Dynamic Max-Consensus and Size Estimation of Anonymous Multi-Agent Networks
11 Dynamic Max-Consensus and Size Estimation ofAnonymous Multi-Agent Networks
Diego Deplano, Mauro Franceschelli, Alessandro Giua
Abstract —In this paper we propose a novel consensus protocolfor discrete-time multi-agent systems (MAS), which solves thedynamic consensus problem on the max value, i.e., the dynamicmax-consensus problem. In the dynamic max-consensus problemto each agent is fed a an exogenous reference signal, the objectiveof each agent is to estimate the instantaneous and time-varyingvalue of the maximum among the signals fed to the network,by exploiting only local and anonymous interactions among theagents. The absolute and relative tracking error of the proposeddistributed control protocol is theoretically characterized and isshown to be bounded and by tuning its parameters it is possibleto trade-off convergence time for steady-state error. The dynamicMax-consensus algorithm is then applied to solve the distributedsize estimation problem in a dynamic setting where the size of thenetwork is time-varying during the execution of the estimationalgorithm. Numerical simulations are provided to corroboratethe theoretical analysis.
Index Terms —Multi-agent systems, dynamic consensus, dis-tributed estimation, network size estimation, anonymous net-works.
I. I
NTRODUCTION
Recent years have seen considerable interest in the designof algorithms to solve the problem of consensus agreementover networked systems. The problem is considered to besolved when the agents agree upon a value while usingonly local information about neighboring agents. Most of theliterature usually considers static agreement values, i.e., agentsare required to converge to a static reference signal. On thecontrary, in the dynamic consensus problem a time-varyingreference signal is associated to each agent, and they arerequired to achieve consensus upon a function of the time-varying reference signals, such as average, median, maximumand so on.While the literature has focused significantly on the dynamicaverage-consensus problem [1], [2], [3], [4], [5], [6], [7], [8],[9], estimating the average is not the only attractive goal. Inparticular, the focus of this work is the the development ofdynamic protocols achieving consensus upon on the max valueamong the reference signals. Applications of dynamic max-consensus protocols mainly reside in the field of distributedsynchronization, such as time-synchronization [10] and target
D. Deplano, M. Franceschelli and A. Giua are withDIEE, University of Cagliari, 09123 Cagliari, Italy. Emails: { diego.deplano,mauro.franceschelli,giua } @unica.it Mauro Franceschelli is corresponding author. This work was supported inpart by the Italian Ministry of Research and Education (MIUR) with thegrant ”CoNetDomeSys”, code RBSI14OF6H, under call SIR 2014 and byRegion Sardinia (RAS) with project MOSIMA, RASSR05871, FSC 2014-2020, Annualit 2017, Area Tematica 3, Linea d’Azione 3.1. tracking [11], and network parameter estimation, such ascardinality [12] and highest/lowest node degree [13].
Literature review.
In the literature the so-called max-consensus problem has been thoroughly investigated. Its ob-jective is to make the states of the agents converge to the max-imum of their initial states. The most popular max-consensusprotocol consists in initializing the network to a set of valuesand let agent update its state at each instant of time by takingthe maximum value among the value of the neighbors’ stateand its own state [14]. The work in [15] proposes conditions toachieve max-consensus and compute convergence rate of theseprotocols for different communication topologies. Only littleeffort has been paid to analyze slightly different but muchmore complicated variations of this problem. In particular,convergence results have only been provided for synchronousswitching topologies [16] and for probabilistic asynchronousfixed frameworks [17]. The contribution of introducing timedelays in the communications is due to [18], while [19] isthe first work allowing noise in the communications. Finally,the case with agents with the possibility to join or leavethe network, so-called open multi-agent systems, is addressedin [20]. If a static consensus protocol is used to performa distributed estimation upon some time-varying quantities,known or measured by the agents, the protocol requires tobe re-initialized in the whole network each time the valueof the function to be estimated changes. Dynamic consensusprotocols has been introduced in order to overcome the issueof re-initializing the network.
Main contribution.
Up to the authors’ knowledge, thereno exists dynamic max-consensus protocol in the literatureand, moreover, the existing results to solve the max-consensusproblem only apply to static networks. Therefore, in this workwe propose two protocols capable of solving the dynamicmax-consensus problem in time-varying networks which areconnected at any time. Furthermore, variants of these protocolsare proposed to solve the dual problem of estimating the minvalue. II. O
PEN D YNAMICAL S YSTEMS
We consider Multi-Agent Systems (MAS) with a time-varying number of agents. These MAS are also called
Open
MAS (OMAS). At any time k ∈ N , let G k = ( V k , E k ) be theundirected graph describing the pattern of interactions amongagents, where V k ⊂ N is the set of active nodes at time k and E k ⊆ ( V k × V k ) is the set of active communication channelsbetween agents in the network. The number n k of activeagents in the network at time k is given by the cardinality a r X i v : . [ m a t h . O C ] S e p of the set V k , i.e., n k = | V k | . Agents p and q are said tobe neighbors at time k if there exists an edge from p to q (and vice versa) at time k , i.e., ( p, q ) , ( p, q ) ∈ E k . A set ofneighbors N pk is associated to each node p at time k , defined as N pk = { q ∈ V k : ( p, q ) ∈ E k } , which represents the agents inthe graph which share a point-to-point communication channelwith agent p .A path π pqk between two nodes p and q in a graph is afinite sequence of m edges e (cid:96) = ( i (cid:96) , j (cid:96) ) ∈ E k that joins node p to node q , i.e., i = p , j m = q and j (cid:96) = i (cid:96) +1 for (cid:96) =1 , . . . , m − . An undirected graph is said to be connected attime k if there exists a path π pqk between any pair of nodes p, q ∈ V k . The diameter of a graph, denoted as δ ( G k ) , isdefined as the longest among the shortest paths among anypair of nodes p, q ∈ V k . For any connected undirected graphit holds δ ( G k ) ≤ n k − .For each time k and agent p ∈ N k , the scalar agent’s state isdenoted x pk ∈ R and its input is denoted u pk , which are definedonly at time instants such that p ∈ V k . More generally, in thispaper we shall call open sequence any sequence { y k : k ∈ N } ,or simply { y k } , where y k ∈ R V k . When the number of nodes n k changes with the time k , it is not possible in general towrite x k +1 as a function solely of x k : therefore, the evolutionof x k does not constitute a ”closed” dynamical system. Thus,one needs to first partition the nodes into three sets: • The departing nodes D k = V k \ V k +1 ; • The arriving nodes A k = V k +1 \ V k ; • The remaining nodes R k = V k ∩ V k +1 .With this notation, we shall define the evolution of each agentdepending on its status, x pk +1 = ∅ if p ∈ D k f a ( x k +1 , V k +1 , E k +1 , u k +1 ) if p ∈ A k f r ( x k , V k , E k , u k ) if p ∈ R k . (1)In other words, since x k +1 must take values in R V k +1 for all k ∈ N , the departing nodes have to be left out, the arrivingnodes need to be initialized when joining to the networkaccording to some rule f a ( · ) and the remaining nodes updatetheir current state according to some rule f r ( · ) . We observethat if the set of agents does not change at time k + 1 , i.e., V k = V k +1 , then we can write in vector form the dynamicsof the corresponding MAS as x k +1 = f ( x k , V k , E k , u k ) . (2)III. P ROBLEM STATEMENT
The dynamic consensus problem in a MAS as in (2) consistsin steering the agents’ states to track a function g ( · ) : R n → R of the time-varying exogenous reference signals u ik such thatthe tracking error e k = max i ∈ V k | x ik − g ( u k ) | (3)converges within a boundary layer for any initial condition.The convergence time T c denotes the time required by theMAS to achieve such a bounded tracking error.If the MAS is open, i.e., agents may join or leave thenetwork, and the natural way to generalize the dynamic consensus problem to OMAS is requiring that at every changeof the network the tracking error (3) decrease before anotherchange, eventually converging within a boundary layer. Sincethe change in the network is a local event, it is not possible tore-initialize the whole network in a distributed way, i.e., theprotocol must be robust to the initial condition. Furthermore,the change may results in a unpredictable discontinuity of thequantity of interests, thus requiring the whole network to agreeupon a value which is uncorrelated to the one at the previousstep. In the light of the above considerations, we have to makesome assumptions on the rate of change of the network andthe inputs.We consider a network of agents whose topology at time k ∈ N is represented by an undirected connected graph G k = ( V k , E k ) satisfying the next assumption. Assumption 1:
There exists an interval time Υ ∈ N suchthat two changes of the network G can not occur in Υ unitsof time, i.e., ∀ k ∈ N : G k − (cid:54) = G k ∗ ⇒ G k = G k +1 , ∀ k ∈ { k , k + Υ } . (cid:4) Each agent i has access to a time-varying external referencesignal u ik ∈ R satisfying the next assumption. Assumption 2:
Each unknown exogenous reference signalis such that its has absolute change is bounded by a constant Π ∈ R + , i.e., | u ik +1 − u ik | ≤ Π , ∀ i ∈ V k , ∀ k ∈ N . (4) (cid:4) Note that, due to Assumptions 1-2, within two changes, theOMAS can be regarded as a MAS with dynamics given in (2),but the inputs keep varying during such an interval. In such aninterval one can define the transient time and the convergencetime as follows.
