Stability and Resilience of Distributed Information Spreading in Aggregate Computing
11 Stability and Resilience of Distributed InformationSpreading in Aggregate Computing
Yuanqiu Mo,
Member IEEE , Soura Dasgupta,
Fellow IEEE and Jacob Beal,
Senior Member IEEE
Abstract —Spreading information through a network of devicesis a core activity for most distributed systems. As such, self-stabilizing algorithms implementing information spreading areone of the key building blocks enabling aggregate computingto provide resilient coordination in open complex distributedsystems. This paper improves a general spreading block inthe aggregate computing literature by making it resilient tonetwork perturbations, establishes its global uniform asymptoticstability and proves that it is ultimately bounded under persistentdisturbances. The ultimate bounds depend only on the magnitudeof the largest perturbation and the network diameter, and threedesign parameters trade off competing aspects of performance.For example, as in many dynamical systems, values leading togreater resilience to network perturbations slow convergence andvice versa.
Index Terms —aggregate computing, multi-agent systems, dis-tributed graph algorithms, nonlinear stability, ultimate bounds.
I. I
NTRODUCTION
Complex networked distributed systems are rapidly becom-ing a feature of many engineering systems. Their stability,dynamics and reliability has acquired paramount importance.Control theorists have embraced this challenge through exten-sive research on the stability of networked control systems,most typically “closed” systems in which a good model of thesystem is available at design time e.g. [1]–[9].An emerging set of alternative challenges is posed by theanalysis and design of complex open systems like tacticalinformation sharing, smart cities, edge computing, personaland home area networks, and the Internet of Things (IoT) [10].These systems also disperse services to local devices, requiredevices to interact safely and seamlessly with nearby brethrenthrough peer to peer information exchange, and to sharetasks. As they are open , however, they must support fre-quent non-centralized changes in the applications and services
This work has been supported by the Defense Advanced Research ProjectsAgency (DARPA) under Contract No. HR001117C0049. The views, opinions,and/or findings expressed are those of the author(s) and should not beinterpreted as representing the official views or policies of the Department ofDefense or the U.S. Government. This document does not contain technologyor technical data controlled under either U.S. International Traffic in ArmsRegulation or U.S. Export Administration Regulations. Approved for publicrelease, distribution unlimited (DARPA DISTAR case 33519, 10/14/20).Y. Mo is with the Institute of Advanced Technology, Westlake Institutefor Advanced Study, Westlake University, Hangzhou 310024, China (email:moyuanqiu@@westlake.edu.cn).S. Dasgupta is with the University of Iowa, Iowa City, Iowa 52242 USA(e-mail: [email protected]). S. Dasgupta is also a Visiting Professorat Shandong Computer Science Center, Shandong Provincial Key Laboratoryof Computer Networks, China.J. Beal is with Raytheon BBN Technologies, Cambridge, MA, USA 02138USA (e-mail: [email protected]) that they host. Current modes of device interactions restricttheir potential by being typically either highly constrainedand inflexible (e.g. single-purpose devices) or by relying onremote infrastructure like cloud services. The former impairsreusability and prevents devices from contributing to multipleoverlapping applications. The latter is centralized with highlatency and lacks the agility to exploit local communication,services and devices.
Aggregate computing , on the other hand, addresses devicecoordination in open systems with a layered approach [10],separating systems into abstraction layers that decomposingsystems engineering into separable tasks, much like the OSImodel does for communication networks, [11]. The layers spanfrom applications to a field calculus (providing distributedscoping of shared information), and an abstract device model(for services such as neighborhood discovery). Between these,a middle layer facilitates resilient device interactions andcomprises three classes of basis set modules that are them-selves distributed graph algorithms: (i) G -blocks that spreadinformation through a network of devices, (ii) C -blocks thatsummarize salient information about the network to be usedby interacting units, and (iii) T -blocks that maintain temporarystate. Prior work [12]–[14], has shown that a broad classof device interactions can be realized by interconnections ofthese three blocks . Our long term research goal is analyze thestability of compositions of these blocks including in feedback.This paper is concerned with a rigorous analysis of the mostgeneral G -block of [14], after making it resilient to networkperturbations .While the empirical assessment of compositions of the G -block with other blocks is encouraging [15]–[17], the formalanalysis of its most general case has been confined to self-stablization [14], and that under the assumption that all stateslie in Noetherian rings and are thus a priori bounded . Unlikeglobal uniform asymptotic stability (GUAS) [18], [19], self-stabilization has no notion of robustness to perturbations, whileperturbations are to be expected in any feedback composition.Thus we improve the generalized G -block to allow removalof the Noetherian assumption, proof of GUAS, and (under anadditional Lipschitz condition) ultimate boundedness in faceof persistent perturbations. Finding ultimate bounds furtheranticipates the development of sophisticated variants of thesmall gain theorem [20], [18] for closed loop analysis.Previously, in [21] and [22], we have performed such ananalysis of the simplest G -block, the Adaptive Bellman-Ford(ABF) algorithm, which estimates distances of nodes from asource set in a distributed manner and (unlike the classicalBellman-Ford algorithm [23]) accommodates underestimates. a r X i v : . [ c s . D C ] F e b n [24] we have analyzed without proof another special case,which generalizes ABF by allowing non-Euclidean distancemetrics, e.g., those that penalize certain routes, and permits broadcast , where each source broadcasts a value like a di-ameter estimate it holds to nearest devices. More features aregiven Section II.A further problem that must be considered in this contextis the rising value problem . All G -block algorithms generateestimates ˆ x i ( t ) that must converge to a value x i . The risingvalue problem is when underestimates ( i.e. ˆ x i ( t ) < x i )may rise very slowly, and this problem affects the G -blocksanalyzed in [22] and [24]. The most general G -block given in[14] removes this problem by treating the estimate generatedby the algorithm in [24] as an auxiliary state ˜ x i ( t ) . Theactual estimate ˆ x i ( t ) is increased by a fixed amount of atleast δ > unless ˜ x i ( t + 1) equals either ˆ x i ( t ) or themaximal element in the Noetherian ring. If either of these twoconditions is violated then ˆ x i ( t + 1) = ˜ x i ( t + 1) . The increaseby δ or more, removes the rising value problem. Howeverthe equality requirement of ˜ x i ( t + 1) = ˆ x i ( t ) introducesfragility to disturbances, since ˜ x ( t + 1) = ˆ x i ( t ) rarely occursunder perturbations and thus ˆ x i ( t ) must persistently rise to themaximal element.We deal with real non-negative numbers rather than Noethe-rian rings and do not assume prior estimate bounds . Insteadwe modify the algorithm in [14] by introducing two additionalparameters, a modulation threshold M and a dead zone value D , that together define regions for improved perturbationtolerance versus regions for fast convergence with δ , and thatreduce to the algorithm in [24] when M = 0 and/or D = ∞ . We show that the improved algorithm is GUAS for all non-negative M and D without the assumption that M is a maxi-mal element . In the case of persistent bounded perturbations,we show that the estimates are ultimately bounded providedthat the dead zone parameter D exceeds a value proportionalto the disturbance bound. A larger D , however, is also lesseffective at mitigating the rising value problem, indicating atrade-off between speed and robustness that is common to mostdynamical systems.In the remainder of the paper, Section II provides thealgorithm, assumptions and motivating applications. SectionIII characterizes stationary points, while Section IV provesGUAS. Section V gives ultimate bounds which are determinedonly by the magnitude of the perturbations and the graph. Section VI discusses design choices, which are affirmed viasimulation in Section VII, and Section VIII concludes.II. A
LGORITHM
In this section, we present a general G -block that spreadsinformation through a network in a distributed fashion. Origi-nally provided in [25] using the language of field calculus, wetranslate it here to one more appropriate of this journal. SectionII-A describes a special case shown to be GUAS in [24],with proofs omitted, plus examples and a shortcoming. SectionII-B then presents a more general algorithm that removesthis deficiency, and Section II-C provides assumptions anddefinitions that will be used for proofs in subsequent sections. A. The Spreading block of [24]
Consider an undirected graph G = ( V, E ) with nodes in V = { , , · · · , N } and edge set E . Nodes i and k are neighbors if they share an edge. The goal of the algorithmis to spread the state x i to node i. Denote N ( i ) as the setof neighbors of i . With ˆ x i ( t ) an estimate of x i , in the t -thiteration, the information spreading in [24], proceeds as: ˆ x i ( t + 1) = min (cid:26) min k ∈N ( i ) { f (ˆ x k ( t ) , e ik ) } , s i (cid:27) , ∀ t ≥ t . (1)The e ik define the structural aspects of G ; e.g. they may bethe edge lengths between neighbors; s i ≥ , which may beeither finite or infinite, is the maximum value that ˆ x i ( t ) canacquire after the initial time. Further, ˆ x i ( t ) ≥ , for all i ∈ V. Function f ( · , · ) must be progressive i.e. for some σ > , f ( a, b ) > a + σ (2)and monotonic in the first variable, i.e. f ( a , b ) ≥ f ( a , b ) , if a ≥ a . (3)and is finite for finite a and b . The initialization in (1) ensuresthat ˆ x i ( t ) ≥ , for all t ≥ t . Define S ∗ as the set of nodeswith finite maximum values s i : S ∗ = { i ∈ V | s i < ∞} . (4)We will assume that this set is nonempty. Evidently, theinformation x i to be spread to node i must be an elementof the vector of stationary values of (1), i.e. obeys x i = min (cid:26) min k ∈N ( i ) { f ( x k , e ik ) } , s i (cid:27) , ∀ i ∈ V. (5)We will prove the less than evident fact that this stationarypoint is unique, finite, and that at least one x i = s i .The simplest example, whose Lyapunov analysis is in [21],[22], is ABF where f ( a, b ) = a + b and s i = 0 or infinity. Theset of i for which s i = 0 are called sources, e ik > is theedge length between nodes i and k , x i represents the distance d i from the set of sources, and with ˆ x i = ˆ d i , (1) becomes ˆ d i ( t + 1) = (cid:40) s i = 0min k ∈N ( i ) (cid:110) ˆ d k ( t ) + e ik (cid:111) s i (cid:54) = 0 . (6) t = 3 x̂ A = x̂ B = x̂ C = x̂ D = Fig. 1. Non-zero s i representing external gateways of a tactical wirelessnetwork: red and purple nodes are high and low speed external links,respectively, while blue are nodes without external links. Black numbersrepresent edge weights e ik , green numbers represent state estimates ˆ x i . After3 rounds, all nodes, including the low-speed link, have converged to routethrough the high-speed link.
