In the recent years protocols for consensus and synchronization in multi-agent networks have been thoroughly studied [1, 2, 3, 4]. Much less studied are “irregular” behaviors, exhibited by many real-world networks, such as e.g. desynchronization  and chaos . An important step in understanding these complex behaviors is to elaborate mathematical models for “partial” or cluster synchronization, or simply clustering [5, 7, 8, 9]. In social influence theory, this problem is known as the community cleavage problem or Abelson’s diversity puzzle [10, 11]: to disclose mechanisms that hinder reaching consensus among social actors and lead to splitting of their opinions into several clusters.
One reason for clustering in multi-agent networks is the presence of “negative” (repulsive, antagonistic) interactions among the agents . Models of signed (or “coopetition”) networks with positive and negative couplings among the nodes describe a broad class of real-world systems, from molecular ensembles  to continental supply chains . Positive and negative relations among social actors can express, respectively, trust (friendship) or distrust (hostility). Negative ties among the individuals may also result from the reactance or boomerang effects, first described in : an individual may not only resist the persuasion process, but even adopt an attitude that is contrary to the persuader’s one.
A simple yet instructive model of continuous-time opinion dynamics over signed networks has been proposed by Altafini [15, 16] and extended to the discrete-time case in . In the recent years, Altafini-type coordination protocols over static and time-varying signed graphs have been extensively studied, see e.g. [18, 19, 20, 21, 22, 23, 24, 25]. It has been shown that under mild connectivity assumptions these models exhibit consensus in absolute value, or modulus consensus: the agents’ opinions agree in modulus yet may differ in signs. The modulus consensus may correspond to the asymptotic stability of the network (the opinions of all individuals converge to zero), usual consensus (convergence of the opinions to the same value, depending on the initial condition) and polarization, or “bipartite consensus”: the agents split into two groups, converging to the opposite opinions.
In the recent works [26, 27] it has been shown that the effect of modulus consensus in the continuous-time Altafini model is in fact a manifestation of a more profound result, concerned with the special class of differential inequalities
where stands for the Laplacian matrix of a time-varying weighted graph. Although the inequality (1) is a seemingly “loose” constraint, any of its bounded solutions (under natural connectivity assumptions) converges to a consensus equilibrium (this property is called consensus dichotomy). This implies, in particular, the modulus consensus in the Altafini model [26, 27]
since the vector of the opinions’ absolute values obeys the inequality (1). In this paper, we extend the theory of differential inequalities to the discrete-time case, where (1) is replaced by the recurrent inequality with being a sequence of stochastic matrices. We establish the consensus dichotomy criteria for these inequalities, which imply the recent results on modulus consensus in the discrete-time Altafini model [17, 23]. We also apply the recurrent inequalities to some problems of matrix theory and the analysis of distributed algorithms for optimization and linear equations solving.
Ii Problem Setup
We start with preliminaries and introducing some notation.
First we introduce some notation. A vector is non-negative () if . Given two vectors , we write (respectively, ) if (respectively, ). The vector of ones is denoted by . Given a matrix , we use to denote the matrix of element-wise absolute values (the same rule applies to vectors). A matrix is stochastic if its entries are non-negative and all rows sum to , i.e. . We use to denote the spectral radius of a square matrix . The standard Euclidean norm of a vector is denoted by .
A non-negative matrix can be associated to a (directed) weighted graph111We assume that the reader is familiar with the standard concepts of graph theory, regarding directed graphs and their connectivity properties, e.g. walks (or paths), cycles and strongly connected components [28, 29]. , whose set of arcs is .
Ii-B Recurrent inequalities and consensus dichotomy.
In this paper, we are interested in the solutions of the following discrete-time, or recurrent, inequality
where is a sequence of vectors and stands for a sequence of stochastic matrices.
dating back to the early works on social influence [33, 34], rational decision making  and distributed optimization . The algorithm (3) may be interpreted as the dynamics of opinions222In the broad sense, “opinion” is just a scalar quantity of interest; it can stand for e.g. a physical parameters or an attitude to some event or issue. formation in a network of agents. At each step of the opinion iteration agent calculates the weighted average of its own opinion and the others’ opinions; this average is used as the new opinion of the th agent . The graph naturally represents the interaction topology of the network at step . Agent is influenced by agent if , otherwise the th agent’s opinion plays no role in the formation of the new agent ’s opinion .