Definition 1:
The transient time T tk ∈ N is the time thatthe network needs to remain unchanged in order achieve adecreasing tracking error (3) from an initial time k ∈ N . Definition 2:
The convergence time T ck ∈ N is the timethat the network needs to remain unchanged in order achievea bounded tracking error (3) from an initial time k ∈ N . (cid:4) In particular, in this work we focus on the so-called dynamicmin/max-consensus problem, i.e., the quantity of interest iseither the minimum u k or the maximum ¯ u k of the theexogenous reference signals, defined as follows u k = min i ∈ V k u ik , ¯ u k = max i ∈ V k u ik . (5) Problem 1:
Consider an OMAS as in (1) satisfying As-sumptions 1-2 and let the quantity of interest g ( · ) be eitherthe maximum ¯ u or the minimum u as in (5).The dynamic min/max-consensus problem consists in de-signing the local interaction rules f a ( · ) , f r ( · ) such that thetracking error (3) satisfies e k +1 < e k for k ∈ [ k + T tk , k + min { T ck , Υ } ] , (6) e k +1 ≤ ε for k ∈ [ k + min { T ck , Υ } , k + Υ] , (7)for any k ∈ N such that G k − (cid:54) = G k and where Protocol 1:
Approximate Dynamic Max-Consensus (ADMC)
Input :
Tuning parameter α ∈ (0 , . Output:
Current state x ik ∈ R for i ∈ V k . for k = 0 , , , . . . each node i doesif i ∈ A k or k = 0 then x ik +1 ← u ik +1 else if i ∈ R k then Gather x jk from each neighbor j ∈ N ik Update the current state according to x ik +1 ← max j ∈N ik (cid:83) { i } (cid:110) x jk − α, u ik (cid:111) • T tk ∈ [0 , min { T ck , Υ } ) is the transient time; • T ck ∈ N is the transient time; • ε ∈ R + is the bound on the tracking error. (cid:4) Problem 1 requires two ingredients for declaring the dynamicmin/max-consensus problem solved, which are the following: • Condition (6): The tracking error must decrease betweentwo changes in the network after a transient time. • Condition (7): The tracking error is bounded if the rateof change of the network is large enough, i.e., Υ ≥ T ck .Objective of this paper is thus to propose local interac-tion protocols (1) for a discrete-time OMAS, which solvethe dynamic consensus problem formalized in Problem 1for anonymous networks, i.e., the agents do not share theiridentification. The proposed protocols are then applied to solvethe distributed online estimation problem of size the time-varying network to which the agents belong.IV. D YNAMIC M AX -C ONSENSUS P ROTOCOLS
In this section we provide two protocols to solve thedynamic max-consenus problem:1)
Approximate Dynamic Max-Consensus (ADMC) Proto-col: it enables the agents to converge to an approximate consensus on the max value without requiring any infor-mation about the network graph.2)
Exact Dynamic Max-Consensus (EDMC) Protocol: itenables the agents to reach an exact consensus on themax value by requiring the knowledge of an upperboundof the network’s graph diameter.For each protocol we prove that Problem 1 is solved for g ( u k ) = ¯ u k as in (5), characterizing the convergence time T ck as in Definition 2 and the bound on the tracking error e ( k ) = max i ∈ V | x i ( k ) − ¯ u ( k ) | . (8) A. Approximate Consensus
In Protocol 1 we detailed the ADMC Protocol, which makesuse of the following local interaction rule: x ik +1 = max j ∈N ik (cid:83) { i } (cid:110) x jk − α, u ik (cid:111) if i ∈ R k u ik +1 if i ∈ A k (9) where α ∈ (0 , is a scalar tuning parameter. At eachiteration all remaining agents gather the state values of theirneighbors and update their state according to (22), whichonly requires local communications, while all arriving agentsinitialize their next state at their own inputs. Note that at theinitial time k = 0 all agents are arriving agents. Theorem 1 (ADMC Protocol: Tracking Error):
Consideran OMAS executing Protocol 1 with tuning parameter α ∈ R + under Assumptions 1-2. Consider a generic time k ∈ N at which the network changes, i.e., G k − (cid:54) = G k .If G k is connected with diameter δ ( G k ) and if α > Π , Υ ≥ δ ( G k ) (10)Problem 1 is solved with transient and convergence times T tk = δ ( G k ) , (11) T ck = max (cid:26) T tk , (cid:24) max { x k − ¯ u k } α − Π (cid:25)(cid:27) , with ¯ u k defined in (5). Moreover, if Υ ≥ T ck , the trackingerror (8) is bounded for any k ∈ [ k + T ck , k + Υ] by e k ≤ ε = ( δ ( G k ) + 1)Π + αδ ( G k ) . (12) Proof:
The proof is given in Appendix.