2e may observe that an ABF stationary point must haveall nodes with s i = 0 also having d i = 0 . If we allow othervalues of s i , then not all members of S ∗ necessarily havestationary states s i . Figure 1 shows one such example, inwhich s i represents transport lag for external gateways in atactical wireless network. In this example, as with (6), f (ˆ x k ( t ) , e ik ) = ˆ x k ( t ) + e ik . (7)Here, node A (red) is a high-speed gateway with s A = 1 ,node D (purple) is a low-speed gateway with s D = 5 , and theothers are non-gateways with s i = ∞ . Through (1), all nodestry to route to external networks through the shortest effectivepath. After 3 rounds, all nodes, including the low-speed link,converge to route through the high-speed link. In this case thestationary state of the low speed link is x D = 4 and does notequal s i = 5 even though s i is finite. Should the high speedlink represented by node A disappear, then the state estimateof D does converge to its maximum value 5, while those ofnodes B and C converge to 7 and 6, respectively, i.e. nodesreroute through the still available low speed link. A BF CE D
A BF CE D
A BF CE D
A BF CE D
A BF CE D
A BF CE D A = x̂ B =x̂ F = x̂ E = x̂ D =x̂ C = Fig. 2. Illustration of the spreading block. In this example, node in redrepresents the source with a maximum value of 0, while nodes in blue have amaximum value of 1. The edge value e ik reflects the success rate of delivery.After 5 rounds, each node finds a path with the smallest failure rate of deliveryto the source. While in the prior examples f ( a, b ) is linear and increasingin b , this need not be the case. A specialization of (1) violatingboth these properties finds the most probable path (MPP)from each node in a network to a source. In this case e ik represents the probability of successful traversal or deliverybetween neighbors i and j . The stationary value x i is the smallest probability of failure in movement from node i tothe source. In this case x i = 0 for sources. For all other nodes x i = min k ∈N ( i ) { − (1 − x k ) e ik } , ∀ i (cid:54) = 1 . The sequence of minimizing nodes k then indicates the MPPfrom node i to any source and can be computed using (1) with f (ˆ x k ( t ) , e ik ) = 1 − (1 − ˆ x k ( t )) e ik . (8)If < σ ≤ e ik < − σ, this is progressive and increasingin ˆ x k ( t ) , though decreasing in e ik . Figure 2 illustrates an example execution, with node A (red) as the source, in whicheach state estimate converges within 5 rounds.A key shortcoming of (1), however, is that underestimatescan rise very slowly in the presence of small e ik . Consider forexample (6) with nodes 1 and 2 having the smallest estimatesand sharing a short edge e. At successive instants ˆ d ( t + 1) =ˆ d ( t ) + e and ˆ d ( t + 1) = ˆ d ( t ) + e , i.e. each rises in smallincrements of e (and as shown in [22]) converge slowly. Thegeneralization below accelerates this slow convergence. B. A more general spreading block
The most general G -block, given in [25], is a generalizationof (10) in that state estimates are updated as ˆ x i ( t + 1) = F (˜ x i ( t + 1) , ˆ x i ( t ) , v i ) (9)with ˜ x i ( t + 1) obeying ˜ x i ( t + 1) = min (cid:26) min k ∈N ( i ) { f (ˆ x k ( t ) , e ik ) } , s i (cid:27) , ∀ t ≥ t , (10)where v i are environmental variables and f ( · , · ) remains pro-gressive and monotonic. The function F ( (cid:96) , (cid:96) , v ) is raising ,i.e. for finite M ≥ , δ > and D ≥ , obeys F ( (cid:96) , (cid:96) , v ) = (cid:40) (cid:96) (cid:96) ≥ M or | (cid:96) − (cid:96) | ≤ Dg ( (cid:96) ) otherwise , (11)where g ( x ) is finite for finite x and obeys g ( x ) ≥ x + δ. (12)Invocation of the second bullet of (11), speeds the initialascent of ˆ x i ( t ) , ameliorating the problem of the slow rise inunderestimates experienced by (1). On the other hand, the firstbullet renders (10) identical to (1). As the second bullet of (11)changes ˆ x i ( t ) , the stationary point of (10-12) is identical tothat of (5). Thus this algorithm spreads the same informationas (1), while accelerating the rise of underestimates . Observealso that D = ∞ and/or M = 0 , reduces (10-12) to (1).The version of (10-12) in [25] sets the dead zone variableas D = 0 . In face of persistent structural perturbations in e ik , l = l cannot be sustained. Consequently, regardless ofthe size of perturbations, with D = 0 , ˆ x i will regularly riseto the limit of the modulation threshold M , then fall, andthen rise again. On the other hand we will show that if D is sufficiently greater than (cid:15) , the bound on the perturbation,then (10-12) will have ultimate bounds proportional to (cid:15). Thisraises an essential trade-off . Too large a D slows convergencethough imparts greater robustness to perturbations. Such acompromise is inherent to most dynamic systems. Slowerconvergence improves noise performance.Another key difference is that [25] assumes that ˆ x i belong toa Noetherian ring with M its maximal element. This implicitlyassumes that the algorithm is a priori bounded. For distanceestimation this means a prior assumption on the diameter,which is unappealing in the context of open systems.The generalized Adaptive Bellman-Ford algorithm (GABF),presented and analyzed without proofs in [26], is a spe-cific example of (9), that is an accelerated ABF. In GABF,3 ̂ A =0
00 1 4 3
00 1 5
00 1
11 1 1
11 1 1 t=0 t=1t=2 t=3t=4
A B C D EA B C D E x̂ B =1 x̂ C =3 x̂ D =2 x̂ E =0 A B C D EA B C D EA B C D E
Fig. 3. Illustration of GABF and sets A ( t ) , E ( t ) , R ( t ) and U ( t ) . Each edgelength in the graph is 1, M = 4 , δ = 1 , D = 0 , s i = 0 for i = A, E and s i = ∞ otherwise. In this case, R (1) = A (1) = { A, B, E } , U (1) = E (1) = { C, D } . f (ˆ x k ( t ) , e ik ) follows (7) with e ik the edge length between i and k , ˆ x k ( t ) the distance estimate of k at time t , s i = 0 if i isa source while s i = ∞ if i is a non-source node. In Figure 3,nodes in red are sources, each edge length in the graph is 1,and numbers in green represent the state estimates. Variables M, D and δ in (11) and (12) are set as , and respectively.Convergence occurs in four rounds. C. Definitions and Assumptions
We define S ( t ) as comprising nodes in S ∗ that acquire theirmaximum values at time t , S ( t ) = { i ∈ S ∗ | ˆ x i ( t ) = s i } , (13)and we say i is a source at time t if i ∈ S ( t ) . The followingassumption holds in this paper. Assumption 1.
Graph G is connected, ∞ > e ik = e ki ≥ e min > , S ∗ defined in (4) in nonempty and s min = min j ∈S ∗ { s j } ≥ , ∀ i ∈ V. (14) Further S min = { i ∈ V | s i = s min } . (15)As in any given iteration the estimated state of a node isobtained by one of the bullets in (11), at each t , we partition V into two sets defined below. Definition 1.
The set A ( t ) (ABF type nodes) comprises allnodes that use the first case in (11) to obtain ˆ x i ( t ) , i.e. in (10), ˆ x i ( t ) = ˜ x i ( t ) . Define the set of extraordinary nodes E ( t ) = V \ A ( t ) to be those that use the second case in (11). The next definition defines a (current) constraining node . Definition 2.
For i ∈ A ( t ) , if ˆ x i ( t ) = s i = x i then i is its owncurrent constraining, or constraining node at t . Otherwise theminimizing k (cid:54) = i in (10) used to find ˆ x i ( t ) , is i ’s constraining node at t . If i ∈ E ( t ) , then i is its own constraining node at t . The constraining node of i at t is said to constrain i at t. III. C
HARACTERIZING S TATIONARY P OINTS
This section characterizes the stationary point of (10-12)which as explained in Section II is identical to the stationarypoint of (1) given in (5). Observe that these comprise two sortsof values. Those where x i = s i . Those where x i < s i . Wecall the former sources and their set is defined as S ∞ = { i | x i = s i } . (16)Evidently x i = s i i ∈ S ∞ min k ∈N ( i ) { f ( x k , e ik ) } i / ∈ S ∞ (17)As shown by example in Section II-A, not all members of S ∗ are sources. To establish the existence of at least one stationarypoint we make a definition. Definition 3.