A similar interpretation can be given to the inequality (2). Unlike the algorithm (3), the opinion of agent at each step of opinion formation is not uniquely determined by the opinions from the previous step, but is only constrained by them . The weight stands for the contribution of agent ’s opinion to this constraint, and in this sense it can also be treated as the “influence” weight. The inequality (2) does not provide the solution’s uniqueness for a given , but only guarantees the existence of an upper bound for the solutions.
Any solution of (2) obeys the inequality
Proposition 1 is proved via straightforward induction on . By definition, ; if then .
Although many solutions of (2) are unbounded from below, under certain assumptions any its bounded solution converges to a consensus equilibrium , where . A similar property, called consensus dichotomy333The term dichotomy originates from ODE theory. A system is dichotomic if any of its solutions either grows unbounded or has a finite limit . has been established in [26, 27] for the differential inequalities (1).
The inequality (2) is said to be dichotomic if any of its bounded (from below) solutions has a limit . It is called consensus dichotomic if these limits are consensus equilibria , where .
Iii Main Results
The first step is to examine time-invariant inequalities (2).
Iii-a A dichotomy criterion for the time-invariant case
In this subsection, we assume that is a constant matrix, whose graph has strongly connected (or strong) components ; in general, arcs between different components may exist (Fig. 1a). A strong component is isolated if no arc enters or leaves it. All strong components are isolated (Fig. 1b) if and only if every arc of the graph belongs to a cycle [28, Theorem 3.2].
The inequality (2) with the static matrix is dichotomic if and only if all the strong components of its graph are isolated and aperiodic444Recall that a graph is aperiodic if the greatest common divisor of its cycles’ lengths (that is also referred to as the graph’s period) is equal to .. The inequality is consensus dichotomic if and only if is strongly connected () and aperiodic, or, equivalently, the matrix is primitive [29, 38].
The proof of Theorem 1, as well as the remaining results of this section, is given in Appendix.
Iii-B Consensus dichotomy in the time-varying case
for any ;
the graph is strongly connected, where .
In other words, removing from the graph all “light” arcs weighted by less than , the remaining subgraph is strongly connected and has self-loops at each node.
The following theorem provides a consensus dichotomy criterion for the case of the time-varying matrix .
The inequality (2) is consensus dichotomic if exists that satisfies the following condition: for any there exists such that .
Notice that for the static matrix one has , so the condition from Theorem 2 means that for some . It can be easily shown that in this case is a strictly positive matrix. On the other hand, if is strictly positive for some , then for sufficiently small . In view of Remark 2 and Theorem 1, in the static case the sufficient condition of consensus dichotomy from Theorem 2 is in fact also necessary, boiling down to the primitivity of .
The condition from Theorem 2 is implied by the two standard assumptions on the sequence .
There exists such that for any
for any ;
for any such that one has .
(Repeated joint strong connectivity) There exists an integer such that the graph is strongly connected for any .
We are going to show that the condition from Theorem 2 holds for and , i.e. for any . Indeed, whenever due to Assumption 1. Supposing that , where , one has , and therefore . Applying this to , one easily notices that is connected to in the graph whenever for some . Assumption 2 implies now that for any .
It should be noticed however that the condition of Theorem 2 may hold in many situations where Assumptions 1 and 2 fail. Even in the static case , the matrix can be primitive yet have zero diagonal entries. The following corollary illustrates another situation where both Assumptions 1 and 2 may fail, whereas Theorem 2 guarantees consensus dichotomy.
Suppose that for any one has , where stands for the primitive matrix and is a set of stochastic matrices, commuting with : . Let the set be infinite. Then the inequality (2) is consensus dichotomic.
Let be so large that is a positive matrix, whose minimal entry equals . For any , we can find such that the sequence contains elements from the set . Since any commutes with , , where
is some stochastic matrix, and thus all entries ofare not less than .
Many sequences , satisfying the conditions of Corollary 2, fail to satisfy Assumptions 1 and 2. For instance, if then the sequence can contain an arbitrary long subsequence of consecutive identity matrices, which violates Assumption 2. Both the matrix and matrices from may have zero diagonal entries, which also violates Assumption 1. The set can also be non-compact, containing matrices with arbitrary small yet non-zero entries.
Iii-C The case of bidirectional interaction
It is known that in the case of bidirectional graphs the conditions for consensus in the network (3) is reached under very modest connectivity assumptions. Under Assumption 1, consensus is reached if and only if the following relaxed version of Assumption 2 holds .
