On Adversary Robust Consensus protocols through joint-agent interactions
OOn Adversary Robust Consensus protocols through joint-agentinteractions
David Angeli and Sabato ManfrediJanuary 10, 2019
Abstract
A generalized family of Adversary Robust Consensus protocols is proposed and analyzed. These aredistributed algorithms for multi-agents systems seeking to agree on a common value of a shared variable,even in the presence of faulty or malicious agents which are updating their local state according to theprotocol rules. In particular, we adopt monotone joint-agent interactions, a very general mechanism forprocessing locally available information and allowing cross-comparisons between state-values of multipleagents simultaneously. The salient features of the proposed class of algorithms are abstracted as a Petri Netand convergence criteria for the resulting time evolutions formulated by employing structural invariants ofthe net.
Algorithms for consensus were introduced in [17] a few decades ago, in the context of distributed optimization,a topic which remains of great interest still today, [18].The role played by propagation of information in achieving consensus among interacting agents, was first high-lighted in the seminal paper [6]. Therein, authors formulated tight and explicit graph-theoretical requirementsfor asymptotic consensus in time-varying linear update protocols, by abstracting the network of agents’ interac-tions and its underlying dynamics as a graph. This sparked a considerable interest of the scientific communityin advancing and applying consensus protocols for multi-agent systems (see i.e. [3, 4, 5] and references therein).Subsequent developments in the theory of nonlinear consensus protocols have formalized and clarified the roleof information spread along the graph of agents’ interaction for more general situations, including second andhigher order agents dynamics, or agents’ states evolving on manifolds [21] or nonlinear interactions [10, 11].More recently, graph theoretical criteria have been similarly developed to encompass asymmetric confidence, asin the case of unilateral interactions [12], or joint-agent interactions, [1].The latter, in particular, account for situations where individual agents impose “filtering thresholds” uponneighbours’ influences by cross-validating their opinions through mutual comparisons that only allow for con-sistent infuences (either from above or below) of two or more neighbouring agents to be enacted upon. Inthis regard, unlike the majority of existing consensus protocols that implicitly assume ‘additive’ dynamics andexhibit variation rates as a disjunctive combination (sum) of neighbors influences, joint-agent interactions allowthe formulation of conjunctive influences, and, respectively, of their additive combination.It is worth stressing that complex contemporary social and engineering systems often need to deal withselfish or malicious users, node faults and attacks ([22]- [26]). In this respect the evaluation of individual andgroup reputation play a focal role for the safety of such systems. In the last years different (centralised anddistributed) algorithms have been proposed to deal with online reputation estimation of both individual ([27]-[32]) and group (clustering all users according to their rating similarities - [33, 34]) to providing incentives tousers acting responsibly and cooperatively.Within this line of investigation, the problem of Adversary Robust Consensus Protocols (ARC-P) was for-mulated in [7], following earlier seminal results in [35]. Therein, Leblanc and coworkers propose and analyzea discrete time protocol which allows n cooperating agents to converge towards a consensus state, within acomplete all-to-all network, even when a subset of agents (of cardinality up to (cid:98) n/ (cid:99) ) is malicious or faulty ,namely it evolves in a completely arbitrary way, with the sole constraint of broadcasting its own state to allremaining agents. The proposed protocol simply orders state values in ascending (or descending order) andremoves F top and lowest values from the ordered list, where F is an apriori fixed bound to the number ofmalicious agents. Then, the average among the remaining values is computed and a standard linear consensusupdate equation is applied. 1 a r X i v : . [ c s . S Y ] J a n ubsequent analysis has been devoted in [9] to the important topic of relaxing the all-to-all topology require-ment and investigating sufficient conditions for Adversary Robust Consensus on the basis of local informationonly, or in the presence of so called Byzantine agents [8], who may, either intentionally or due to faulty con-ditions, communicate different state values to different neighbours. A related line of investigation assumes thepresence of trusted nodes, [2].The interconnection topology is interpreted, in such context, as a specific type of switching (time-varying)linear consensus, arising through the application of the so called sorting function, its composition with the reducing function (responsible for discarding highest