Corollary 1 (ADMC Protocol: Steady State Error):
Consider an OMAS executing Protocol 1 with tuningparameter α ∈ R + under Assumption 1. Consider a generictime k ∈ N at which the network changes, i.e., G k − (cid:54) = G k and let the inputs be constant for k ∈ [ k , k + Υ] .If G k is connected with diameter δ ( G k ) and if α > Π , Υ ≥ δ ( G k ) Problem 1 is solved with transient and convergence times T tk = δ ( G k ) , (13) T ck = max (cid:26) T tk , (cid:24) max { x k − ¯ u k } α (cid:25)(cid:27) , with ¯ u ( k ) defined in (5). Moreover, if Υ ≥ T ck , the trackingerror (8) is fixed for any k ∈ [ k + T ck , k + Υ] e k ≤ ε ss = αδ ( G k )) . (14) Proof:
Since the inputs are constant for k ∈ [ k , k + Υ] ,Assumption 2 is satisfied with Π = 0 . Therefore, we can applyTheorem 1 and specialize transient and convergence timesgiven in (13) along with the tracking error given in (12) for
Π = 0 , completing the proof of the corollary. Furthermore thebound is now a strict condition. In fact, since the inputs areconstant, the network reaches an equilibrium and so the steadystate error does not change over time.From the result of Theorem 1 it follows that, accordingto (12), to minimize the absolute estimation error we need tochoose α ≈ , α > Π ≥ . On the other hand, α determinesthe convergence time T according to (13), with smaller valuesof α giving a greater convergence time. Thus, the value of α trades-off estimation error and convergence time.It follows that a pragmatic design criterion for the choice of α is to first fix the desired steady-state error and then choosethe largest α which allows to satisfy the error performanceconstraint while minimizing the convergence time. Protocol 2:
Exact Dynamic Max-Consensus (EDMC)
Input :
Network’s diameter upperbound ∆ ∈ N . Output:
Current state x i ∆ ( k ) ∈ R for i ∈ V . for k = 0 , , , . . . each node i doesif i ∈ A k or k = 0 thenfor (cid:96) = 0 , , . . . , ∆ do x i(cid:96)k +1 ← u i(cid:96)k +1 else if i ∈ R k then Gather [ x j k , . . . , x j ∆ k ] from each neighbor j ∈ N i Update the current states according to x i k +1 ← u ik for (cid:96) = 1 , . . . , ∆ do x i(cid:96)k +1 ← max j ∈N ik (cid:83) { i } x j ( (cid:96) − k B. Exact Consensus
In Protocol 2 we detailed the EDMC Protocol, which makesuse of the following local interaction rule x i k +1 = (cid:40) u ik if i ∈ R k u ik +1 if i ∈ A k (15) x i(cid:96)k +1 = max j ∈N ik (cid:83) { i } x j ( (cid:96) − k if i ∈ R k u ik +1 if i ∈ A k with (cid:96) = 1 , . . . , ∆ and ∆ ∈ N is an upperbound on the thediameter of the underlying comunication network, i.e., δ ≥ δ ( G k ) at any time k ∈ N . At each iteration all agents gather thestate values of their neighbors and update their state accordingto (28), which only requires local communications, while allarriving agents initialize their next state at their own inputs.Note that at the initial time k = 0 all agents are arrivingagents. Theorem 2 (EDMC Protocol: Tracking Error):
Consideran OMAS executing Protocol 2 under Assumptions 1-2.Consider a generic time k ∈ N at which the networkchanges, i.e., G k − (cid:54) = G k .If graph G k is connected with diameter δ ( G k ) and if Υ ≥ ∆ ≥ δ ( G k ) , (16)Problem 1 is solved with transient and convergence times T ck = T tk = ∆ . (17)Moreover, if Υ ≥ T ck , the tracking error (8) is bounded forany k ∈ [ k + T ck , k + Υ] by e k ≤ ε = (∆ + 1)Π . (18) Proof:
The proof is given in Appendix.
Corollary 2 (EDMC Protocol: Steady State Error):
Consider an OMAS executing Protocol 2 under Assumption 1.Consider a generic time k ∈ N at which the networkchanges, i.e., G k − (cid:54) = G k and let the inputs be constant for k ∈ [ k , k + Υ] . Protocol 3:
Approximate Dynamic Min-Consensus (ADmC)
Input :
Tuning parameter α ∈ (0 , . Output:
Current state x ik ∈ R for i ∈ V k . for k = 0 , , , . . . each node i doesif i ∈ A k or k = 0 then x ik +1 ← u ik +1 else if i ∈ R k then Gather x jk from each neighbor j ∈ N ik Update the current state according to x ik +1 ← min j ∈N ik (cid:83) { i } (cid:110) x jk + α, u ik (cid:111) If graph G k is connected with diameter δ ( G k ) and if Υ ≥ ∆ ≥ δ ( G k ) , Problem 1 is solved with transient and convergence times T ck = T tk = ∆ . (19)Moreover, if Υ ≥ T ck , the tracking error (8) is bounded forany k ∈ [ k + T ck , k + Υ] by e k = ε ss = 0 . (20) Proof:
The proof is similar to the one of Corollary 1.V. D
YNAMIC M IN -C ONSENSUS P ROTOCOLS
In this section we provide two protocols to solve thedynamic min-consenus problem:1)
Approximate Dynamic Min-Consensus (ADmC) Protocol:it enables the agents to converge to an approximate consensus on the min value without requiring any infor-mation about the network graph.2)
Exact Dynamic Min-Consensus (EDmC) Protocol: it en-ables the agents to reach an exact consensus on the minvalue by requiring the knowledge of an upperbound ofthe network’s graph diameter.For each protocol we prove that Problem 1 is solved for g ( u k ) = u k as in (5), characterizing the convergence time T ck as in Definition 2 and the bound on the tracking error e ( k ) = max i ∈ V | x i ( k ) − u ( k ) | . (21) A. Approximate Consensus
In Protocol 3 we detailed the ADmC Protocol, which makesuse of the following local interaction rule: x ik +1 = min j ∈N ik (cid:83) { i } (cid:110) x jk + α, u ik (cid:111) if i ∈ R k u ik +1 if i ∈ A k (22)where α ∈ (0 , is a scalar tuning parameter. At eachiteration all remaining agents gather the state values of theirneighbors and update their state according to (22), whichonly requires local communications, while all arriving agents initialize their next state at their own inputs. Note that at theinitial time k = 0 all agents are arriving agents. Theorem 3 (ADMC Protocol: Tracking Error):
Consideran OMAS executing Protocol 3 with tuning parameter α ∈ R + under Assumptions 1-2. Consider a generic time k ∈ N at which the network changes, i.e., G k − (cid:54) = G k .If G k is connected with diameter δ ( G k ) and if α > Π , Υ ≥ δ ( G k ) (23)Problem 1 is solved with transient and convergence times T tk = δ ( G k ) , (24) T ck = max (cid:40) T tk , (cid:38) max (cid:8) u k − x k (cid:9) α − Π (cid:39)(cid:41) , with ¯ u k defined in (5). Moreover, if Υ ≥ T ck , the trackingerror (8) is bounded for any k ∈ [ k + T ck , k + Υ] by e k ≤ ε = ( δ ( G k ) + 1)Π + αδ ( G k ) . (25) Proof:
The proof is given in Appendix.
Corollary 3 (ADMC Protocol: Steady State Error):
Consider an OMAS executing Protocol 3 with tuningparameter α ∈ R + under Assumption 1. Consider a generictime k ∈ N at which the network changes, i.e., G k − (cid:54) = G k and let the inputs be constant for k ∈ [ k , k + Υ] .If G k is connected with diameter δ ( G k ) and if α > Π , Υ ≥ δ ( G k ) Problem 1 is solved with transient and convergence times T tk = δ ( G k ) , (26) T ck = max (cid:40) T tk , (cid:38) max (cid:8) u k − x k (cid:9) α (cid:39)(cid:41) , with ¯ u ( k ) defined in (5). Moreover, if Υ ≥ T ck , the trackingerror (8) is bounded for any k ∈ [ k + T ck , k + Υ] by e k ≤ ε ss = αδ ( G k )) . (27) Proof:
Since the inputs are constant for k ∈ [ k , k + Υ] ,Assumption 2 is satisfied with Π = 0 . Therefore, we can applyTheorem 3 and specialize transient and convergence timesgiven in (26) along with the tracking error given in (25) for
Π = 0 , completing the proof of the corollary.
