) s (cid:54) = j ϕ ls ( q, p ) x s ( p ) by noticing that x s ( p ) ≤ y n ( p ) for any s (cid:54) = j . Lemma 2:
Suppose that for some k ≥ and m > k onehas Φ( m, k ) ∈ S ε . Then for any i < n one has y i +1 ( m ) ≤ (1 − ε ) y n ( k ) + εy i ( k ) . (19) Proof:
The definition of the set J i ( k ) and (2) implythat x j ( m ) ≤ (1 − ε ) y n ( k ) + εy i ( k ) for any j ∈ J i ( k ) since ϕ jj ( m, k ) ≥ ε and x j ( k ) ≤ y i ( k ) . Since the graph G ε [Φ( m, k )] is strongly connected, there exist some j ∈ J i ( k ) and l ∈ J ci ( k ) such that ϕ lj ( m, k ) ≥ ε , and thus x l ( m ) ≤ (1 − ε ) y n ( k ) + εy i ( k ) due to (2). This entails (19)since the set J i ( k ) ∪ { l } contains i + 1 elements.The statement of Lemma 2 retains its validity, replacingthe condition from Theorem 2 by the assumptions of The-orem 3. However, in this situation m = m ( k, i ) should bechosen in a different way and depends on both i and k . Lemma 3:
Let Assumptions 1 and 3 hold and the commu-nication be bidirectional w ij ( k ) > ⇐⇒ w ji ( k ) > . Thenfor all k ≥ , i < n there exists m = m ( i, k ) > k such that y i +1 ( m ) ≤ (1 − δ ) y n ( k ) + δy i ( k ) , (20)where δ > is the constant from Assumption 1. Proof:
For a fixed k ≥ and i < n , we denote forbrevity J = J i ( k ) and J c = J ci ( k ) . Assumption 3 impliesthe existence of s ≥ k such that w jl ( s ) > (and thus w lj > ) for some j ∈ J and l ∈ J c . Let m − stand for the minimum of such s (that is, m ≥ k + 1 ). Since w jl ( s ) = w lj ( s ) = 0 for any s = k, . . . , m − , one has ϕ jl ( m − , k ) = ϕ lj ( m − , k ) = 0 for any pair j ∈ J, l ∈ J c . Hence x j ( m −
1) = (cid:88) r ∈ J ϕ jr ( m − , k ) x r ( k ) ≤ y i ( k ) . Applying (18) to p = m − and j = l ∈ J , one has x j ( m ) ≤ (1 − δ ) y n ( m − δx j ( m − ≤ (1 − δ ) y n ( k )+ δy i ( k ) . At thesame time, there exist j ∈ J, l ∈ J c such that w lj ( m − ≥ δ .In view of (18), x l ( m ) ≤ (1 − δ ) y n ( m −
1) + δx j ( m − ≤ (1 − δ ) y n ( k ) + δy i ( k ) . This entails (20) since the set J i ( k ) ∪ { l } contains i + 1 elements. A. Proofs of Theorems 2 and 3
Consider a bounded solution x ( k ) of (2) and its ordering y ( k ) . The inequality (2) implies, obviously, that y n ( k +1) ≤ y n ( k ) , and therefore there exists the limit y ∗ =lim k →∞ y n ( k ) . Our goal is to show that y j ( k ) −−−−→ k →∞ y ∗ forany j , provided that the assumptions of either Theorem 2or Theorem 3 are valid. The proof is via induction on j = n, n − , . . . , . For j = n the statement holds. Supposethat y j ( k ) → y ∗ for j = i + 1 , . . . , n ; we are now goingto prove that y i ( k ) → y ∗ as k → ∞ . Since y i ( k ) ≤ y n ( k ) ,it suffices to show that lim k →∞ y i ( k ) ≥ y ∗ . Suppose, on thecontrary, that lim k →∞ y i ( k ) < y ∗ , that is, there exist a sequence k s −−−→ s →∞ ∞ and q > , such that y i ( k s ) −−−→ s →∞ y ∗ − q .As implied by Lemma 2 (respectively, Lemma 3), underthe assumptions of Theorem 2 (respectively, Theorem 3), asequence m s > k s and a constant ε > exist such that y i +1 ( m s ) ≤ εy i ( k s ) + (1 − ε ) y n ( k s ) . Passing to the limit as s → ∞ , one arrives at y ∗ = lim s →∞ y i +1 ( m s ) ≤ (1 − ε ) y ∗ + ε ( y ∗ − q ) = y ∗ − εq, which is a contradiction. Thus y i ( k ) → y ∗ as k → ∞ , whichproves the induction step. Therefore, the solution convergesto a consensus equilibrium x ( k ) −−−−→ k →∞ y ∗ n . (cid:4) B. Proof of Theorem 1
The sufficiency part is immediate from Remark 2 andTheorem 2. Indeed, if the graph G [ W ] is strongly connectedand aperiodic, then W d is a strictly positive matrix for some d , so the condition of Theorem 2 holds: Φ( k + d, k ) = W d ∈ S ε for some ε > . Hence the inequality (2) isconsensus dichotomic. If the graph G [ W ] is constituted by s > isolated and aperiodic strongly connected components,then (2) is dichotomic, reducing to s independent consensusdichotomic inequalities of lower dimensions.To prove necessity, consider a dichotomic inequality (2)with W ( k ) ≡ W and let w ij > , that is, i is connected to j in the graph G [ W ] . Let the set J include node j and allnodes that are reachable from j by walks. We are going tohow that i ∈ J , that is, i and j belong to the same strongcomponent. Suppose, on the contrary, that i (cid:54)∈ J and let x r ( k ) = ( − k , r = i,M, r ∈ J, − , r (cid:54)∈ J ∪ { i } , r = 1 , . . . , n. Here M is chosen sufficiently large so that ( M + 1) w ij > .It can be easily shown that the vector x ( k ) is a solution to (2).Indeed, for any r ∈ J and q (cid:54)∈ J one obviously has w rq = 0 (otherwise, q would be reachable from j via l ). Therefore, M = x r ( k ) = (cid:80) nq =1 w rq x q ( k ) = (cid:80) q ∈ J w rq x q ( k ) . For any r (cid:54)∈ J ∪{ i } we have x r ( k ) = min q x q ( k ) ≤ (cid:80) nq =1 w rq x q ( k ) .Finally, x i ( k ) ≤ ≤ M w ij − (1 − w ij ) = M w ij − (cid:80) q (cid:54) = j w iq ≤ w ij x j ( k )+ (cid:80) q (cid:54) = j w iq x q ( k ) = (cid:80) nq =1 w iq x q ( k ) .Since x ( k ) is bounded yet does not converge, one arrives atthe contradiction with the assumption of dichotomy.Hence for the dichotomy it is necessary that any arcconnects nodes from the same strong components, whereastwo different components have no arcs between them. Noticethat the dichotomy (respectively, consensus dichotomy) ofthe inequality (2) implies that any solution of the equa-tion (3) converges to an equilibrium (respectively, to aconsensus equilibrium). Applying the standard convergenceand consensus criteria for the static consensus protocol (3)(see e.g. [11]), one shows that dichotomy is possible onlywhen all strong components of G [ W ]]