and lowest values) and finally by averaging the entries ofthe vector obtained. It turns out, however, that similar types of agents interactions can also be recast withinthe framework of joint-agent interactions.The formalism of joint-agent interactions is, in this respect, even more flexible and may, for instance, allowto partition neighbours of every agent in several subgroups, to be suitably sorted, reduced and averaged whileadding (possibly with different weights) the influences resulting from distinct subgroups as a final step. Thistype of rules for processing local information results in consensus protocols which allow different levels of trustattributed to different set of neighbors and, generally speaking, break the simmetry implicit in the use of asingle sorting and reducing function.The extended class of intrinsically nonlinear consensus protocols afforded by the use of joint-agent inter-actions can be conveniently described and characterized, from a topological point of view, as bipartite graphs ,and more specifically Petri Nets. It turns out that structural notions, developed in the context of Petri Nets toascertain their liveness as Discrete Event Systems, play a crucial role in characterizing the ability of a networkof agents to reach consensus regardless of initial conditions [1].The specific details of the conditions needed for this to happen will be illustrated in a subsequent Section.Nevertheless, it is intuitive that if, on one hand, application of conjunctive filtering conditions among neighbour-ing agents limits the spread of information across the network (and therefore, if not done carefully may preventconsensus from happening at all), on the other, it only allows “trustworthy” information to be propagated, andtherefore may result (if carefully deployed) in Adversary Robust Consensus protocols. In this paper we addressthe issue of when a network of agents, with arbitrary (and possibly asymmetric) interconnection topology (al-lowing for instance differentiated trust levels among neighboors) exhibits the ability to reach consensus despitea subset of its agents being either faulty or malicious , viz. able to influence other nodes according to theirindividual state-value but, in fact, upgrading their position in a completely arbitrary fashion.Just to illustrate the potential of the approach, we present below simulations referring to an all-to-all networkof 5 agents, involving linear interactions, or a similar network entailing joint-agent interactions. Our theoryallows to prove that, in the latter network, robustness can be achieved allowing any set of 2 out of 5 agents tobe malicious or faulty, and still guaranteeing the remaining healthy agents will retain the ability to reach exactconsensus. In particular, we simulate the following linear network:˙ x i = (cid:88) j (cid:54) = i a ij ( x j − x i ) , i, j ∈ { . . . } , (1)for some a ij > x i = (cid:88) J ⊂{ ... }\{ i } : | J | =3 f J → i ( x ) (2)where the function f J → i : R n → R is defined below: f J → i ( x ) = max j ∈ J min { x j − x i , } + min j ∈ J max { x j − x i , } . We pick the initial condition [35 , , , , (cid:48) and run the two consensus protocols assuming that agents 4 and5 are faulty and follow the apriori fixed time evolutions: x ( t ) = 15 + cos(3 t ) −
19 + t sin(3 t )3 + t x ( t ) = 20 + sin(2 t )4 + t (cid:16) t ) − (cid:17) . These agents effectively act as exogenous disturbance inputs for the remaining 3 agents, and, from the practicalpoint of view, may be regarded as faulty agents or malicious agents trying to disrupt the consensus. In the2igure 1: Linear consensus protocol subject to faulty agents 4 and 5case of equation (1), as it is expected due to linearity and addivity of interactions, the exogenous disturbances x and x are able to spread their influence to the remaining agents and effectively prevent the remainingagents to asymptotically reach consensus (see Fig. 1). In the case of equation (2), instead, nonlinear joint-agentinteractions allow the three“healthy” agents, 1 , , x and x , (see Fig. 2). In other words, while the faulty agents may, to acertain extent, affect the final consensus value reached, they are unable to disrupt it. The aim of this note is to derive necessary and sufficient conditions to characterize when networks of agentsimplementing joint-agent interactions may be able to achieve robust consensus in the presence of possibly malicious or faulty agents. In particular, we study networks described by the following class of nonlinearfinite-dimensional differential equations: ˙ x = f ( x ) (3)where x ∈ R n is the state vector, and f : R n → R n is a Lipschitz continous function, describing the updatelaws of each agent as a function of its own and neighbours’ state values. For convenience we ask that f j bemonotonically non-decreasing with respect to all x i ( i (cid:54) = j ) so that the resulting flow is monotone with respectto initial conditions, once the standard order induced by the positive orthant is adopted. This assumption,while not essential, can to a certain extent simplify the