B. Exact Consensus
In Protocol 4 we detailed the EDmC Protocol, which makesuse of the following local interaction rule x i k +1 = (cid:40) u ik if i ∈ R k u ik +1 if i ∈ A k (28) x i(cid:96)k +1 = min j ∈N ik (cid:83) { i } x j ( (cid:96) − k if i ∈ R k u ik +1 if i ∈ A k with (cid:96) = 1 , . . . , ∆ and ∆ ∈ N is an upperbound on the thediameter of the underlying comunication network, i.e., δ ≥ δ ( G k ) at any time k ∈ N . At each iteration all agents gather the Protocol 4:
Exact Dynamic Min-Consensus (EDmC)
Input :
Network’s diameter upperbound ∆ ∈ N . Output:
Current state x i ∆ ( k ) ∈ R for i ∈ V . for k = 0 , , , . . . each node i doesif i ∈ A k or k = 0 thenfor (cid:96) = 0 , , . . . , ∆ do x i(cid:96)k +1 ← u i(cid:96)k +1 else if i ∈ R k then Gather [ x j k , . . . , x j ∆ k ] from each neighbor j ∈ N i Update the current states according to x i k +1 ← u ik for (cid:96) = 1 , . . . , ∆ do x i(cid:96)k +1 ← min j ∈N ik (cid:83) { i } x j ( (cid:96) − k state values of their neighbors and update their state accordingto (28), which only requires local communications, while allarriving agents initialize their next state at their own inputs.Note that at the initial time k = 0 all agents are arrivingagents. Theorem 4 (EDmC Protocol: Tracking Error):
Consideran OMAS executing Protocol 4 under Assumptions 1-2.Consider a generic time k ∈ N at which the networkchanges, i.e., G k − (cid:54) = G k .If graph G k is connected with diameter δ ( G k ) and if Υ ≥ ∆ ≥ δ ( G k ) , (29)Problem 1 is solved with transient and convergence times T ck = T tk = ∆ . (30)Moreover, if Υ ≥ T ck , the tracking error (8) is bounded forany k ∈ [ k + T ck , k + Υ] by e k ≤ ε = (∆ + 1)Π . (31) Proof:
The proof is in Appendix.
Corollary 4 (EDmC Protocol: Steady State Error):
Consider an OMAS executing Protocol 4 under Assumption 1.Consider a generic time k ∈ N at which the networkchanges, i.e., G k − (cid:54) = G k and let the inputs be constant for k ∈ [ k , k + Υ] .If graph G k is connected with diameter δ ( G k ) and if Υ ≥ ∆ ≥ δ ( G k ) , Problem 1 is solved with transient and convergence times T ck = T tk = ∆ . (32)Moreover, if Υ ≥ T ck , the tracking error (8) is bounded forany k ∈ [ k + T ck , k + Υ] by e k = ε ss = 0 . (33) Proof:
The proof is similar to the one of Corollary 3.
Transient Time Convergence Time Tracking Error Steady State Error Conditions T tk T ck ε ε ss ADMC δ ( G k ) max (cid:26) T tk , (cid:24) max { x k − ¯ u k } α − Π (cid:25)(cid:27) ( δ ( G k ) + 1)Π + αδ ( G k ) αδ ( G k ) α > Π & Υ ≥ δ ( G k ) ADmC δ ( G k ) max (cid:40) T tk , (cid:38) max (cid:110) u k − x k (cid:111) α − Π (cid:39)(cid:41) ( δ ( G k ) + 1)Π + αδ ( G k ) αδ ( G k ) α > Π & Υ ≥ δ ( G k ) EDMC ∆ ∆ (∆ + 1)Π 0 Υ ≥ ∆ ≥ δ ( G k ) EDmC ∆ ∆ (∆ + 1)Π 0 Υ ≥ ∆ ≥ δ ( G k ) Table IT
ABELLA RIASSUNTIVA DEI RISULTATI
VI. D
YNAMIC N ETWORK ’ S S IZE E STIMATION
In this section we introduce two interesting problems inwhich the dynamic max-consensus protocols can be applied.The first problem is the one of size-estimation of anonymousnetworks in which nodes can leave or join the network, thusleading to a time-varying quantity (the network’s size) to beestimated. The second problem is the distributed computationof the Fiedler vector, which is the eigenvector correspondingto the smallest non-trivial eigenvalue of the graph’s Laplacianmatrix.
A. Size-Estimation of Anonymous Network
Here we extend the strategy proposed in [21] for staticnetworks to time-varying networks, i.e., nodes can leave orjoin the network at any time. The approach in [21] is totallydistributed and based on statistical inference concepts and canbe briefly summarized as follows:1) Nodes independently generate a vector of M independentrandom numbers from a known distribution;2) Nodes distributedly compute a specific function f of allthese numbers through a consensus algorithm;3) Each node infers the network size exploiting the statisticalproperties of the so computed quantities.In particular, we consider the max-consensus scenario, i.e.,when the function f of the randomly generated numbers tobe estimated is the maximum of them. Differently from [22],in our case nodes are allowed to leave and join the network.Thus, we extend the above scheme by adding the followingrules: • ADMC and EDMC Protocols are used as consensusalgorithm in step ; • When a node join the network, generate a new randomnumber.If the network is static (no nodes leave or join the network) theproblem is the one considered with all reference signals stay-ing constant. Since our algorithm is robust to re-initialization,every time a node leaves or joins the network, the algorithmis able to converge to the new set of inputs. Intuitively, therate at which nodes leave or join the network is correlated tothe rate of change of the maximum value of the numbers andthus one may possibly have some critical scenarios. Here, wejust make the simple assumptions that our protocols can run asufficiently high number of iterations such that an equilibriumis reached after each change of the network. We formalize thisconcept in the following assumption.