As the graph is connected, there is a path fromevery node to every other node. Define P ji to be the set ofall paths from j to i , including j = i . Denote such a path P ∈ P ji , e.g. l → l → · · · , → l L = i , by the ordered set P = { j = l , l , · · · , l L = i } . In particular the path from i to i , will be the solitary node: P ii = . (18) Consider the recursion, x ∗ l k ( P ) = (cid:40) s l k k = 0 f ( x ∗ l k − ( P ) , e l k − l k ) k ∈ { , · · · , L } . (19) Define x ji as the smallest value x ∗ i ( P ) can have among allthe paths from j to i , i.e. x ji = min P∈P ji { x ∗ i ( P ) } . (20) Further define ¯ x i = min j ∈ V { x ji } . (21)This sequence mimics the second case of (17) sans mini-mization. The ¯ x i are uniquely determined by the structure ofthe graph and will provide a characterization of S ∞ and the x i . Key points stemming from the fact that f ( a, b ) > a are:(i) Only sequences that commence at a node with finitemaximum values yield finite x ∗ k , i.e. x ji = ∞ iff j / ∈ S ∗ . (ii) One has x ij = s i iff j = i and further ¯ x i = s i , iff x ji ≥ s i , ∀ j (cid:54) = i. (22)(iii) Given an ¯ x i (cid:54) = s i there is a j (cid:54) = i , P i ∈ {P ji } and k ∈ N ( i ) , the penultimate node in P i such that ¯ x i = f ( x ∗ k ( P i ) , e ik ) . (23)Either for this k, ¯ x k = s k or there are m , P k ∈ {P mk } and l ∈ N ( k ) , the penultimate node in P k such that ¯ x k = f ( x ∗ l ( P k ) , e lk ) . (24)4iv) Because f ( · , · ) is progressive, in the sequence (19), x ∗ l k ( P ) > x ∗ l k − ( P ) .The next lemma concerns the scenario in (iii). Lemma 1.
Under assumption 1 consider the quantities definedin (iii) above. Suppose ¯ x i (cid:54) = s i and ¯ x k (cid:54) = s k . Then i / ∈ P k . Proof.
To establish a contradiction suppose i ∈ P k . Thenbecause of (19-21) ¯ x i ≤ x ∗ i ( P k ) < ¯ x k ≤ x ∗ k ( P i ) . On the otherhand as k ∈ P i , ¯ x i > x ∗ k ( P i ) leading to a contradiction. (cid:4) We now show that ¯ x i obey a recursion like (5), thus provingthat they represent a stationary point. Lemma 2.
Under Assumption 1, then ¯ x i in Definition 3 obeys: ¯ x i = min (cid:26) min k ∈N ( i ) f (¯ x k , e ik ) , s i (cid:27) , ∀ i ∈ V. (25) Proof.
From Definition 3, in particular, (20) and (22) and therecursion in (19), and the fact that s i = x ii there holds: ¯ x i = min (cid:26) min j ∈ V \{ i } { x ji } , s i (cid:27) = min (cid:26) min j ∈ V \{ i } (cid:26) min P∈P ji { x ∗ i ( P ) } (cid:27) , s i (cid:27) = min (cid:26) min j ∈ V \{ i } (cid:26) min P∈P ji (cid:26) min k ∈N ( i ) { f ( x ∗ k ( P ) , e ki ) } (cid:27)(cid:27) , s i (cid:27) . (26)The nature of the recursion in (19) ensures that for every k ∈N ( i ) and P ∈ P ji there is a ¯ P ∈ P jk such that in (26), theminimizing x ∗ k ( P ) equals the minimizing x ∗ k ( ¯ P ) . Thus, ¯ x i = min (cid:26) min j ∈ V \{ i } (cid:26) min P∈P jk (cid:26) min k ∈N ( i ) { f ( x ∗ k ( P ) , e ki ) } (cid:27)(cid:27) , s i (cid:27) . As from Lemma 1, the minimizing path P jk cannot include i, ¯ x i = min (cid:26) min j ∈ V (cid:26) min P∈P jk (cid:26) min k ∈N ( i ) { f ( x ∗ k ( P ) , e ki ) } (cid:27)(cid:27) , s i (cid:27) . As f ( · , · ) is monotonically increasing in the first argument,from (20) and (22), (25) is proved by ¯ x i = min (cid:26) min k ∈N ( i ) (cid:26) f (cid:18) min j ∈ V, P∈P jk { x ∗ k ( P ) } , e ki (cid:19)(cid:27) , s i (cid:27) = min (cid:26) min k ∈N ( i ) { f (¯ x k , e ki ) } , s i (cid:27) . (27) (cid:4) Thus we have established the existence of at least one sta-tionary point. To establish its uniqueness we make a definition.
Definition 4.
In (17), if x i = s i , then we say that i is itsown true constraining node. Otherwise, any minimizing k inthe second bullet of (17) is a true constraining node of i . As i may have more than one true constraining node, its set oftrue constraining nodes is designated as C ( i ) . As f ( · , · ) is progressive we have that, x i > x k , ∀ k ∈ C ( i ) and i / ∈ S ∞ . (28)The following lemma catalogs some crucial properties of trueconstraining nodes and their implications to stationary points. Lemma 3.
Consider x = [ x , · · · , x n ] T whose elements obey(5). Then the following hold under Assumption 1. (A) Considerany sequence of nodes, without loss of generality { , , · · · , l } such that i + 1 ∈ C ( i ) as defined in Definition 4. Then thissequence is finite and its last element is in S ∞ , defined in(16). (B) The set S ∞ is nonempty. (C) The set S min ⊂ S ∞ . (D) All x i are finite.Proof. Due to (28) the chain in (A) cannot have cycles. Asthere are only N nodes it must end, and the last element l mustbe its own true constraining node i.e. l ∈ S ∞ . This proves (A),and also (B). Without loss of generality suppose s = s min .To establish a contradiction, suppose / ∈ S ∞ . Then from (A)there is a sequence of nodes starting from 1 and terminatingin j ∈ S ∞ , such that each is the true constraining node of itspredecessor. Thus from (28) x j = s j < s = s min , violatingthe definition of s min , proving (C). To prove (D) consider i (cid:54) = 1 . As the graph is connected there is a path from to i , comprising nodes { l → l → · · · l k = i. } Then from(17) for each n ∈ { , · · · , k } there holds x l n ≤ f ( x l n − , e l n ,l n − ) . Due to the fact that f ( a, b ) is finite for finite a, b, x l n is finiteif x l n − is finite. The result follows as x is finite. (cid:4) We make another definition for proving uniqueness of thestationary point and convergence of the algorithm.
11 1
Fig. 4. Illustration of graph where S ∞ is not a subset of F . Here s = 0 , s = 1 and s = ∞ . All edge lengths are and f ( a, b ) = a + b. In thiscase x = 0 , x = 1 and x = 1 . Here ∈ S ∞ as x = s . However, as x = x + 1 , ∈ F . Definition 5.
We call a path from a node i to j ∈ S ∞ ashortest path, if it starts at i , ends with j ∈ S ∞ , and eachnode in the path is a true constraining node of its predecessor.We call a shortest path from i the longest shortest path if ithas the most nodes among all shortest paths of i . The set F i is the set of nodes whose longest shortest paths to the sourceset have i + 1 nodes. We call D ( G ) the effective diameter of G if the longest shortest path among all i ∈ V has D ( G ) nodes. From Lemma 3, the effective diameter is always finite. Ifa node i has two shortest paths, one with two and the otherwith three nodes, then i / ∈ F but i ∈ F . It is temptingto believe that F = S ∞ . However, the scenario of Figure 4provides a counterexample. In this case s = 0 , s = 1 and s = ∞ . All edge lengths are and f ( a, b ) = a + b. In thiscase x = 0 , x = 1 and x = 1 . Here ∈ S ∞ as x = s . However, as x = x + 1 , ∈ F . The following lemma exposes a key property of the sets F i . emma 4. Under the conditions of Lemma 3, consider F i given in Definition 5. If for some k ∈ { , · · · , D ( G ) − } , F k is nonempty then every node in F k has a true constrainingnode in F k − . Further S min ⊂ F ⊂ S ∞ . Proof.
Consider any i ∈ F k . From Definition 5, startingfrom i there is a sequence containing k + 1 nodes to a j ∈ S ∞ in which each node is the true constraining nodeof its predecessor. Suppose the second node in this sequenceis l . By definition l is a true constraining node of i. Also bydefinition l ∈ F m , where m ≥ k − . If m > k − , then forsome M > k , i ∈ F M . This contradicts the assumption that i ∈ F k . Thus l ∈ F k − . By definition, every node in F isits own true constraining node as otherwise it will belong tosome F i , i > . Thus from Definition 4, F ⊂ S ∞ . Finally consider j ∈ S min . By definition s j = s min ≤ s i for all i. If j ∈ F k , with k > , then there is a sequencestarting from j to an l ∈ S ∞ , such that each node is thetrue constraining of its predecessor. Thus from the progressiveproperty of f ( · , · ) , s min = s j ≥ x j > s l , establishing acontradiction. Thus j ∈ F and S min ⊂ F . (cid:4) Lemma 5.