analysis and the definition of interaction among agents.Many authors, in recent years, have elaborated conditions under which solutions of (3) asymptotically convergetowards equilibriums of the following form: lim t → + ∞ ϕ ( t, x ) = ¯ x (4)for some ¯ x ∈ R , where is the vector of all ones in R n . When this occurs for all solutions, and regardless ofinitial conditions, we say that system (3) achieves global asymptotic consensus .In this note, however, we consider a more general situation in which the state vector x is partitionedinto two subvectors, x H and x F , associated to healthy and faulty agents respectively. Accordingly, we denote f ( x ) = [ f H ( x ) (cid:48) , f F ( x ) (cid:48) ] (cid:48) and wish to characterize under what assumptions solutions of˙ x H = f H ( x H , x F ) (5)asymptotically converge to equilibria of the type:lim t → + ∞ x H ( t ) = ¯ x H x H (0) and all exogenous input signals x F ( · ). A formal definition follows. Definition 1
We say that network (3) achieves robust consensus in the face of faults in F ⊂ N , if, partitioningthe state vector according to F and H := N \ F yields: lim t → + ∞ ϕ H ( t, x H (0) , x F ( · )) = ¯ x H for all x H (0) , and all uniformly bounded exogenous input x F ( · ) , (where ϕ H ( t, x H (0) , x F ( · )) denotes the solutionof (5) at time t from initial condition x H (0) and input x F ( · ) ). In practice, for a given net, we will be interested in considering several possible combinations of faulty agents(corresponding to several choices of F ) and, for each one of them, verify conditions for asymptotic convergencetowards consensus of the remaining healthy agents H .In order to characterize the flow of information needed for achieving such kind of behaviour, we recall thenotion of joint agent interaction , as proposed in [1]. Definition 2
We say that a group of agents I ⊂ N jointly influences agent j ∈ N \ I if for all compact intervals K ⊂ R there exists a positive definite function ρ , such that, for all x I , x j ∈ K it holds:sign ( x I − x j ) f j ( x j + ( x I − x j ) e I ) ≥ ρ ( | x I − x j | ) . (6) We denote this by the following shorthand notation: I → j . Notice that influence from I to j , denoted as I → j , is monotone (in its first argument I ) with respect toset-inclusion. In particular, if j ∈ N \ ˜ I we have: I → j and I ⊂ ˜ I ⇒ ˜ I → j. For this reason, it is normally enough to consider minimal influences alone. We say that I influences j and thatthis influence is minimal if there is no ˜ I (cid:40) I such that ˜ I → j . Our goal is to derive characterizations of a graph theoretical nature regarding the ability of networks withjoint-agent interactions to exhibit robust consensus, in the face of faults or malicious attacks. We adopt, tothis end, the formalism introduced in [1]. In particular, we represent multiagent networks as Petri Nets. These4 A A A A A A A Figure 3: Petri Nets associated to network of interactions (7) and (8)are a type of bipartite graph, used to model Discrete Event Systems, and can be conveniently adopted in thepresent study. In fact, a rich literature on structural invariants for Petri Nets already exists, including softwarelibraries to compute them as well as complexity analysis of the available algorithms.An (ordinary) Petri Net is a quadruple { P, T, E I , E O } , where P and T are finite sets (with P ∩ T = ∅ ) re-ferred to as places and transitions , respectively. These are nodes of a directed bipartite graph. In fact, directededges are of two types: E I ⊂ T × P connecting transitions to places and E O ⊂ P × T connecting places totransitions.In our context places represent agents while transitions stand for interactions among them. More closely,to each agent i ∈ N there exists a unique associated place p i ∈ P . Furthermore, if agents in J ⊂ N jointlyinfluence agent i , this is denoted as J → i and, provided this interaction is minimal, it is represented graphicallyby a single transition t ∈ T , with edges ( p j , t ) ∈ E O for all j ∈ J and a single edge ( t, p i ) in E I . Notice thatevery transition can be assumed to only afford exactly one outgoing edge, unlike in general Petri Nets. As anexample, we show in Fig. 3 the graphical representation of the Petri Nets associated to the list of interactions { } → , { , } → , (7)and, next to it, for a kind of ring topology with 5 agents and the following list of minimal joint agent interactions: { , } → , { , } → , { , } → , { , } → , { , } → . (8)The next concepts will be crucial in characterizing, from the topological point of view, networks that guaranteeasymptotic convergence towards consensus. The set of input transitions for a place p , is denoted as I ( p ) = { t ∈ T : ( t, p ) ∈ E I } , and, similarly for a set of places S ⊂ P , its input transitions are: I ( S ) = { t ∈ T : ∃ p ∈ S : ( t, p ) ∈ E I } . Simmetrically, output transitions are denoted as: O ( S ) = { t ∈ T : ∃ p ∈ S : ( p, t ) ∈ E O } . Definition 3