Protocol 5:
Dynamic Size-Estimation (DSE)
Input :
Number of random numbers p ∈ N . Output: ˆ n ik ← − p (cid:80) pj =1 log( x ijk ) for i ∈ V k . for k = 0 , , , . . . each node i doesif i ∈ A k or k = 0 then u ik +1 ← rand ([0 , p ) for j = 1 , . . . , p do Update state x ijk +1 according to either Protocol 1or Protocol 2 with inputs [ u jk , . . . , u njk ] Assumption 3:
The minimum time Υ between two changesof the network ensured by Assumption 1 is greater or equalthan the convergence time T ck of the employed protocol, i.e., Υ ≥ T ck Under Assumption 3, we are able to estimate and track thetime varying size of the network without any synchronizationamong the agents, since no re-initialization is required byADMC and EDMC Protocols. Our strategy is formalized inProtocol 5, where the following notation is used (we omit herethe time-dependency ( k ) ): • Each agent i generates p random numbers u i ∈ [0 , p ; • x ijk denotes the i -th agent’s estimation of the maximumamong all u (cid:96)jk for (cid:96) ∈ V at time k ; • ˆ n ik is the i -th agent estimation of the size of the networkbased on estimations x i(cid:96)k for (cid:96) ∈ P at time k . Theorem 5:
Consider an OMAS executing Protocol 5 withparameter p ∈ N , p > , under Assumptions 1-2-3. Considera generic time k ∈ N at which the network changes, i.e., G k − (cid:54) = G k .If it is employed Protocol 1 under the conditions of Corol-lary 1, then the expected value E [ˆ n ik ] at a steady state, i.e., k ∈ [ k + T ck , k + Υ] is E (cid:2) ˆ n ik (cid:3) = ε p − e εnp ( np ) p Γ(1 − p, εnp ) , (34)with ε = δ ( G k ) α .If it is employed Protocol 2 under the conditions of Corol-lary 2, then the expected value E [ˆ n ik ] at a steady state, i.e., k ∈ [ k + T ck , k + Υ] is E (cid:2) ˆ n ik (cid:3) = npp − (35) Proof:
The proof is given in Appendix. . . . k max i u ik x ik . . . .
81 Time k εε ss e k Figure 1. Evolution of a MAS evolving according to Protocol 1. . . . k max i u ik x ik . . . .
81 Time k εε ss e k Figure 2. Evolution of a MAS evolving according to Protocol 2.
VII. N
UMERICAL SIMULATIONS
To illustrate the performance of the proposed protocol,simulation results are given in this section. First, we substan-tiate stability and error bounds of the proposed protocols bysimulating a worst-case scenario network with line topology.Second, we applied these protocols in the context of distributedsize estimation of time-varying networks, in which nodes canjoin and leave over time. A discussion of pros and cons of theproposed protocols is provided.
A. Network with line topology
For the sake of clarity and without loss of generality, in thissubsection we limit the simulations to [ k , k + Υ] for any k ∈ N in which the topology remains unchanged, accordingto Assumption 1. This allows us to show how the protocolsteer the agents to track the time-varying maximum ¯ u k valueamong the inputs u ik , proving the results on the transient andconvergence times and on the bound on the tracking errorgiven in Theorems 1-2 and Corollaries 1-2. Dual simulationsfor the dynamical min-consensus problem are omitted.We simulate a network of n = 6 agents with line topology.The choice of the line topology is instrumental to run simu-lations in the worst case scenario. In fact, for line graphs theinformation takes exactly δ ( G ) = n − steps to flowthrough the network, thus maximizing the error for a fixednumber of agents.Figures 1-2 show evolution of the state variables (dashed redlines) and maximum among the time-varying inputs (blue line)when ADMC and EDMC protocols (given in Protocol 1-2, re-spectively) are run over the MAS, respectively. State variablesare initialized at x (0) = [0 , . , . , . , . , T while inputsare initialized at u (0) = [0 . , . , . , . , . , . T . Allinputs remain constant except for the -th component, which is time-varying with respect to the following u ( k ) u (0) if k < u (0) − Π if k ∈ [60 , u (80) if k ∈ [80 , u (0) + Π if k ∈ [100 , u (140) if k ≥ , (36)with Π = 0 . being the absolute change according toAssumption 2.1) ADMC Protocol under Theorem 1: • Input parameter α = 0 . ; • Transient time T tk = 5 ; • Convergence time T ck = 34 ; • Bound on the tracking error ε = 0 . ; • Bound on the steady state error ε ss = 0 . .2) EDMC Protocol under Theorem 2 • Input parameter ∆ = 5 ; • Transient time T tk = 5 • Convergence time T ck = 5 • Bound on the tracking error ε = 0 . ; • Bound on the steady state error ε ss = 0 . B. Size Estimation
We chose to run simulations of size estimation over scale-free networks [23], [24]. A scale-free network is a networkwhose degree distribution follows a power law, at least asymp-totically. That is, the fraction P ( k ) of nodes in the networkhaving k connections to other nodes goes for large values of k as P ( k ) ∼ k − γ where parameter γ ∈ R typically is in the range [2 , .Such networks are known to be ultrasmall , as proved in[25], meaning that their diameter scales very slow with thedimension of the network, behaving as d ∼ ln ln N. We randomly generated a scale-free network by means ofBarabsiAlbert (BA) model proposed in [23]. This algorithmgenerates random scale-free networks using a preferentialattachment mechanism given an initial small network, nonecessarily scale-free. We use as initial network a line networkof nodes, and then we run the algorithm until a network of n = 100 nodes is generated. This network has a diameterof the order of the original small network, i.e., d ∼ . Inorder to simulate nodes leaving and joining the network whilekeeping connectivity and scale-free structure of the graph, werandomly deactivate or activate the last m ≤ nodes addedto the network by the algorithm every steps.Fig. 3 shows the estimation of the size of a network bymethod proposed in [22] by means of Protocol 5 which makesuse of one dynamic max-consensus protocols proposed inSection IV, i.e., the ADMC protocol given in Protocol 1 andthe EDMC protocols given in Protocol 2.VIII. C ONCLUSIONS