Under the conditions of Lemma 3, with F i definedin Definition 5, F i (cid:54) = ∅ , ∀ i ∈ { , , · · · , D ( G ) − } , (29) and for all i ∈ { , , · · · , D ( G ) − } each node in F i +1 hasa true constraining node in F i .Proof. We first show by induction that for each k ∈{ , · · · , D ( G ) − } , F k is nonempty. From Definition 5, F D ( G ) − (cid:54) = ∅ , initiating the induction. Now suppose for some L ∈ { , · · · , D ( G ) − } , F L (cid:54) = ∅ . Then from Lemma 4 everymember of F L has a true constraining node in F L − , making F L − (cid:54) = ∅ . Further, again from Lemma 4 every member of F has a true constraining node in F making the latter nonempty.Then Lemma 4 proves the result. (cid:4) We can now prove the uniqueness of the stationary point.
Theorem 1.
Under the conditions of Lemma 3, x i = ¯ x i , defined in Definition 3, represents the unique stationary pointobeying (5). Further the source set is given by: S ∞ = { i ∈ V | ¯ x i = s i } . (30) Proof.
From Lemma 2 x i = ¯ x i is a stationary point and from(16), (30) is the corresponding source set.Call ¯ x = [¯ x , · · · , ¯ x n ] T and consider a potentially differentstationary point x = [ x , · · · , x n ] T . As constraining nodes,source sets and the sets F i depend on the stationary point,in this proof we will distinguish them with the additionalargument of the stationary point, e.g. C ( i, x ) .We first assert that for all i, x i ≥ ¯ x i . To establish acontradiction suppose for some x i < ¯ x i . From Lemma 3,there is a j ∈ S ∞ ( x ) ⊂ S ∗ and a sequence of nodes i = l → · · · → l L = j such that x l k +1 = f (cid:0) x l k , e l k ,l k +1 (cid:1) . (31)From Definition 3 this means x ij ≤ x i < ¯ x i , violating the definition of ¯ x i . Thus indeed x i ≥ ¯ x i . As from Lemma 4 for all j ∈ { , · · · , D ( G ) − } , F j (¯ x ) (cid:54) = ∅ , we use induction to show that x i = ¯ x i , for all i ∈ F j (¯ x ) . Consider any i ∈ F (¯ x ) . As F (¯ x ) ⊂ S ∞ (¯ x ) , ¯ x i = s i . As bydefinition, s i ≥ x i ≥ ¯ x i = s i , one must have x i = s i . To sustain the induction assume that for some ≤ L < D ( G ) − , x k = ¯ x k , for all k ∈ F L (¯ x ) . To establish acontradiction suppose for some i ∈ F L +1 (¯ x ) , x i (cid:54) = ¯ x i . ByLemma 5 there is a k ∈ {C ( i, ¯ x ) (cid:84) F L (¯ x ) } , By the inductionhypothesis, ¯ x k = x k . Then as k is a neighbor of i , from (17) x i = min l ∈N ( i ) { f ( x l , e il ) } ≤ f ( x k , e ik ) = f (¯ x k , e ik ) = ¯ x i . (32)As x i ≥ ¯ x i , one obtains, x i = ¯ x i . (cid:4) Thus we have characterized the stationary point given by(5) and proved its uniqueness.IV. G
LOBAL UNIFORM ASYMPTOTIC STABILITY
Having established the existence and uniqueness of thestationary point x i , in this section, we prove that the stateestimates ˆ x i ( t ) yielded by (9)-(12), globally, uniformly con-verge to these x i , in graphs without perturbations i.e. when e ik and s i do not change. The key steps of the proof are:(a) We show that all underestimates are eventually elimi-nated, i.e. for all i, and sufficiently large t, ˆ x i ( t ) ≥ x i . This is done by using the progressive property of f ( · , · ) and the second case of (11), which causes ˆ x i to increase.(b) We show that once underestimates are eliminated, themoment a source node i ∈ S ∞ ⊃ F invokes thefirst bullet of (11), i.e. lies in A ( t ) , it converges to s i . Similarly if at a time t (cid:48) and beyond, F to F L definedin Definition 5 have converged, if i ∈ F L +1 invokes thefirst case of (11) then ˆ x i ( t ) converges to x i . (c) We then show that over every finite interval, the firstbullet of (11) must be invoked at least once as eachinvocation of the second case of (11) by i increases ˆ x i by δ, . Thus the repeated use of the second bullet by i will eventually induce ˆ x i > M , forcing i to use the firstbullet of (11).(d) To show (a) we define two time varying sets that partition V . The first, R ( t ) , the set of nodes rooted to sources,comprises elements of S ( t ) , the source set at time t ,and all nodes constrained by elements of R ( t − . Weshow that for all i ∈ R ( t ) , ˆ x i ( t ) ≥ x i . Nodes in thesecond unrooted set U ( t ) must also eventually have nostate estimates that are underestimates. A. Key lemmas underlying (b,c)
This section is dedicated to key lemmas that underlie (b)and (c). The first lemma provides and proves a key mechanismbehind (b). Specifically, should after a time t no neighbor ofa node i ever carry underestimates and its true constrainingnode converges, then for all t > t ˜ x i ( t ) in (10) equals x i . Lemma 6.
Consider (10) and (11) under Assumption 1 anda node i and a time t such that the following hold for all ≥ t . If i ∈ S ∞ , ˆ x j ( t ) ≥ x j for all j ∈ N ( i ) . If i / ∈ S ∞ ,(i) m ˆ x j ( t ) ≥ x j for all j ∈ N ( i ) ; and (ii) with k a trueconstraining node of i, ˆ x k ( t ) = x k . Then for all t > t , ˜ x i ( t ) defined in (10) equals x i . Proof.
Suppose i ∈ S ∞ , then i is its own true constrainingnode. Then from Definition 2 and (5), and the fact that f ( a, b ) is strictly increasing in a , from (10) there holds for all t ≥ t ˜ x i ( t + 1) = min (cid:26) min j ∈N ( i ) { f (ˆ x j ( t ) , e ij ) } , s i (cid:27) = min (cid:26) min j ∈N ( i ) { f ( x j , e ij ) } , s i (cid:27) = s i = x i . If i / ∈ S ∞ then from Definition 2 and (ii) x i = min j ∈N ( i ) { f ( x j , e ij ) } = f ( x k , e ik )= f (ˆ x k ( t ) , e ik ) , ∀ t ≥ t . Further from (i) min j ∈N ( i ) { f ( x j , e ij ) } ≤ min j ∈N ( i ) { f (ˆ x j ( t ) , e ij ) } ∀ t ≥ t . As k ∈ N ( i ) one thus has that min j ∈N ( i ) { f (ˆ x j ( t ) , e ij ) } = f ( x k , e ik ) , = x i t ≥ t . By definition i / ∈ S ∞ implies that x i < s i . Thus from (10)for all t ≥ t , ˜ x i ( t + 1) equals min (cid:26) min j ∈N ( i ) { f (ˆ x j ( t ) , e ij ) } , s i (cid:27) = min { x i , s i } = x i . (cid:4) In view of Definition 1 under the conditions of Lemma 6,if at any t > t , i ∈ A ( t ) (cid:84) S ∞ , then ˆ x i ( t ) = ˜ x i ( t ) = s i . Thenext lemma shows that if after t defined in Lemma 6, i everenters A ( t ) then ˆ x i ( t ) converges immediately to x i . Lemma 7.
Consider (9-12). Suppose the conditions of Lemma6 hold, and for some t > t , i ∈ A ( t ) defined in Definition1. Then for all t ≥ t , ˆ x i ( t ) = x i . Proof.
In view of Lemma 6, we need to show that for all t ≥ t , i ∈ A ( t ) . Use induction. By hypothesis, i ∈ A ( t ) . Now suppose for some t ≥ t > t , i ∈ A ( t ) . Then fromDefinition 1 and Lemma 6, ˆ x i ( t ) = ˜ x i ( t ) = x i . Further, alsofrom Lemma 6, ˜ x i ( t + 1) = x i = ˆ x i ( t ) . Thus from (10) andthe first bullet of (11), ˆ x i ( t + 1) = ˜ x i ( t + 1) , i.e. i ∈ A ( t + 1) . (cid:4) Thus under the conditions of Lemma 6 all it takes for ˆ x i ( t ) to converge after t is for i to enter the ABF set. The nextlemma bounds the time, described in (c), for this to happen. Lemma 8.
Under the conditions of Lemma 6, consider (9)-(12). Suppose ˆ x i ( t ) = m i for some t > t . Define t = t + 1 + min (cid:26)(cid:24) M − m i δ (cid:25) , (cid:27) . Then for all t > t , ˆ x i ( t ) = x i . Proof.
Suppose ˆ x i ( t ) (cid:54) = x i . Then from Lemma 7, i ∈ E ( t ) .Now suppose i ∈ E ( t ) for all t ≤ t ≤ t (cid:48) . In this case from the second bullet of (11), t (cid:48) ≤ t . From the first bullet of (12) thismeans i ∈ A ( t + 1) . Then Lemma 7 proves the result. (cid:4)
So if no neighbor of a non-source i carries an underestimateand at least one of its true constraining nodes has converged,then ˆ x i ( t ) converges the moment i enters A ( t ) , which it mustin a time quantified in Lemma 8. The same is true if thestates of all neighbors of a source i have exceeded s i , albeitunder a weaker condition. The next subsection proves a keyproperty that facilitates this convergence: the eventual removalof underestimates noted in (a) at the start of this section. B. Disappearance of underestimates
We first define the two time varying sets U ( t ) and R ( t ) mentioned in (d) at the beginning of the section. Definition 6.