A non-empty set of places S ⊂ P is called a siphon if I ( S ) ⊂ O ( S ) . A siphon is minimal if noproper subset is also a siphon. Informally, in a group of agents that correspond to a siphon, any influence needs to come (at least in part)from within the group. In [1], a characterization of the ability of agents to asymptotically converge towardsconsensus (regardless of their initial conditions) is provided. This feature, called structural consensuability , isshown to be equivalent to the requirement that any pair of siphons in the associated Petri Net have non-emptyintersection.To address robustness questions, within the same set-up of joint agent interactions, an externsion of theconcept of siphon is needed. The following is, to the best of our knowledge, an original definition:
Definition 4
A non-empty set of places S ⊂ P is an F -controlled siphon, if: I ( S ) ⊂ O ( S ) ∪ O ( F ) . F = ∅ . Union of F -controlled siphonsis again an F -controlled siphon and, in particular, if a set is an F -controlled siphon it is also an F -controlledsiphon for all F ⊇ F . We call the set F the switch of siphon S . Informally, this terminology is adopted asmalicious agents in F may prevent healthy agents in S from increasing (or decreasing) their own state values.This is indeed achievable by malicious agents simply broadcasting values which are either below the minimumor, respectively, above the maximum of all values within the siphon.The following notions are appropriate to characterize occurrence of robust consensus. Definition 5
We say that a Petri Net fulfills robust consensuability with respect to faults in F ⊂ N if H := N \ F is a siphon and for all pairs of controlled siphons S , S and associated switches F , F ⊂ F , we have thefollowing: S ∩ S = ∅ ⇒ F ∩ F (cid:54) = ∅ . (9)It is worth pointing out that robust consensuability, when F = ∅ , boils down to structural consensuability asdefined in [1]. Also, a direct comparison with previously existing conditions for consensuability in networkswhere all influences are single-agent influences is not possible, as the condition would never be fulfilled. It isin fact use of joint-agent interactions and cross validations that make adversary robust consensus achievable.On the other hand, we believe that our conditions boil down to those proposed in [9] when only interactionsobtained through a sorting and reducing function are allowed. In the following Section we state the main result and clarify the steps of its proof. To this end, in order toallow dynamical properties of a multiagent system to be derived on the basis of structural conditions fulfilledby the associated Petri Net, it is important to establish a closer link between the considered equations and theassociated Petri Net. For any transition t ∈ T , denote by: I ( t ) := { p ∈ P : ( p, t ) ∈ E O } and by j ( t ) the unique place such that ( t, j ( t )) belongs to E I . In particular, for a given Petri Net { P, T, E I , E O } we consider non-decreasing locally Lipschitz functions F i : R | I ( p i ) | → R , with F i (0 , , . . . ,
0) = 0, and such that F i is strictly increasing in each of its arguments in 0. These are employed to define networks of equations:˙ x i = F i ( f I ( t ) → i ( x ) , f I ( t ) → i ( x ) , . . . , f I ( t | O ( pi ) | ) → i ( x )) , O ( p i ) = { t , . . . t | O ( p i ) | } . (10)A typical example arises when F i ( f ) = (cid:80) k α k f k , for some choice of coefficients α k >
0. Equation (10) is,however, more general and allows non-additive agents’ infuences. As an example of a non-additive function F i ,one may consider for instance the map F i ( f ) = min k ∈{ ,..., | O ( p i ) |} f k + max k ∈{ ... | O ( p i ) |} f k .Composition of the above maps with monotonic increasing functions, such as saturations or (odd) powers arealso legitimate choices, i.e. F i ( f ) = (cid:80) k α k sat( f k ) or F i ( f ) = ( (cid:80) k α k f k ) .We are now ready to state our main result and, later, to discuss the technical steps of its derivation. Theorem 1
Consider a cooperative network of agents as in (3) and let N be the Petri Net associated to its setof minimal joint agent interactions. Consider a partition of N into two disjoint subgroups F, H ⊂ N , whichrepresent the Faulty and the Healthy agents (respectively), along with the projected dynamics, (5). Then, robustconsensus is achieved among the agents in H provided N fulfills robust consensuability with respect to faultyagents in F . It is worth pointing out that the result assumes faulty agents are following arbitrary continuous evolutions andthat these are reliably broadcast to all neighbouring agents. This hypothesis cannot model the situation inwhich malicious agents intentionally communicate different evolutions to different neighbors. Agents with thisability are usually referred to as Byzantine agents, and Byzantine consensus protocols exhibit robustness to suchkind of threats. Notice that the ability of malicious agents of differentiating the information sent to neighborsmay disrupt consensus even when robust consensuability is fulfilled. An example of this situation is later shownin Section 5.We start the technical discussion by generalizing Proposition 11 in [1].