We have proposed, and characterized in terms time and errorconvergence, a distributed protocol for multi-agent systems to k n k x ik (EDMC) x ik (ADMC) Figure 3. Dynamic Size Estimation of a Network by means of Protocols 5. effectively dealing with the problem of tracking the maximumof a set of positive time-varying input reference signals. Twostrengths of the proposed protocol are the following: 1) theability to track the maximum reference signal even it isstrictly lower than all states variables; 2) the robustness toinitialization, meaning that the protocol is ensured to worksfor any initialization of the state variables. A weakness ofthis protocol is that exact consensus is never reached, evenwith constant reference inputs, thus avoiding the chance toreach a zero error. In the view of this weakness, we aim toimprove the proposed protocol by means of locally distributedand time-varying tuning parameters to ensure convergence toa consensus state. A
PPENDIX
Proof of Theorem 1:
Let us denote the maximum and,respectively, the minimum, among all agents’ states, as ¯ x k = max i ∈ V k x ik , x k = min i ∈ V k x ik . It follows that the tracking error (8) satisfies e k = max i ∈ V k | x ik − ¯ u k | = max {| ¯ x k − ¯ u k | , | x k − ¯ u k |} . (37)Without loss of generality, we consider a generic time k ∈ N at which a change of network occurs, i.e., G k − (cid:54) = G k and Υ ≥ T ck . For k ∈ [ k , k + Υ] the network is static. Now, forthe sake of simplicity, we assume the following statements tohold, which are later proven: x k ≥ ¯ u k − ( δ ( G k )+1)Π − αδ ( G k ) , k ∈ [ k + δ ( G k ) , k +Υ] . (38) ¯ x k ≤ ¯ u k + Π , k ∈ [ k + T (cid:48) , k + Υ] , (39)with T (cid:48) = (cid:24) max { x k − ¯ u k , } α − Π (cid:25) . (40)Relations (38)-(39) are both satisfied for T ck = max { δG k ) , T (cid:48) } . This is the convergence timebecause it allows to prove the boundedness of the trackingerror. In fact, for k ∈ [ k + T ck , k + Υ] it holds x ik ∈ [¯ u k − ( δ ( G k ) + 1)Π − αδ ( G k ) , ¯ u k + Π] . and a bound on the tracking error can be computed from (37)as e k = max i ∈ V k | x ik − ¯ u k | = max {| ¯ x k − ¯ u k | , | x k − ¯ u k |}≤ max { Π , ( δ ( G k ) + 1)Π + αδ ( G k ) }≤ ( δ ( G k ) + 1)Π + αδ ( G k ) , which corresponds to the bound given by the theorem. Thisproves that condition (7) of Problem 1 is achieved.Now, we proceed to prove that condition (6) of Problem 1 isachieved. For k ∈ [ k + δ ( G k ) , k + T ck ] , relation (38) holdstrue but relation (39) does not, thus we use the following ¯ x k +1 = max i ∈ V k x ik +1 (41) = max i ∈ V k max j ∈N ik ∪{ i } { x jk − α, u ik } = max i ∈ V k { x ik − α, u ik } = max { ¯ x k − α, ¯ u k } , which is due to Protocol 1 and its interaction rule given ineq. (22). The difference in the tracking error can be computedfrom (37) as e k +1 − e k = max i ∈ V k | x ik +1 − ¯ u k +1 | − max i ∈ V k | x ik − ¯ u k | = max {| ¯ x k +1 − ¯ u k +1 | , | x k +1 − ¯ u k +1 |}− max {| ¯ x k − ¯ u k | , | x k − ¯ u k |} . Observing that | x k +1 − ¯ u k +1 | = | x k − ¯ u k | by (38), it issufficient to prove that | ¯ x k +1 − ¯ u k +1 | ≤ | ¯ x k − ¯ u k | in order toprove e k +1 − e k ≤ . We compute | ¯ x k +1 − ¯ u k +1 | = | max { ¯ x k − α, ¯ u k } − ¯ u k +1 | = max {| ¯ x k − α − ¯ u k +1 | , | ¯ u k − ¯ u k +1 |} = max {| ¯ x k − α − ¯ u k ± Π | , Π }≤ max {| ¯ x k − ¯ u k − α | + Π , Π }≤ | ¯ x k − ¯ u k − α | + Π For k ∈ [ k + δ ( G k ) , k + T ck ] the tracking error is surelylarger than the bound ( δ ( G k ) + 1)Π + αδ ( G k ) and thus alsothan ≥ max { α, Π } , thus one can derive ¯ x k − ¯ u k − α + Π ≤ ¯ x k − ¯ u k − α + Π ≤ α ≥ Π . Since by assumption it holds (10), i.e., α > Π then e k +1 < e k and this proves that condition (6) of Problem 1 is achieved andthe transient time is given by T tk = δ ( G k ) . Therefore, Problem 1 is solved. To complete the proof, weproceed by proving the veracity of inequalities (39)-(38). • Proof of eq. (38). It trivially holds x ik ≥ x k ∀ i ∈ V k , ∀ k ∈ N . At time k we define the set V = (cid:8) i ∈ V k : x ik = ¯ x k (cid:9) denote the set of agents whose state at time k is the maximumamong all others. Let us now consider the set V of one-hopneighbors of nodes in set V at time k + 1 . Formally, V = { i ∈ V k : ( i, j ) ∈ E, j ∈ V } . Thus, for all i ∈ V , the state update rule (22) reduces to x ik +1 = max { ¯ x k − α, u ik } , because all agents i ∈ V have a neighbor j ∈ V with statevalue x jk +1 = ¯ x k . Thus, exploiting eq. (39) we can write x ik +1 ≥ ¯ x k − α, ∀ i ∈ V . (42)By induction, define the set V (cid:96) = (cid:40) i ∈ V k : ( i, j ) ∈ E, j ∈ (cid:96) − (cid:91) s =0 V s (cid:41) . It easily follows that, since the longest shortest path betweentwo nodes in a connected graph is at most equal to its diameter δ ( G k ) , for (cid:96) = δ ( G k ) it holds V δ ( G ) ≡ V k . Therefore x ik + δ ( G k ) ≥ ¯ x k − δ ( G k ) α ∀ i ∈ V k . For k ∈ [ k + δ ( G k ) , k + Υ] we can combine the previousinequality with (39) and (43), leading to x ( k ) ≥ ¯ x k − δ ( G k ) − δ ( G k ) α ≥ ¯ u k − δ ( G k ) − − δ ( G k ) α ≥ ¯ u k − ( δ ( G k ) + 1)Π − αδ ( G k ) , which proves the veracity of eq. (38) • Proof of eq. (39). Under Assumption 2, at the generictime k + T with T ∈ [0 , Υ] it holds ¯ u k = ¯ u k + T ≥ ¯ u k − T Π (43)and thus by (41) it follows ¯ x k + T +1 = max { ¯ x k − T α, ¯ u k + T } . Under condition (10), one derives that the inputs vary slowerthan the agents’ states, and therefore there exists a time k + T after which the system reaches the input, i.e., ¯ x k − T α < ¯ u k − T Π . Solving for T , we obtain T (cid:48) as in (40). Thus, for k ∈ [ k + T (cid:48) , k + Υ] , recalling (43), the dynamics of ¯ x ( k ) is given by ¯ x k = ¯ u k − ≤ ¯ u k + Π , proving the veracity of eq. (39) and completing the proof ofthe theorem. Proof of Theorem 2:
At time k , we define the set V = (cid:26) i ∈ V k : x i k = max j ∈V k x j k (cid:27) . Since by Protocol 2 it holds x i k = u ik − , then V = (cid:26) i ∈ V k : x i k = max j ∈ V k u jk − (cid:27) . Let us now consider time k + 1 and the set V of one-hopneighbors of nodes in set V . Formally, V = { i ∈ V k : ( i, j ) ∈ E, j ∈ V } . The state update rule (28) for i ∈ V , (cid:96) = 1 reduces to x i k +1 = max j ∈N ik ∪{ i } x j k = max j ∈ V k u jk − = ¯ u k − , because all agents i ∈ V have a neighbor j ∈ V with statevalue x j k = ¯ u k − . By induction, for (cid:96) ≥ define V (cid:96) = (cid:40) i ∈ V k : ( i, j ) ∈ E, j ∈ (cid:96) − (cid:91) s =0 V s (cid:41) , and therefore for all i ∈ V (cid:96) it holds x i(cid:96)k + (cid:96) = ¯ u k − . By noticing that V ∆ = V δ ( G ) ≡ V k , we infer that for all i ∈ V k and for any time k ∈ [ k + ∆ , k + Υ] with Υ ≥ ∆ ,it holds x i ∆ k = ¯ u k − ∆ − , (44)which proves that transient and convergence times coincideand they are equal to the upperbound ∆ , i.e., T ck = T tk = ∆ .Furthermore, by Assumption 2 it follows ¯ u k ∈ [¯ u k − ∆ − − (∆ + 1)Π , (45) ¯ u k − ∆ − + (∆ + 1)Π] . Finally, exploiting (44) and (45), we conclude that for any k ∈ [ k + ∆ , k + Υ] the tracking error (8) is bounded by e k = max i ∈ V k | x i ∆ k − ¯ u k | ≤ (∆ + 1)Π , completing the proof. Proof of Theorem 3:
We start the proof with a trivialstatement u k = − ¯ v k , v k = − u k . (46)Now, consider an OMAS executing Protocol 1 with state y ∈ R n and inputs v k , thus agents’ local interaction rule is y ik +1 = max j ∈N ik (cid:83) { i } (cid:110) y jk − α, v ik (cid:111) if i ∈ R k v ik +1 if i ∈ A k Given the initial condition y k = − x k , we have that for any k ∈ [ k , k + Υ] it holds for i ∈ R k y ik +1 = max j ∈N ik j (cid:83) { i } (cid:110) − x jk − α, − u ik (cid:111) , = − min j ∈N i (cid:83) { i } (cid:110) x jk + α, u ik (cid:111) and then by invoking (46) it follows y ik +1 = − x ik +1 , ∀ i ∈ V k . It is trivial to notice that due to the above relation one canderive transient and convergence time by substitution into (13) T tk = δ ( G k ) ,T ck = max (cid:26) δ ( G k ) , (cid:24) max { y k − ¯ v k } α − Π (cid:25)(cid:27) , = max (cid:40) δ ( G k ) , (cid:38) max (cid:8) u k − x k (cid:9) α − Π (cid:39)(cid:41) , and the tracking errror by (12) as follows e k = max i ∈ V k | x ik − u k | = max i ∈ V k | − y ik + ¯ v k | = max i ∈ V k | y ik − ¯ v ( k ) |≤ ( δ ( G ) + 1)Π + αδ ( G )) . of Theorem 1, completing the proof. Proof of Theorem 4:
We start the proof with a trivialstatement u k = − ¯ v k , v k = − u k . (47)Now, consider an OMAS executing Protocol 2 with state y ∈ R n and inputs v k , thus agents’ local interaction rule is y i k +1 = (cid:40) v ik if i ∈ R k v ik +1 if i ∈ A k y i(cid:96)k +1 = min j ∈N ik (cid:83) { i } y j ( (cid:96) − k if i ∈ R k v ik +1 if i ∈ A k Given the initial condition y k = − x k , we have that for any k ∈ [ k , k + Υ] it holds for i ∈ R k and (cid:96) = 1 , . . . , ∆ y i(cid:96)k +1 = max j ∈N i (cid:83) { i } y j ( (cid:96) − k = max j ∈N i (cid:83) { i } − x j ( (cid:96) − k = − min j ∈N i (cid:83) { i } x j ( (cid:96) − k . and then by invoking (47) it follows y ik +1 = − x ik +1 , ∀ i ∈ V k . It is trivial to notice that due to the above relation one canderive transient and convergence time by substitution into (17) T tk = T ck ∆ and the tracking error by (18) as follows e k = max i ∈ V k | x ik − u k | = max i ∈ V k | − y ik + ¯ v k | = max i ∈ V k | y ik − ¯ v ( k ) |≤ (∆ + 1)Π . of Theorem 2, completing the proof. Proof of Theorem 5:
Since by Assumption 1 the networkremains unchanged for k ∈ [ k , k + Υ] , then by Protocol 5the inputs of the agents V k are constant in this interval. Thus,in the following we omit the dependence of all variables from k . For the purpose of the proof, we recall some basic conceptson order statistics []. Consider the sample u j , . . . , u nj consist-ing of the j -th numbers generated by the agents i = 1 , . . . , n .The j -th smallest value is called the j -th order statistic ofthe sample. Let us denote ¯ u j the n -th order statistics of thesample, i.e., the maximum value ¯ u j = max i ∈ V u ij , ∀ j = 1 , . . . , p. All u ij are i.i.d. with probability density function p ( a ) andand probability distribution function P ( a ) given by p ( a ) = (cid:40) ≤ a ≤ otherwise , P ( a ) = (cid:40) a ≤ a ≤ otherwise , while the probability density function of the n -th order statisticis given by p n ( a ) = nP n − ( a ) . (48)Consider now the sample obtained by the n -th order statis-tics of each random number generated by the agents, i.e., ˜ u = { ¯ u , . . . , ¯ u p } . Variables ¯ u j in the sample ˜ u depend on theparameter n . The likelihood function L ( n | ˜ u ) is equal to theprobability that the particular outcome ˜ u given the parameter n and, since all variables in the sample are i.i.d. random variableswith probability density function (48), then it can be computedas the product of the probability density functions, i.e., L ( n | ˜ u ) = p (cid:89) j =1 p n (¯ u j ) = n p p (cid:89) j =1 ¯ u n − j . In practice, it is often convenient to work with the naturallogarithm of the likelihood function, called the log-likelihood L ∗ ( n | ˜ u ) = ln ( L ( n | ˜ u )) = ln n p p (cid:89) j =1 ¯ u n − j = ln ( n p ) + ln p (cid:89) j =1 ¯ u n − j = ln ( n p ) + p (cid:88) j =1 ln (cid:0) ¯ u n − j (cid:1) = p ln ( n ) + ( n − p (cid:88) j =1 ln (¯ u j ) . The maximum likelihood estimate (MLE) is the valuewhich maximizes log-likelihood L ∗ ( n | x n ) , thus giving thebest estimate of n from the sample ˜ u . By putting to zero thederivative in n , we can find an expression for the MLE, M LE = − p (cid:80) pj =1 ln (¯ u j ) . (49)Equation (49) represents the best way to estimate the size n ofthe network trough inference by the maximum values among the numbers generated by the agents. However, variables ¯ u j are not known exactly at each node, and thus the best an agentcan do for the estimation of n is to implement the following (cid:92) M LE i = − p (cid:80) pj =1 ln ( x ij ) , ∀ i ∈ V. (50)It is necessary to understand how such the error arising from x ij (cid:54) = ¯ u j affects the estimation of n . We start our discussiontaking into consideration the employment of Protocol 1. • Discussion for Protocol 1 : By Corollary 1 the steady stateerror in the estimating of ¯ u j is bounded by the following e j = max i ∈ V (cid:12)(cid:12) x ij − ¯ u j (cid:12)(cid:12) ≤ δ ( G ) · α = ε. (51)A fundamental consideration, resulting from the constructingproof of Theorem 1 and Corollary 1, is that at steady statethe estimation x ij of agent i of the quantity ¯ u j is always anunderestimation, i.e., x ij ≤ ¯ u j , ∀ i ∈ V. With this consideration in mind, it is easy to realize thatthe worst case is when at least one agent underestimates allvariables ¯ u j with maximum error (55). Thus, we consider sucha worst case scenario by assuming that ∃ i ∈ V : x ij = ¯ u j − ε, ∀ j = 1 , . . . , p. (52)Under condition (52), the worst MLE estimation (cid:92) M LE ∗ isgiven by approximated MLE given by (cid:92) M LE ∗ = − p (cid:80) pj =1 ln (¯ u j − ε ) . One can prove that condition (52) implies the worst casescenario since the distance max i ∈ V | M LE − (cid:92) M LE i | is max-imized. It is straightforward to notice that (cid:92) M LE ∗ ≤ M LE ,thus we manipulate the above expression to get a lowerboundas follows (cid:92)