The set of nodes rooted to sources is R ( t +1) = S ( t + 1) (cid:83) P ( t + 1) with S ( t + 1) as in (13) and P ( t + 1) comprising those whose constraining node at t + 1 is in R ( t ) .Further R ( t ) = S ( t ) . The unrooted set is U ( t ) = V \ S ( t ) . Evidently, U ( t + 1) (cid:84) S ( t + 1) = ∅ . As every node musthave a constraining node at every t , and members of R ( t + 1) are either in S ( t + 1) or are constrained by members of R ( t ) ,each member of U ( t + 1) must be constrained at time t + 1 by one of U ( t ) . Thus U ( t ) = ∅ = ⇒ U ( t + 1) = ∅ . (33)Sets in definitions 1 and 6 are exemplified through GABF inFigure 3. In this case, S (0) = { A, E } as ˆ x i (0) = 0 = s i for i = A or E . At t = 1 , ˜ x B (1) = ˆ x A (0)+ e AB = ˆ x B (0) = 1 , as D = 0 , node B will take A as the current constraining nodeand use the first bullet of (11) to update its estimate, leadingto B ∈ A (1) ∩ R (1) . Meanwhile, as ˜ x D (1) = ˆ x E (0) + e DE =1 (cid:54) = ˆ x (0) , D = 0 and ˆ x D (0) < M , node D will update itsestimate using the second bullet of (11) and take itself as thecurrent constraining node, then D ∈ E (1) ∩ U (1) .We will now show that underestimates in U ( t ) must even-tually disappear. To this end define ˆ x min ( t ) = min j ∈U ( t ) { ˆ x j ( t ) } if U ( t ) (cid:54) = ∅ , (34)Define: x max = max k ∈ V { x k } , (35)and T = (cid:24) x max − ˆ x min ( t )min { σ, δ } (cid:25) . (36) Lemma 9.
Consider (9)-(11) under Assumption 1, with U ( t ) , ˆ x min ( t ) , x max and T defined in Definition 1, (34), (35) and(36), respectively. Then (37) and (38) hold while U ( t ) (cid:54) = ∅ : ˆ x i ( t ) ≥ ˆ x min ( t ) + min { σ, δ } ( t − t ) , ∀ i ∈ U ( t ) (37) ˆ x i ( t ) ≥ x max ≥ x i , ∀ i ∈ U ( t ) and ∀ t ≥ t + T. (38) Proof.
Because of (33), U ( t ) is nonempty only on a singlecontiguous time interval commencing at t . Thus, from (35)and (36), (38) will hold if (37) holds.We prove (37) by induction in t ≥ t . It clearly holds for t = t . Thus suppose it holds at some t ≥ t . If U ( t + 1)
7s empty then it remains so for all future values. So assume U ( t + 1) (cid:54) = ∅ i.e. U ( t ) (cid:54) = ∅ . Suppose i ∈ U ( t + 1) is suchthat ˆ x i ( t + 1) = ˆ x min ( t + 1) . From the remark after Definition6, j the current constraining node of i is in U ( t ) . Suppose i ∈ E ( t + 1) defined in Definition 1, then from Definition 2, j = i . The induction hypothesis and (12) yield: ˆ x i ( t + 1) = ˆ x min ( t + 1) ≥ ˆ x j ( t ) + δ ≥ ˆ x min ( t ) + min { σ, δ } (39) ≥ ˆ x min ( t ) + min { σ, δ } ( t + 1 − t ) . If i ∈ A ( t + 1) , then, i / ∈ S ( t + 1) , i.e. ˆ x i ( t + 1) (cid:54) = s i . Thus, ˆ x i ( t + 1) =ˆ x min ( t + 1) = f (ˆ x j ( t ) , e ij ) ≥ ˆ x j ( t ) + σ ≥ min { σ, δ } ( t + 1 − t ) . (cid:4) We now show that after t + T all ˆ x i ( t ) are overestimates. Lemma 10.
Under the conditions of Lemma 9, ˆ x i ( t ) ≥ x i , ∀ i ∈ V, and t ≥ T + t . (40) Proof.
We will first show by induction that whenever R ( t ) given in Definition 6 is nonempty, ˆ x i ( t ) ≥ x i for all i ∈ R ( t ) . Then as U ( t ) = V \R ( t ) , the result will follow from Lemma 9.If R ( t (cid:48) ) (cid:54) = ∅ , then there is a t ≤ t ≤ t (cid:48) such that R ( t ) (cid:54) = ∅ ,for all t ≤ t ≤ t (cid:48) and R ( t ) = S ( t ) . Clearly by definition of S ( t ) , ˆ x i ( t ) = s i ≥ x i , for all i ∈ S ( t ) = R ( t ) . If t (cid:48) = t , then all elements of R ( t (cid:48) ) carry overestimates.If t (cid:48) > t then use induction on t ≤ t ≤ t (cid:48) . Suppose x i ≤ ˆ x i ( t ) for some t ≤ t < t (cid:48) , and all i ∈ R ( t ) . Considerany i ∈ R ( t +1) . Then from Definition 6, either i ∈ S ( t +1) , inwhich case the result holds, or j the current constraining nodeof i is in R ( t ) . Then by the induction hypothesis, ˆ x j ( t ) ≥ x j . As ˆ x i ( t +1) (cid:54) = s i , and f ( a, b ) is increasing in a, if i ∈ A ( t +1) ,there follows: ˆ x i ( t + 1) = f (ˆ x j ( t ) , e ij ) ≥ f ( x j , e ij ) ≥ x i . If i ∈ E ( t + 1) then it is its own true constraining node and i ∈ R ( t ) . Thus by the induction hypothesis, ˆ x i ( t ) ≥ x i . Thusfrom (12) ˆ x i ( t + 1) ≥ ˆ x i ( t ) + δ > x i . (cid:4) Thus we have established (a) described at the beginning ofthis section. In the next section we prove GUAS.
C. Proof of convergence
Define the smallest stationary value in F i as x i min = min j ∈F i { x j } . (41)From Lemma 5, we have x = s min . Define a sequence T i = max (cid:26) , (cid:24) M − x i min δ (cid:25)(cid:27) + 2 . (42)We then have the main result of this section, provingthe convergence of each ˆ x i ( t ) to x i . Specifically, we willshow by induction that with T defined in (36), for all i ∈{ , · · · , D ( G ) − } , the elements of F , · · · , F i , defined inDefinition 5, converge by the time T + (cid:80) ij =0 T j . Theorem 2.
Consider (9) - (11) under Assumption 1, with T i and T defined in (42) and (36), respectively. Then ∀ i ∈ V , ˆ x i ( t ) = x i , ∀ t > t + T + D ( G ) − (cid:88) i =0 T i . (43) Proof.
We will prove by induction that for every L ∈{ , · · · , D ( G ) − } , ˆ x i ( t ) = x i , ∀ t ≥ t + T + L (cid:88) j =0 T j and i ∈ L (cid:91) j =0 F j (44)Then the result will follow as the F j partition V. Consider i ∈ F and t > t + T + T . From Lemma 4, i ∈ S ∞ . As from Lemma 10, ˆ x k ( t ) ≥ x k , for all t > t + T and k ∈ V , i satisfies the conditions of Lemma 6 and thus ofLemma 8. Thus, from Lemma 8, (44) holds for L = 0 .Suppose (44) holds for some N ∈ { , · · · , D ( G ) − } .Consider i ∈ F N +1 . By Lemma 5, i has a true constrainingnode k ∈ F N . By the induction hypothesis ˆ x k ( t ) = x k for all t > t + T + (cid:80) Nj =0 T j , and ˆ x l ( t ) ≥ x l , for all l ∈ V . Thusfrom Lemma 10, this i satisfies the conditions of Lemma 6and thus of Lemma 8. Thus, from Lemma 8, (44) holds for L = N + 1 , completing the proof. (cid:4) In fact one can show that this theorem also holds with T = max (cid:26) , (cid:24) M − min { δ + s min , x max } δ (cid:25)(cid:27) + 2 , (45)as ˆ x i ( t + T ) ≥ min { δ + s min , x max } for all i ∈ F . The factthat the time elapsed between the initial time t and the timeto converge is independent of t proves GUAS.V. R OBUSTNESS UNDER PERTURBATIONS
In this section, we prove that (9) is ultimately boundedunder persistent perturbations in the e ij , albeit with someadditional assumptions. In particular, the dead zone parameter D must exceed a value proportional to the magnitude ofthe perturbation. Otherwise, with probability one ˆ x i ( t ) willpersistently rise to M. This value is provided in this section.Proofs of this section are in the appendix.The first additional assumption extends the monotonicityproperty to the second argument of f ( · , · ) . Given that thisargument represents edge lengths in most applications, this is areasonable assumption. As is also standard in most robustnessanalysis, we also impose a Lipschitz condition. Assumption 2.
The function f ( · , · ) is monotonically increas-ing with respect to its second argument, i.e. f ( a, b ) obeys f ( a, b ) ≥ f ( a, b ) , if b ≥ b . (46) Further, there exist L i > , such that | f ( a, b ) − f ( a, b ) | ≤ L | b − b | (47) and | f ( a , b ) − f ( a , b ) | ≤ L | a − a | (48)The perturbations on e ij are modeled as, e ij ( t ) = e ij + (cid:15) ij ( t ) (49)8ith | (cid:15) ij ( t ) | ≤ (cid:15) < e min , (50)where e min is defined in Assumption 1. Notice that theperturbations need not be symmetric, i.e. we permit (cid:15) ij ( t ) (cid:54) = (cid:15) ji ( t ) . (51)Such perturbations could reflect noise, localization error, or (ifcoherent) movement of devices. In this case, (10) becomes ˜ x i ( t + 1) = min (cid:26) min k ∈N ( i ) { f (ˆ x k ( t ) , e ik ( t )) } , s i (cid:27) . (52)We define a shrunken graph , for deriving bounds on underes-timates. It corresponds to the smallest possible values of e ij . Definition 7.