Lemma 1
Let H be a siphon of N . Consider a network of equations (10) and let x H denote the state vectorof agents in H , along with the corresponding equations ˙ x H ( t ) = f H ( x H ( t ) , x F ( t )) (11)6 s introduced in (5). Then, for any c ∈ R , the sets: ¯ X c := { x H ∈ R | H | : x H ≤ c H } , X c := { x H ∈ R | H | : x H ≥ c H } are robustly forward invariant for any bounded input signal x F ( · ) . Proof
Let x F ( · ) take value in the compact set K and h ∈ H be any agent whose associated state value fulfills x h = c . To prove invariance of ¯ X c we need to show f h ( x H , x F ) ≤ x F ∈ K . This condition, in fact,amounts to f ( x H , x F ) ∈ T C x ( ¯ X c ) for all x H ∈ ∂ ¯ X c and all x F ∈ K . This, in turn implies forward invariance of¯ X c by Nagumo’s Theorem. Let O ( p h ) = { t , t , . . . , t | O ( p h ) | } . Since h is a siphon, for all t i in O ( p h ) there exists˜ h ∈ I ( t i ) ∩ H . Hence: f I ( t i ) → h ( x ) ≤ f I ( t i ) → h (¯ x H H , x F ) = 0 . By monotonicity of F h then:˙ x h = F h ( f I ( t ) → h ( x ) , f I ( t ) → h ( x ) , . . . , f I ( t | O ( ph ) | ) → h ( x )) ≤ F h (0 , , . . . ,
0) = 0 . This completes the proof of the Lemma.The rest of the Section is devoted to illustrate the main technical steps of the proof.
Proof
Let x F ( t ) be an arbitrary bounded, continuous signal. Assume, in particular, that x F ( t ) ∈ K for somecompact set K ⊂ R | F | . Pick any initial agent distribution x H (0) and define the evolution of healthy agentsin H according to the equation (5). In particular, we denote the solution x H ( t ) := ϕ H ( t, x H (0) , x F ( · )), for all t ≥
0. Moreover, we let: ¯ x H := max h ∈ H x h x H := min h ∈ H x h . Since H is a siphon, by Lemma 1, we see that for all t ≥ t ≥ x H ( t ) ∈ ¯ X ¯ x H ( t ) ⇒ x H ( t ) ∈ ¯ X ¯ x H ( t ) . In particular then, ¯ x H ( t ) ≥ ¯ x H ( t ), viz. ¯ x H is monotonically non-increasing. As expected, a symmetricargument shows that x H ( t ) is monotonically non-decreasing. Therefore x H ( t ) is uniformly bounded and thelimits ¯ x ∞ H := lim t → + ∞ ¯ x H ( t ) x ∞ H := lim t → + ∞ x H ( t ) , (12)exist finite. For future reference, it is convenient to define the convex-valued differential inclusion given below:˙ z ∈ F H ( z ) := co (cid:32) (cid:91) x F ∈ K { f H ( z, x F ) } (cid:33) . (13)Due to compactness of K , and Lipschitz continuity of f H , F H is a Lipschitz continuous set-valued map. Inparticular, x H ( t ), is also a (bounded) solution of (13). Consider next the associated ω -limit set, which, byboundedness of x H ( t ), is non-empty and compact:Ω H := (cid:26) x ∈ R | H | : ∃ { t n } + ∞ n =1 : lim n → + ∞ t n = + ∞ and x = lim n → + ∞ x H ( t n ) (cid:27) . (14)Notice that, by definition, for any z H ∈ Ω H we have ¯ z H = ¯ x ∞ H and z H = x ∞ H . As is well known, Ω H is a weaklyinvariant set for the differential inclusion (13). Selecting any element ˜ z H in Ω H , there exists at least one viablesolution ˜ z H ( t ) of (13), such that ˜ z H ( t ) ∈ Ω H , for all t . Notice that, by Lipschitzness of F H , the sets M ( t ) := { h ∈ H : ˜ z h ( t ) = ¯ x ∞ H } , and m ( t ) := { h ∈ H : ˜ z h ( t ) = x ∞ H } are monotonically non-increasing with respect to set-inclusion and, trivially, non-empty for all t ≥