M LE ∗ = − p (cid:80) pj =1 ln (¯ u j − ε )= − p (cid:80) pj =1 ln (cid:16) ¯ u j (1 − ε ¯ u j ) (cid:17) = − p (cid:80) pj =1 (cid:104) ln ¯ u j + ln (cid:16) − ε ¯ u j (cid:17)(cid:105) ≥ − p (cid:80) pj =1 [ln ¯ u j + ln (1 − ε )] ≥ − p (cid:80) p(cid:96) =1 (ln ¯ u j − ε ) ≥ p (cid:80) pj =1 ( − ln ¯ u j ) + pε ≥ p (cid:80) pj =1 ( − ln ¯ u j ) + ε (53)At the denominator of (53) we can recognize the term γ = 1 p p (cid:88) j =1 − ln ¯ u j . Since ( − ln ¯ u j ) with j = 1 , . . . , p are p i.i.d. exponentialrandom variables with rate n , their averaged sum γ is knownto be a gamma random variable with shape α = p and rate β = pn . Therefore, ˆ n is the reciprocal of a shifted gammavariable, whose probability density function g ( · ) is given by g ( a ) = ( np ) p ( p − a p − e − npa We can now use the law of the unconscious statistician tocalculate the expected value of (cid:92)
M LE ∗ by E [ (cid:92) M LE ∗ ] = (cid:90) ∞ f ( x ) g ( x ) dx, (54)where f ( x ) = 1 / ( x + ε ) . Solution to (54) can be computedthrough any solver, giving the following E [ (cid:92) M LE ∗ ] = ε p − e εnp ( np ) p Γ(1 − p, εnp ) , where Γ( p, x ) is known as the upper incomplete gammafunction. We point out that this expression holds for n, p ∈ N and ε ∈ R such that n ≥ , p > and ε ≥ . This completesthe first part of the proof. • Discussion for Protocol 2 : By Corollary 2 the steady stateerror in the estimating of ¯ u j is bounded by the following e j = max i ∈ V (cid:12)(cid:12) x ij − ¯ u j (cid:12)(cid:12) = 0 = ε. (55)Solution to (54) for ε = 0 can be computed by any solver,giving the following E [ (cid:92) M LE ∗ ] = npp − . We point out that this expression holds for n, p ∈ N and ε ∈ R such that n ≥ , p > and ε ≥ . This completes the firstpart of the proof. This result is coherent to the expected valuewith zero error given in [21], thus proving (35) and confirmingthat (34) is a generalization for ε ≥ .R EFERENCES[1] D. P. Spanos, R. Olfati-Saber, and R. M. Murray, “Dynamic consensuson mobile networks,” in
IFAC world congress , 2005, pp. 1–6.[2] R. A. Freeman, P. Yang, and K. M. Lynch, “Stability and convergenceproperties of dynamic average consensus estimators,” in
Proceedingsof the 45th IEEE Conference on Decision and Control , Dec 2006, pp.338–343.[3] M. Zhu and S. Mart´ınez, “Discrete-time dynamic average consensus,”
Automatica , vol. 46, no. 2, pp. 322 – 329, 2010.[4] E. Montijano, J. I. Montijano, C. Sags, and S. Mart´ınez, “Robust discretetime dynamic average consensus,”
Automatica , vol. 50, no. 12, pp. 3131– 3138, 2014.[5] M. Franceschelli and A. Gasparri, “Multi-stage discrete time and ran-domized dynamic average consensus,”
Automatica , vol. 99, pp. 69 – 81,2019.[6] H. Bai, R. A. Freeman, and K. M. Lynch, “Robust dynamic averageconsensus of time-varying inputs,” in , Dec 2010, pp. 3104–3109.[7] B. V. Scoy, R. A. Freeman, and K. M. Lynch, “A fast robust nonlin-ear dynamic average consensus estimator in discrete time,” , vol. 48, no. 22, pp. 191 – 196, 2015.[8] S. S. Kia, J. Cort´es, and S. Mart´ınez, “Dynamic average consensusunder limited control authority and privacy requirements,”
InternationalJournal of Robust and Nonlinear Control , vol. 25, no. 13, pp. 1941–1966, 2015.[9] M. Franceschelli and P. Frasca, “Proportional dynamic consensus in openmulti-agent systems,” in , Dec 2018, pp. 900–905. [10] Z. Dengchang, A. Zhulin, and X. Yongjun, “Time synchronization inwireless sensor networks using max and average consensus protocol,” International Journal of Distributed Sensor Networks , vol. 2013, 032013.[11] A. Petitti, D. Di Paola, A. Rizzo, and G. Cicirelli, “Consensus-baseddistributed estimation for target tracking in heterogeneous sensor net-works,” in , Dec 2011, pp. 6648–6653.[12] R. Lucchese, D. Varagnolo, J. Delvenne, and J. Hendrickx, “Networkcardinality estimation using max consensus: The case of bernoulli trials,”in , Dec2015, pp. 895–901.[13] T. Borsche and S. A. Attia, “On leader election in multi-agent controlsystems,” in , May 2010,pp. 102–107.[14] R. Olfati-Saber and R. M. Murray, “Consensus problems in networksof agents with switching topology and time-delays,”
IEEE Transactionson Automatic Control , vol. 49, no. 9, pp. 1520–1533, Sep. 2004.[15] B. M. Nejad, S. A. Attia, and J. Raisch, “Max-consensus in a max-plus algebraic setting: The case of fixed communication topologies,”in , Oct 2009, pp. 1–7.[16] B. M. Nejad, S. A. Attia, and J. Raisch, “Max-consensus in a max-plus algebraic setting: The case of switching communication topologies,”
IFAC Proceedings Volumes , vol. 43, no. 12, pp. 173 – 180, 2010, 10thIFAC Workshop on Discrete Event Systems.[17] F. Iutzeler, P. Ciblat, and J. Jakubowicz, “Analysis of max-consensusalgorithms in wireless channels,”
IEEE Transactions on Signal Process-ing , vol. 60, no. 11, pp. 6103–6107, Nov 2012.[18] S. Giannini, A. Petitti, D. Di Paola, and A. Rizzo, “Asynchronousmax-consensus protocol with time delays: Convergence results andapplications,”
IEEE Transactions on Circuits and Systems I: RegularPapers , vol. 63, no. 2, pp. 256–264, Feb 2016.[19] S. Zhang, C. Tepedelenliolu, M. K. Banavar, and A. Spanias, “Maxconsensus in sensor networks: Non-linear bounded transmission andadditive noise,”
IEEE Sensors Journal , vol. 16, no. 24, pp. 9089–9098,Dec 2016.[20] M. Abdelrahim, J. M. Hendrickx, and W. P. M. H. Heemels, “Max-consensus in open multi-agent systems with gossip interactions,” in , 2017,pp. 4753–4758.[21] D. Varagnolo, G. Pillonetto, and L. Schenato, “Distributed size estima-tion in anonymous networks,”
IEEE Transactions on Automatic Control(submitted) , 2011.[22] H. Terelius, D. Varagnolo, and K. H. Johansson, “Distributed sizeestimation of dynamic anonymous networks,” in , Dec 2012, pp. 5221–5227.[23] R. Albert and A.-L. Barab´asi, “Statistical mechanics of complexnetworks,”
Rev. Mod. Phys. , vol. 74, pp. 47–97, Jan 2002. [Online].Available: https://link.aps.org/doi/10.1103/RevModPhys.74.47[24] L. A. N. Amaral, A. Scala, M. Barth´el´emy, and H. E. Stanley, “Classesof small-world networks,”
Proceedings of the National Academy ofSciences , vol. 97, no. 21, pp. 11 149–11 152, 2000.[25] R. Cohen and S. Havlin, “Scale-free networks are ultrasmall,”