(Infinite joint strong connectivity) The graph is strongly connected, where
The following result extends this consensus criterion to the condition of consensus dichotomy in the inequality (2).
The relaxation of Assumption 1 in Theorem 3 remains a non-trivial open problem. To the best of the authors’ knowledge, the same applies to usual consensus algorithms (3): most of the existing results for consensus in discrete-time switching networks [31, 30, 32, 2] rely on Assumption 1 or at least require uniformly positive diagonal entries .
Iv Examples and Applications
In this section we apply the criteria from Section III to the analysis of several multi-agent coordination protocols.
Iv-a Modulus consensus in the discrete-time Altafini model
Here the matrix satisfies the following assumption.
For any , the matrix has non-negative diagonal entries , and the modulus matrix is stochastic.
The non-diagonal entries in (5) may be both positive and negative. Considering the elements as “opinions” of agents, the positive value can be treated as trust or attraction among agents and . In this case, agent shifts its opinion towards the opinion of agent . Similarly, the negative value stands for distrust or repulsion among the agents: the th agent’s opinion is shifted away from the opinion of agent . The central question concerned with the model (5) is reaching consensus in absolute value, or modulus consensus .
We say that modulus consensus is established by the protocol (5) if the coincident limits exist
The absolute values obey the inequalities
In particular, if Assumption 1 holds, then modulus consensus is ensured by the repeated strong connectivity (Assumption 2), which can be relaxed to the infinite strong connectivity (Assumption 3) if the network is bidirectional . Theorem 4 includes thus the results of Theorems 2.1 and 2.2 in . As discussed in Section III, the condition from Theorem 2 holds in many situations where Assumption 1 fails, e.g. may be a constant primitive matrix with zero diagonal entries. Unlike consensus algorithms (3), where the gains are design parameters, the social influence (or “social power”) of an individual over another one depends on many uncertain factors , and the uniform positivity of the non-zero gains may become a restrictive assumption.
In general, the assumptions of Theorems 2 and 3 do not guarantee the exponential convergence rate to the equilibrium, which is provided by Assumptions 1 and 2 [19, 25]. In the case of exponential convergence, an additional criterion has been established in [19, 25] (see also Theorem 2.3 in ), allowing to distinguish between “degenerate” modulus consensus (asymptotic stability of the linear system (5)) and polarization. In the latter case, the agents split into two “hostile camps” , and the opinions of agents from converge to , where for almost all . If or , then polarization reduces to usual consensus of opinions.
Iv-B Substochastic matrices and the Friedkin-Johnsen model
A non-negative matrix is called substochastic if . We say that the th row of is a deficiency row of if the latter inequality is strict
. Unlike a stochastic matrix, always having an eigenvalue at, a substochastic square matrix is usually Schur stable . Theorem 1 allows to give an elegant proof of the Schur stability criterion for substochastic matrices [40, 41].
Let be the graph of a substochastic square matrix and is the subset of its nodes, corresponding to the deficiency rows of . If any node either belongs to the set , or is reachable from it in via some walk, then .
Consider the matrix , defined by
Obviously, is stochastic and when . Hence in the graph each node is connected to any other node and to itself, and hence is aperiodic. The condition of Lemma 1 implies that is also strongly connected. Choosing an arbitrary non-negative vector , the vectors are non-negative for any and satisfy the inequality (2) with . Thanks to Theorem 1, , where . It remains to notice that
is not an eigenvector ofsince , and hence . Thus as for any , which implies the Schur stability of since any vector is a difference of two non-negative vectors.
Notice that Lemma 1 implies the following well-known property of substochastic irreducible matrices : if is strongly connected then is either stochastic or Schur stable. The condition from Lemma 1 is not only sufficient but also necessary for the Schur stability . Lemma 1 implies the condition of opinion convergence in the Friedkin-Johnsen model of opinion formation [42, 10, 41]
Here is a stochastic matrix of influence weights, and is a diagonal matrix of the agents’ susceptibilities to the social influence , . Without loss of generality, one may suppose that ; in this case agent is stubborn (often it is assumed  that ). Another extremal case is , which means that agent “forgets” its initial opinion and iterates the usual procedure of opinion averaging . If , then agent is “partially stubborn” or prejudiced [11, 43]: such an agent adopts the others’ opinions, however it is “attached” to its initial opinion and factors it into every opinion iteration.