0. Hence,there exists some finite τ ≥ M ( t ) = M ( τ ) and m ( t ) = m ( τ ) for all t ≥ τ . Moreover, for all suchvalues of t , we see that: ˙˜ z h ( t ) = 0 ∀ h ∈ M ( τ ) , (15)and similarly ˙˜ z h ( t ) = 0 ∀ h ∈ m ( τ ) . (16)7o prove asymptotic consensus, we need to show M ( τ ) ∩ m ( τ ) (cid:54) = ∅ . To this end, we claim that there exists F M ⊂ F such that M ( τ ) is an F M controlled siphon, as we argue next by contradiction.Should this not happen, at least some h would exist in M ( t ) and I ⊂ N such that I → h and still I ∩ ( M ( t ) ∪ F ) = ∅ . In particular then, ¯˜ z I ( t ) := max i ∈ I ˜ z i ( t ) < ¯ x ∞ H and this violates (15) by virtue of definition(6) as for all x F ∈ K : f h (˜ z H ( t ) , x F ) ≤ f h (¯ x ∞ H H + (¯˜ z I ( t ) − ¯ x ∞ H ) e I , x F ) ≤ − ρ (¯ x ∞ H − ¯˜ z I ( t )) < . A similar argument can be used to show that m ( t ) is an F m controlled siphon.Consider next any switch pairs F M , F m ⊂ F such that M ( τ ) and m ( τ ) are, respectively, an F M and F m controlled siphon. Assume, without loss of generality, F M and F m minimal with respect to set inclusion (amongsimilar siphons’ switches).In the following we argue by contradiction considering the case M ( τ ) ∩ m ( τ ) = ∅ . By structural consen-suability this implies F M ∩ F m (cid:54) = ∅ and we may pick ¯ f ∈ F M ∩ F m . By minimality of F M and F m , moreover,taking out ¯ f from them violates the definition of controlled siphon, viz. there exist h M ∈ M ( τ ) and h m ∈ m ( τ )(distinct from each other), such that for some joint interactions I M → h M and I m → h m we see that I M ∩ ( M ( t ) ∪ ( F M \{ ¯ f } )) = ∅ , (17)and, similarly, I m ∩ ( m ( t ) ∪ ( F m \{ ¯ f } )) = ∅ . (18)Since H is a siphon, however, f h M ( z, x F ) ≤ z ∈ Ω H and all x F ∈ K . Moreover condition (17) yields: x ¯ f < ¯ x ∞ H ⇒ f h M (˜ z H ( t ) , x F ) ≤ − ρ (¯ x ∞ H − max { ¯˜ z I M ( t ) , x ¯ f } ) < . Similarly, f h m ( z, x F ) ≥ z ∈ Ω H and all x F ∈ K . In addition, x ¯ f > x ∞ H ⇒ f h m (˜ z H ( t ) , x F ) ≥ ρ (min { ˜ z I m ( t ) , x ¯ f } − x ∞ H ) > . Notice that, whenever x ∞ H < ¯ x ∞ H , we have ( −∞ , ¯ x ∞ H ) ∪ ( x ∞ H , + ∞ ) = R and therefore, f h M (˜ z H ( t ) , x F ) − f h m (˜ z H ( t ) , x F ) ≤ − ρ (min (cid:8) ¯ x ∞ H − max { ¯˜ z I M ( t ) , ( x ∞ H + ¯ x ∞ H ) / } , min { ˜ z I m ( t ) , ( x ∞ H + ¯ x ∞ H ) / } − x ∞ H (cid:9) ) . As a consequence: ( e h M − e h m ) (cid:48) F H (˜ z H ( t ) , x F ) ≤ − ρ (min (cid:8) ¯ x ∞ H − max { ¯˜ z I M ( t ) , ( x ∞ H + ¯ x ∞ H ) / } , min { ˜ z I m ( t ) , ( x ∞ H + ¯ x ∞ H ) / } − x ∞ H (cid:9) ) < x F ∈ K . This, however, contradicts either (15) or (16).Notice that, in the proof of Theorem 1, it is crucial that the position communicated by malicious agents toall of its neighbors are consistent. If not, malicious agents could more easily prevent consensus by sendingdifferentiated signals to individual agents, and a correspondingly stronger notion of structural consensuabilitywould be needed. We consider next an example with 9 agents arranged in a 3 × { , , } := N . In particular then N = N × N . We consider the following interconnectiontopology. For all ( i, j ) ∈ N × N , we have two joint-agent interactions:( N \{ i } ) × { j } → ( i, j ) { i } × ( N \{ j } ) → ( i, j ) . The associated Petri Net is shown in Fig. 4. Notice that, by construction, whenever an agent belongs to asiphon, somebody from the same column and row also needs to be within the siphon. Let, for a set Σ ⊂ N × N ,Σ i denote the elements of Σ belonging to { i } × N and, by Σ j the elements of Σ in N × { j } . We see that Σ8 , , , , , , , , , Figure 4: Petri Net associated to joint-agent interactions(a) , , , , , , , , , (b) , , , , , , , , , Figure 5: Siphons of minimal support (gray)(non-empty) is a siphon if and only if | Σ i | ≥ i ∈ N such that | Σ i | > | Σ j | ≥ j ∈ N such that | Σ j | >