Given a graph G , define G − as a shrunkenversion of G such that, ∀ i ∈ V and j ∈ N ( i ) in G , e ij becomes e − ij in G − : With (cid:15) defined in (50) e − ij = e ij − (cid:15). (53) Also consider (9) implemented on this shrunken graph, i.e. ˆ X i ( t + 1) = F ( ˜ X i ( t + 1) , ˆ X i ( t ) , v i ) , ˆ X i (0) ≤ ˆ x i (0) . (54) with ˜ X i ( t + 1) obeying ˜ X i ( t + 1) = min (cid:26) min k ∈N ( i ) (cid:110) f (cid:16) ˆ X k ( t ) , e − ik (cid:17)(cid:111) , s i (cid:27) (55) As G − satisfies the same assumptions as G and is perturbationfree, we define X = [ X , · · · , X N ] as the unique stationarypoint in G − to which (54) converges. Further, S −∞ and D ( G − ) denote the source set and the effective diameter of G − . The unique stationary point in G − obeys X i = min (cid:26) min k ∈N ( i ) f (cid:0) X k , e − ik (cid:1) , s i (cid:27) , ∀ i ∈ V. (56)Specifically, from (16) and (17), the source set in G − obeys S −∞ = { i | X i = s i } , (57)and the stationary point obeys: X i = s i i ∈ S −∞ min k ∈N ( i ) { f ( X k , e − ik ) } i / ∈ S −∞ . (58)Evidently, the following holds in G − : s min ≤ X i ≤ s i , ∀ i ∈ V. (59)Define X max = max k ∈ V { X k } , (60)and T − = (cid:24) X max − ˆ x min ( t )min { δ, σ } (cid:25) (61)The next lemma shows that the lower bound in U ( t ) willexceed X max after t + T − . Lemma 11.
Consider (9), (11) and (52), with U ( t ) , ˆ x min ( t ) , X max and T − defined in Definition 6, (34), (60) and (61),respectively. Then (62) and (63) hold while the set U ( t ) (cid:54) = ∅ : ˆ x i ( t ) ≥ ˆ x min ( t ) + min { σ, δ } ( t − t ) , ∀ i ∈ U ( t ) (62) ˆ x i ( t ) ≥ X max ≥ X i , ∀ i ∈ U ( t ) and ∀ t ≥ t + T − (63)We now turn to R ( t ) and prove that under perturbations allestimates in R ( t ) are lower bounded by their correspondingstationary values in G − . Lemma 12.
Consider (9), (11) and (52), with A ( t ) , E ( t ) , R ( t ) and X i as in definitions 1, 6 and 7, respectively. There holds: ˆ x i ( t ) ≥ X i , ∀ i ∈ R ( t ) . (64)Consequently, with Lemma 11 and Lemma 12, there holds: ˆ x i ( t ) ≥ X i , ∀ i ∈ V, ∀ t ≥ t + T − . (65)By quantifying the relation between the stationary point in G − and that in G in the next lemma, we will show that after t + T − all estimates are lower bounded.To this end we define the following function: W ( L , D ) = D − (cid:88) i =0 L i = (cid:40) L D − L − L (cid:54) = 1 D L = 1 , (66)and the summation is zero if the lower limit exceeds the upper. Lemma 13.
Under Assumption 2, with (cid:15) , W ( · ) and D ( G − ) defined in (50), (66) and Definition 7, respectively. Then forall i ∈ V and t ≥ T − + t , there holds: x i ≤ X i + W ( L , D ( G − ) − L (cid:15). (67)With F i defined in Definition 5, define X i min as X i min = min j ∈F i { X j } (68)Note that S min is a subset of S −∞ as well as F , thus X = s min . Define a sequence T − i = max (cid:26) , (cid:24) M − X i min δ (cid:25)(cid:27) + 2 . (69)Then we have the following lemma that recognizes that tobehave acceptable under perturbations the dead zone D in(11) must be sufficiently large. Lemma 14.
Consider (9), (11) and (52) under Assumption 2,with (cid:15) , W ( · ) defined in (50), (66), respectively. Suppose D in(11) obeys D ≥ ( W ( L , D ( G − ) −
1) + W ( L , D ( G ) − L (cid:15) (70) and at a time t (cid:48) ≥ t + T − defined in (61), for some L ∈{ , , · · · , D ( G ) − } ˆ x i ( t ) ≤ x i + W ( L , L ) L (cid:15), ∀ i ∈ F L , ∀ t ≥ t (cid:48) . (71) Then with T − i define in (69), there holds: ˆ x i ( t ) ≤ x i + W ( L , L + 1) L (cid:15), ∀ i ∈ F L +1 , t ≥ t (cid:48) + T − L +1 . The next theorem proves that the algorithm is ultimatelybounded under bounded persistent perturbations and providesan upper bound on the time to attain the ultimate bound.
Theorem 3.
Under the conditions of Lemma 14, for all i ∈ V and t ≥ t + T − + (cid:80) D ( G ) − i =0 T − i , | ˆ x i ( t ) − x i | ≤ (cid:15)L max (cid:8) W ( L , D ( G ) − , W ( L , D ( G − ) − (cid:9) . (cid:15) , the magnitude of the disturbance. Define T − = max (cid:26) , (cid:24) M − min { δ + s min , X max } δ (cid:25)(cid:27) + 2 . (72)One can show that Theorem 3 holds for a tighter time boundif one uses T − defined in (72). This is so as one can provethat ˆ x i ( t + T − ) ≥ min { δ + s min , X max } for all i ∈ F .VI. D ESIGN CHOICES AND DISCUSSION
Theorem 3 verifies the intuitively clear requirement that thedead zone D should grow proportionally to the disturbancebound (cid:15). However, as this is a worst-case analysis, it masksthe full effects of parameters M , δ , and D. Looking beyondworst-case analysis, however, we can find that choosing theseparameters involves tradeoffs between the convergence speedof underestimates and overestimates.The convergence of underestimates is upper bounded by T in (36), which is in turn determined by (37), and thusconservatively by the smaller of σ and δ . In practice, if σ is small and the first bullet of raising is invoked too oftenthen underestimates rise slowly, i.e. the rising value problemwill persist. If the second bullet of (11) is invoked at mosttimes and δ (cid:29) σ then underestimates decline fast. Large D orsmall M makes this less likely and slows convergence, whilea large δ ≥ M speeds convergence by reducing T and T i . For the convergence of overestimates, T i gives the worstcase time to invoke the first case of (11), whereupon allelements in F i converge forthwith. The worst case analysisquantifies T i by how long it takes for ˆ x i ( t ) to exceed M andassumes that the second clause of (11) is invoked until thishappens. With a large D , however, this time shortens as thefirst bullet is likely to be invoked more quickly.In most cases, the need to alleviate the rising value problemis more compelling as overestimates in algorithms like plainABF converge in at most D ( G ) − steps. Accordingly, thedesirability of a smaller dead zone D competes with therequirement of resilience to persistent perturbations as quanti-fied by (70). This of course is common to most dynamicalsystems where faster convergence generally comes at theprice of reduced resilience. We note, however, the followingappealing fact: both the ultimate bound and the required D aredetermined exclusively by the perturbation magnitude (cid:15) andthe effective diameters of the original and shrunken graph. Complementarily, note that in the special case of the algo-rithm in [24], we effectively have M = 0 and D = ∞ . In thiscase the second bullet of (11) is never invoked. Accordingly,a small σ leads to large T and T − and the rising valueproblem. In particular, the algorithm remains GUAS withthe same ultimate bound as (70) is automatically satisfied.Overestimates however, converge quickly as T i = T − i = 2 . VII. S
IMULATIONS
In this section, we empirically confirm the results presentedin the prior sections through simulations. We first investi-gate the effect of parameters in the general spreading blockconfigured as GABF, then compare with the performance of (a) ∆ + ( t ) (b) ∆ − ( t ) Fig. 5. Convergence time for (a) the greatest overestimate ∆ + ( t ) and (b) theleast underestimate ∆ − ( t ) without perturbations, with M = 5 , D = 0 and δ is varying from . M to M in steps of . M . The solid line representsthe average value of 100 trials, the dotted and dashed lines represent upperand lower envelopes, respectively. In (b) solid and dotted lines of δ = 0 . M and δ = M overlap, and dashed lines of δ = 0 . M , δ = 0 . M , δ = 0 . M and δ = M overlap. ABF in the presence of persistent perturbations. Finally, weillustrate the applicability of the general spreading block tomore complex cases with an example of a non-Euclideandistance metric. Except where otherwise noted, all simulationsuse 500 nodes, randomly distributed in a 4 km × √ km (i.e., the longest possible distance for the simulated space). A. Effect of parameters
We begin with an empirical investigation of the designchoices and parameter effects discussed in Section VI, usingGABF, as defined in Section II-B, as an example to demon-strate the impact of these parameters on convergence speed.Progress toward convergence may be measured using thegreatest overestimate ∆ + ( t ) and least underestimate ∆ − ( t ) : ∆ + ( t ) = max (cid:104) , max i { ∆ i ( t ) } (cid:105) (73) ∆ − ( t ) = max (cid:104) , − min i { ∆ i ( t ) } (cid:105) . (74)where ∆ i ( t ) = ˆ d i ( t ) − d i the distance estimation error ofnode i . Then ∆ + ( t ) = ∆ − ( t ) = 0 indicates that all distanceestimates converge to their true distances at time t .We start with δ , which controls how quickly M is reached.Figure 5 shows the results of 100 runs using M = 5 , D = 0 ,and δ varying from . M to M in steps of . M . Theaverage D ( G ) is 18.8. With a fixed M , both ∆ + ( t ) and10 a) ∆ + ( t ) (b) ∆ − ( t ) Fig. 6. Convergence time for (a) the greatest overestimate ∆ + ( t ) and (b)the least underestimate ∆ − ( t ) without perturbations, with δ = M = 5 and D = 0 , . K, . K, K and K , using K = ( D ( G ) + D ( G − ) − (cid:15) . Thesolid line represents the average value of 100 trials, the dotted and dashedlines represent upper and lower envelopes, respectively. In (b) dashed lines of D = 0 , . K and . K overlap. ∆ − ( t ) converge slower with a smaller δ , since a smaller δ means estimates exceed M later. The corresponding averageconvergence times are 61.2, 54.1, 38.1, 32.6 and 24.1.The dead zone value D , on the other hand, has opposingeffects on ∆ + ( t ) and ∆ − ( t ) . Figure 6 shows the results of100 runs using δ = M = 5 and D = 0 , . K, . K, K and K , using K = ( D ( G ) + D ( G − ) − (cid:15) . In these simulations,the average value of (cid:15) is . × − km and the averagevalues of D ( G ) and D ( G − ) are 17.9 and 26.8, respectively.For ∆ + ( t ) , a large D (e.g., D = 4 K ) or a small D (e.g., D = 0 ) accelerates convergence. In the former case GABFacts more like ABF in which case D ( G ) will converge within D ( G ) − rounds [22]. In the latter case the second bulletof (11) will be frequently