If the substochastic matrix is Schur stable, then the opinion vector in (7) converges to the equilibrium
By noticing that the graphs and differ only by the structure of self-loops (recall that unless and ), Lemma 1 implies the following.
Iv-C Constrained consensus
In this subsection, we consider another application of the recurrent inequalities case, related to the problem of constrained or “optimal” consensus that is closely related to distributed convex optimization [44, 45, 46] and distributed algorithms, solving linear equations [47, 48, 49].
For any closed convex set and the projection operator can be defined, mapping a point to the closest element of , i.e. . This implies that (Fig. 2) and
The distance is a convex function.
Consider a group of discrete-time agents with the state vectors . Each agent is associated with a closed convex set (e.g., the set of minima of some convex function). The agents’ cooperative goal is to find some point . To solve this problem, various modifications of the protocol (3) have been proposed. We consider the following three algorithms
Here stands for the sequence of stochastic matrices. The protocol (10) has been proposed in the influential paper  (see also ), dealing with distributed optimization problems. The special cases of protocols (11) and (12) naturally arise in distributed algorithms, solving linear equations, see respectively [47, 48] and ; a randomized version of (12) has been also examined in .
Due to the page limit, we give only an outline of the proof. By assumption, there exists some . Denote , and let . Under Assumptions 1 and 2, to prove the constrained consensus (13) it suffices to show  that
Applying (9) to , , , one gets
In this paper, we have examined a class of recurrent inequalities (2), inspired by the analysis of “modulus consensus” in signed networks. Under natural connectivity assumptions the inequality is shown to be consensus dichotomic, that is, any of its solution is either unbounded or converges to consensus. Besides signed networks, we illustrate the applications of this profound property to some problems of matrix theory and distributed optimization algorithms.
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Although the results of Theorems 1-3 resemble usual criteria of convergence for consensus protocols (3), their proofs are based on different techniques, which are inspired by the analysis of differential inequalities from [26, 27]. Given a solution of (2), let be the permutation of the indices , sorting the elements of in the ascending order. In other words, the numbers satisfy the inequalities
We also introduce the sets and (where ). Recall that stands for the evolutionary matrix of the linear equation (3).
We start with the following simple proposition.
For any , and the inequality holds as follows
In particular, for the case where one has
The proof is immediate from the inequalities
by noticing that for any .
Suppose that for some and one has . Then for any one has
The definition of the set and (2) imply that for any since and . Since the graph is strongly connected, there exist some and such that , and thus due to (2). This entails (19) since the set contains elements.
The statement of Lemma 2 retains its validity, replacing the condition from Theorem 2 by the assumptions of Theorem 3. However, in this situation should be chosen in a different way and depends on both and .
For a fixed and , we denote for brevity and . Assumption 3 implies the existence of such that (and thus ) for some and . Let stand for the minimum of such (that is, ). Since for any , one has for any pair . Hence
Consider a bounded solution of (2) and its ordering . The inequality (2) implies, obviously, that , and therefore there exists the limit . Our goal is to show that for any , provided that the assumptions of either Theorem 2 or Theorem 3 are valid. The proof is via induction on . For the statement holds. Suppose that for ; we are now going to prove that as . Since , it suffices to show that . Suppose, on the contrary, that , that is, there exist a sequence and , such that .
Passing to the limit as , one arrives at
which is a contradiction. Thus as , which proves the induction step. Therefore, the solution converges to a consensus equilibrium .
-B Proof of Theorem 1
The sufficiency part is immediate from Remark 2 and Theorem 2. Indeed, if the graph is strongly connected and aperiodic, then is a strictly positive matrix for some , so the condition of Theorem 2 holds: for some . Hence the inequality (2) is consensus dichotomic. If the graph is constituted by isolated and aperiodic strongly connected components, then (2) is dichotomic, reducing to independent consensus dichotomic inequalities of lower dimensions.
To prove necessity, consider a dichotomic inequality (2) with and let , that is, is connected to in the graph . Let the set include node and all nodes that are reachable from by walks. We are going to show that , that is, and belong to the same strong component. Suppose, on the contrary, that and let
Here is chosen sufficiently large so that . It can be easily shown that the vector is a solution to (2). Indeed, for any and one obviously has (otherwise, would be reachable from via ). Therefore, . For any we have . Finally,