0. In particular, minimal siphons fulfill the equality rather than the strict inequality andare essentially of two kinds, as shown in Fig. 5. It is straightforward to see that, despite having siphons ofcardinality strictly smaller than half of the size of the group (4 < / , N × N \{ (2 , } is a siphon. Moreover, any siphon of the full Petri Netthat does not contain (2 ,
2) is also a ∅ -controlled siphon when F = { (2 , } . Next, we look for (2 , ,
2) acts as a switch. It can be seen that Σ is a (2 ,
2) controlledsiphon if (and only if) Σ ∪ { (2 , } is a siphon. This direct implication is true for all Petri Nets, but the converseneed not hold in general. In particular, then, only two types of controlled siphons can be identified (up topermutations), as shown in Fig. 6.Because of this, for any pair of ∅ or (2 ,
2) controlled siphons Σ , Σ , with associated switches F , F it istrue that (Σ ∪ F ) ∩ (Σ ∪ F ) (cid:54) = ∅ . , , , , , , , , , (b) , , , , , , , , , Figure 6: (2 , , , , , , , , , , (b) , , , , , , , , , Figure 7: { (2 , , (3 , } -controlled siphons of minimal support (gray)And, since by definition Σ i ∩ F i = ∅ , then the following is true:Σ ∩ Σ = ∅ ⇒ F ∩ F (cid:54) = ∅ . Hence, robust structural consensuability is fulfilled and one may expect the 8 healty agents to reach asymp-totic consensus despite the exogenous disturbance coming from agent (2 , , , F = { (2 , , (3 , } . As we already characterized s ∅ controlled siphons and siphons controlled byswitch of cardinality one, we need only look for F -controlled siphons. Any set Σ such Σ ∪ F is a siphon is alsoan F -controlled siphon. In addition, the network exhibits two types of F -controlled siphons that do not fulfillsuch condition. These are shown in Fig. 7.Notice that, { (2 , } is an F -controlled siphon. On the other hand, the set { (1 , , (1 , , (3 , , (3 , } is an ∅ -controlled siphon. These two controlled siphons and their associated switches have both empty intersection.Hence, robust structural consensuability does not hold for this choice of faulty agents. Indeed agents in F have the ability to prevent consensus between the agents in the siphons described above. For instance, onemay take initial conditions x , (0) = x , (0) = x , (0) = x , (0) = 1. For a ∅ -controlled siphon this results insolutions fulfilling x , ( t ) = x , ( t ) = x , ( t ) = x , ( t ) = 1 for all t . At the same time, one may let the maliciousagents fulfill x , ( t ) = x , ( t ) = 0 so that, any solution with x , (0) = 0 would result in x , ( t ) = 0 identically,thus preventing consensus. A similar issue arises, letting agents x , , x , , x , , x , be initialized with negativevalues, while x , and x , oscillate at some higher values as shown in Fig. 8.Due to simmetry of the considered network, it follows that selection of any two faulty agents result in theconditions for robust consensuability to be violated.We emphasize that the considered network is not robust with respect to Byzantine malicious agents. Inparticular, in the simulation we show the result of agent (2 ,
2) broadcasting higher values to agents (1 , , , , x , (0) = 0 and does not10igure 8: Malicious agents (2 ,
2) and (3 ,
3) disrupt consensus.Figure 9: Byzantine agent (2 ,