invoked such that underestimateswill be eliminated more quickly with a large M , and thus T defined in (36) becomes smaller. Both of these phenomena areseen in these simulations, with the average convergence timeof ∆ + ( t ) being 23.2, 67.9, 99.3, 34.0 and 6.5. As for ∆ − ( t ) ,A large D always has a negative impact on the convergencespeed of ∆ − ( t ) since the behavior of GABF is more like ABFin this situation, where the rising value problem [22], [24] willcause the underestimates rise very slowly. Here, the averageconvergence time is 2, 49.3, 310.0, 535.9 and 631.7. Note thatthese underestimates are more vulnerable to the change of D :while the time to convergence of ∆ + ( t ) is roughly 60 roundsfaster by increasing D from . K to K , that of ∆ − ( t ) maybe hundreds of rounds slower under such a change. Overallconvergence time is thus regulated by ∆ + ( t ) for small D andby ∆ − ( t ) for large D , with the joint average convergence time (a) ∆ + ( t ) (b) ∆ − ( t ) Fig. 7. Convergence time for (a) the greatest overestimate ∆ + ( t ) and (b) theleast underestimate ∆ − ( t ) without perturbations, with D = 0 , δ = M , and M varying from 4.108 to 4.124. The solid line represents the average value of100 trials, the dotted and dashed lines represent upper and lower envelopes,respectively. In (b) dashed lines of all different M overlap, and dotted linesof M = 4 . and . overlap. of 23.2, 67.9, 310.0, 535.9 and 631.7.Finally, the impact of M may be separated out from theother two parameters by setting D = 0 and δ = M . In thiscondition, small changes in M can result in large changesin convergence rate. Figure 7 illustrates this for 100 runs ofGABF, with D = 0 , δ = M , and M increasing from 4.108to 4.124 in steps of 0.004. In these simulations, the average D ( G ) is 18.5. Here, convergence unconditionally improveswith higher M : the average convergence times of ∆ + ( t ) are92.9, 72.6, 55.1, 32.0 and 23.7 rounds, while those of ∆ − ( t ) are 87.3, 63.0, 43.5, 14.4 and 2 rounds. Even though T i definedin (42) satisfies T i = 3 for all different M , as we have set δ = M , overestimates disappear more quickly with larger M because a larger δ helps the time T defined in (36), afterwhich all states are overestimates becoming smaller by (39)in Lemma 9. Underestimates converge more quickly once M is greater than both the largest true distance and initial distanceestimate. In this case, all nodes acquire overestimates in thefirst round by invoking the second bullet of (11), and thus theunderestimates converge in only 2 rounds. B. Robustness against persistent perturbations
As discussed in Section VI, robustness against perturbationshould be controlled primarily by parameter D . In particular, ( D ( G ) − (cid:15) and ( D ( G − ) − (cid:15) are the ultimate bounds of ∆ + ( t ) and ∆ − ( t ) under perturbations, respectively [22] (i.e., L W ( L , D ( G ) − and L W ( L , D ( G − ) − in Theorem14, respectively), and thus ( D ( G ) + D ( G − ) − (cid:15) is the11 -4 -2 (a) ∆ + ( t ) and ( D ( G ) − (cid:15) -5 -4 -3 -2 -1 (b) ∆ − ( t ) and ( D ( G − ) − (cid:15) Fig. 8. Robustness against perturbation for GABF with various values of D , compared over 100 trials with ultimate bounds and with ABF: (a) meanvalues of ∆ + ( t ) and mean value of ultimate bound ( D ( G ) − (cid:15) , and (b)mean values of ∆ − ( t ) and mean value of ultimate bound ( D ( G − ) − (cid:15) .Edge lengths are perturbed by measurement errors uniformly distributedbetween 0 and e min . Parameters for GABF are set as δ = M = √ , D = 0 , . K, . K, . K, . K, K and K , where K = ( D ( G ) + D ( G − ) − (cid:15) . In (a), lines of D = 0 , . K and . K overlap. minimum value of D to guarantee the robustness of GABFunder perturbations.Figure 8 illustrates this for 100 runs of GABF, comparingthis ultimate boundedness with ABF and with GABF using δ = M = √ and D = 0 , . K, . K, . K, . K, K or K , where K = ( D ( G ) + D ( G − ) − (cid:15) , (cid:15) = 0 . e min .Perturbation is injected as asymmetric noise in the estimated e ij , such that measurement errors (cid:15) ij ( t ) defined in (49) followa uniform distribution between 0 and . e min in each round.In these simulations, the average value of e min defined in (50)is . × − km, D ( G ) = 18 . and D ( G − ) = 20 . on average.The results show the tradeoffs in ∆ + ( t ) versus ∆ − ( t ) withGABF. For ∆ − ( t ) , ABF is constrained by the rising valueproblem [22], [24] such that ∆ − ( t ) needs a much longer timethan ∆ + ( t ) to drop below than its ultimate bound. With GABF,lower values of D increase the speed of convergence, with D = 0 achieving the fastest time. For ∆ + ( t ) , on the otherhand, ABF converges extremely quickly, while GABF does notconverge at all for low values of D . In this case, GABF with D = 0 , . K, . K, . K and . K will not be ultimatelybounded, while the average time for ABF and GBAF with D = K and K to drop below the ultimate bounds follows859.3, 196.5 and 300.1 rounds. Further, the average time forABF and GABF with D = K or K to reach the bottomis 5750 and 2406 rounds, respectively. Combining both, wefind that GABF outperforms ABF under perturbations whenthe dead zone value D is equal to or slightly larger thanthe minimum required value defined in (70). Thus, when the (a) The spreading block of (1)(b) The more general spreading block of (9)-(11)Fig. 9. In this example, 400 nodes are randomly located in a × field. There is a source at the red asterisk located at (0.3, 0.3), and themiddle of the field is a . × . radiation zone. Color representsdegree of contamination, with a logarithmic scale. While both spreadingblock and the general spreading block can achieve the shortest availablepath, the fast convergence of the general spreading block greatly reducestotal contamination. general spreading block is under perturbation, D should beset as ( W ( L , D ( G − ) −
1) + W ( L , D ( G ) − L (cid:15) definedin (70) of Lemma 14 in order to guarantee robustness andmeanwhile attain a fast convergence speed. Observe thoughthe floor is much below the theoretical ultimate bound, andeven with D = . K , ∆ + ( t ) though persistently rising fromthe floor rises only up to the unltimate bound. C. Non-Euclidean Distance
Finally, we illustrate how the general spreading block canaccommodate non-Euclidean distance metrics. Figure 9 showsexample of a nonlinear f ( · , · ) in (1), for nodes to computepaths minimizing exposure to a hazard. In this scenario, 500nodes are randomly distributed in a × field, andcommunicate over a 0.6 km radius. A source is located at (0.3,0.3). In the middle of the area, there exists a . × . radiation zone centered at (1.95, 1.95). Define M as the set ofnodes in the radiation zone, a node i is radioactive if i ∈ M or i has ever taken a radioactive node as its constraining node. f (ˆ x k ( t ) , e ik ) in (1) obeys f (ˆ x k ( t ) , e ik ) = ˆ x k ( t )+ e ik , k / ∈ M ,where e ik is the edge length between node i and k . When k ∈M , f (ˆ x k ( t ) , e ik ) = h (ˆ x k ( t ) + 1000 e ik ) , where h ( a ) = a . a > and a otherwise. Further, s i defined in (1) is 0 if i is a source and ∞ otherwise. In each round a node i willreceive 100 ∼
120 units of radiation dose if it is radioactiveand 0 ∼ D = 0 , δ = M > x max in the generalspreading block. In both cases, nodes outside the radiationzone never cross the zone due to the high cost, and nodesinside the zone take the shortest path to exit the zone. However,the degree of contamination is greatly reduced when using thegeneral spreading block with appropriately chosen parameters,due to the much faster time of convergence to a safe path.VIII. C
ONCLUSION
We have improved a general algorithm for spreading in-formation across a network of devices by making it resilientto perturbations and by removing a prior boundedness as-sumption. This algorithm, a key building block for aggregatecomputing and applicable to a wide range of distributedsystems, has parameters that remove the rising value problemthat appears in some of its special cases, such as ABF. UnlikeABF, however, the general spreading algorithm covers a muchwider class of uses and application, such as dealing with non-Euclidean distance metrics. We have proven global uniformasymptotic stability for this algorithm and provide ultimatebounds in face of persistent network disturbances using anadditional Lipschitz condition. Notably, the ultimate boundsdepend only on the largest perturbation and structural networkproperties. Finally, we provide design guidelines for the threenew parameters, demonstrating how algorithm parametershave competing effects on performance.These results are a crucial stepping stone in our long termgoal of determining stability conditions for feedback intercon-nections of aggregate computing blocks, using possibly newsmall gain theorems, [20], or equivalent techniques, [27], likethe passivity theorem and its variants, [28]. Progress in thisprogram has broad applicability for the engineering of resilientdistributed systems. R
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Proof of Lemma 11:
From Definition 6, (33) still holds if(10) is replaced by (52). Thus U ( t ) can be nonempty only ona single contiguous time interval commencing at t .We prove (62) by induction. As it holds for t = t , suppose(62) holds for some t ≥ t . Consider i ∈ U ( t + 1) with ˆ x i ( t +1) = ˆ x min ( t +1) . Then U ( t ) (cid:54) = ∅ and j the constraining13ode of i is in U ( t ) . If i ∈ E ( t + 1) in Definition 1 then fromDefinition 2, j = i . From the induction hypothesis and (12) ˆ x i ( t + 1) = ˆ x min ( t + 1) ≥ ˆ x j ( t ) + δ ≥ ˆ x min ( t ) + min { σ, δ }≥ ˆ x min ( t ) + min { σ, δ } ( t + 1 − t ) . If i ∈ A ( t +1) ∩U ( t +1) , then, i / ∈ S ( t +1) , i.e. ˆ x i ( t +1) (cid:54) = s i . From the induction hypothesis and (52) we have ˆ x i ( t + 1) =ˆ x min ( t + 1) = f (ˆ x j ( t ) , e ij ( t )) ≥ ˆ x j ( t ) + σ ≥ min { σ, δ } ( t + 1 − t ) . Further with (60) and (61), (63) follows.