2) disrupts consensus.change his position, while it sends the value +2 and − Adversarily Robust Consensus was first introduced by LeBlanc and coworkers in [7]. This is proposed, initially,for all-to-all networks and later extended in [9] to networks with more general topologies. We start this Sectionby highlighting how the all-to-all topology considered in [7] can be seen as a specific type of symmetric joint-agent interaction. Similar considerations apply when the set of neighbours of each agent is a proper subset of { , , . . . , n } but, for the sake of simplicity, this is not illustrated in detail. Let ¯ σ k ( x ) denote the k -th largestentry in x and, similarly, σ k ( x ) the k -th smallest entry in x . We see that,¯ σ k ( x ) = max J ⊂N : | J | = k min k ∈ J x k σ k ( x ) = min J ⊂N : | J | = k max k ∈ J x k . σ k ( x ) = σ n +1 − k ( x ), and therefore for any integer F with n − F ≥ F + 1 we see n − F (cid:88) k = F +1 ¯ σ k ( x ) = n − F (cid:88) k = F +1 σ k ( x ) . Consider next the protocol described by the following set of equations:˙ x i = − x i + (cid:80) n − Fk = F +1 ¯ σ k ( x ) n − F . (19)This is, essentially, a continuous-time version of the algorithm proposed in [7], where each agent is directedtowards the average of the n − F agents’ opinions of intermediate value (as achieved in [7] by using the sortingand reducing maps). It is easy to see that (19) is a monotone cooperative network, moreover, we claim that forall J of cardinality F + 1 and any agent i it holds J → i . To this end, let J be a subset of cardinality F + 1 andlet x J > x i be the common value associated with agents in J . All agents not in J , including agent i have value x i , instead. Clearly ¯ σ k ( x ) = x J for all k = 1 . . . F + 1 and ¯ σ k ( x ) = x i for all k = F + 2 . . . n . In particular, then:˙ x i = − x i + x J + ( n − F − x i n − F = x J − x i n − F , which proves a joint influence of agents in J towards i from above. Similar results hold when x J < x i . Moreover, J → i is a minimal influence. In fact, any proper subset of J consists of at most F elements and therefore,assuming their value is x J while x i is the value of other agents, a simple computation shows that˙ x i = − x i + ( n − F ) x i n − F = 0 . thus ruling out the possibility of joint-influences for sets of agents of cardinality F or lower. Similar computationscan be carried out when the network is not of all-to-all type, and each agent has a specific set of neighboursthat get sorted, reduced and averaged upon. This paper has explored tight necessary and sufficient conditions for continuous-time Adversary ConsensusProtocols of networks with joint-agent interactions of arbitrary topology. This captures, as a particular case,the notion of ARC consensus studied in the discrete-time case by Leblanc and co-workers, using sorting and selection maps. Consensus is achieved in the face of agents that behave as arbitrary bounded disturbances,and are only constrained to broadcasting the same information to all of the neighbours they have an influenceupon. In this respect, the problem of
Byzantine consensus, where agents may maliciously or unintenionally senddifferent information to distinct neighbours is an interesting open question for further research. Conditions areformulated in the language of Petri Nets, in particular making use of the notion of controlled siphon , in whichfaulty agents play the role of a ‘switch’ capable of disabling some influences by suitably positioning itself aboveor below the value of agents within the same joint-agent interaction. An example is presented to illustrate theapplicability of the considered results. This does not fall within the class of networks considered in [9] sinceeach agent is only has two distint group of neighbours (vertical and horizontal ones in the picture) which aretreated separately when cross-validating information in joint-agent interactions). In particular, the equationsconsidered can never be achieved by means of sorting and selection functions.
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