Proof of Lemma 12: If R ( t ) (cid:54) = ∅ , then ∃ t , t such that ∀ t ≤ t ≤ t ≤ t and R ( t ) = S ( t ) . As ˆ x i ( t ) = s i for all i ∈ S ( t ) = R ( t ) , from (59) the result holds for t = t .Suppose ˆ x i ( t ) ≥ X i for some t ≤ t < t and all i ∈ R ( t ) .Consider any i ∈ R ( t + 1) . From Definition 6, either i ∈S ( t + 1) in which case the rsult holds, or or i is constrainedby some j ∈ R ( t ) . If i ∈ A ( t + 1) , then by the inductionhypothesis, ˆ x j ( t ) ≥ X j . As i / ∈ S ( t + 1) , there follows: ˆ x i ( t + 1) = f (ˆ x j ( t ) , e ij ( t )) ≥ f ( X j , e − ij ) (75) ≥ X i (76)where (75) uses e ij ( t ) ≥ e − ij for all t and the fact that f ( · , · ) isincreasing in each argument, and (76) uses (58). If i ∈ E ( t +1) ,then i is its own constraining node and i ∈ R ( t ) . Thus by ourinduction hypothesis, ˆ x i ( t ) ≥ X i . From (12), ˆ x i ( t + 1) ≥ ˆ x i ( t ) + δ > X i . (77) Proof of Lemma 13:
Consider nodes n , n , · · · , n T suchthat n ∈ S −∞ , and for all i ∈ { , . . . , T − } , n i is a trueconstraining node of n i +1 in G − . Each node in G − is in onesuch sequence. As from Definition 7, T ≤ D ( G − ) − , theresult will follow if x n i − X n i ≤ W ( L , i ) L (cid:15), ∀ i ∈ { , · · · , T } . (78)Evidently, x n ≤ s n = X n . Suppose (78) holds for some i ∈ { , · · · , T − } . As n i and n i +1 are neighbors in both G and G − , n i is a true constraining node of n i +1 in G − and x n i ≤ X n i + W ( L , i ) L (cid:15) by our induction hypothesis, x n i +1 ≤ f ( x n i , e n i n i +1 ) ≤ f ( X n i + W ( L , i ) L (cid:15), e n i n i +1 ) ≤ f ( X n i , e n i n i +1 ) + L W ( L , i ) L (cid:15) (79) = f ( X n i , e − n i n i +1 + (cid:15) ) + L W ( L , i ) L (cid:15) (80) ≤ f ( X n i , e − n i n i +1 ) + L (cid:15) + L W ( L , i ) L (cid:15) (81) = X n i +1 + W ( L , i + 1) L (cid:15) (82)where (79) uses (48), (80) uses (53), (81) uses (47), and (82)uses the fact that n i is a true constraining node of n i +1 in G − . Proof of Lemma 14:
Consider any i ∈ F L +1 . Because of (11)and (12), there is a t (cid:48) < t ≤ t (cid:48) + T − L +1 such that i ∈ A ( t ) .This is so as i ∈ E ( t ) implies ˆ x i ( t + 1) ≥ ˆ x i ( t ) + δ and atsome time in the interval ( t (cid:48) , t (cid:48) + T − L +1 ] , ˆ x i ( · ) > M . From Lemma 5, there is a j ∈ F L that is a true constraining nodeof i in G . Thus ˆ x j ( t − ≤ x j + W ( L , L ) L (cid:15) by (71). Then ˆ x i ( t ) = min (cid:26) min k ∈N ( i ) { f (ˆ x k ( t − , e ik ( t − } , s i (cid:27) ≤ f (ˆ x j ( t − , e ij ( t − ≤ f ( x j + W ( L , L ) L (cid:15), e ij + (cid:15) ) (83) ≤ f ( x j , e ij ) + L W ( L , L ) L (cid:15) + L (cid:15) (84) = x i + W ( L , L + 1) L (cid:15) (85)where (83) uses (49), (50) and (71), (84) uses (47) and (48).Similarly, as (71) holds for all t ≥ T − + t for all j ∈ F L , ˜ x ( t + 1) ≤ x i + W ( L , L + 1) L (cid:15). (86)As t > t + T − , (65) implies that ˆ x k ( t ) ≥ X k for all k ∈ V .As f ( · , · ) is monotonically increasing in both its argumentsand X i ≤ s i , (56) implies that ˜ x i ( t + 1) = min (cid:26) min k ∈N ( i ) { f (ˆ x k ( t ) , e ik ( t )) } , s i (cid:27) ≥ min (cid:26) min k ∈N ( i ) (cid:8) f (cid:0) X k , e − ik (cid:1)(cid:9) , s i (cid:27) = X i , i.e. [ X i , x i + W ( L , L + 1) L (cid:15) ] contains both ˆ x i ( t ) and ˜ x i ( t + 1) . Then (70) and Lemma 13 yield | ˜ x i ( t + 1) − ˆ x i ( t ) | ≤ | x i + W ( L , L + 1) L (cid:15) − X i |≤ W ( L , D ( G − ) − L (cid:15) + W ( L , L + 1) L (cid:15) ≤ D, i.e, ˆ x i ( t + 1) = ˜ x i ( t + 1) . An induction proves the result. Proof of Theorem 3:
From Lemma 13 and (65) ˆ x i ( t ) − x i ≥ − W ( L , D ( G − ) − L (cid:15) ∀ t ≥ t + T − , (87)proving the lower bound on ˆ x i ( t ) − x i implicit in the theoremstatement. To prove the upper bound we will first show that ˆ x i ( t ) ≤ x i = x i + W ( L , , ∀ i ∈ F , t ≥ t + T − + T − . (88)Then the repeated application of Lemma 14 will prove that ˆ x i ( t ) − x i ≤ W ( L , D ( G ) − L (cid:15) ∀ t ≥ t + T − + D ( G ) − (cid:88) i =0 T − i and thus the theorem.Consider i ∈ F . As i ∈ E ( t ) implies ˆ x i ( t + 1) ≥ ˆ x i ( t ) + δ from (11), (12) and (69), there is a t + T − < t ≤ t + T − + T − such that i ∈ A ( t ) . As F ⊂ S ∞ , from (17) ˆ x i ( t ) = ˜ x i ( t ) ≤ s i = x i . (89)As t > t + T − , it follows from (65) that ˆ x k ( t ) ≥ X k forall k ∈ V . As f ( · , · ) is monotonically increasing in both itsarguments and X i ≤ s i , we obtain ˜ x i ( t + 1) = min (cid:26) min k ∈N ( i ) { f (ˆ x k ( t ) , e ik ( t )) } , s i (cid:27) ≥ min (cid:26) min k ∈N ( i ) (cid:8) f (cid:0) X k , e − ik (cid:1)(cid:9) , s i (cid:27) = X i (90)14here (90) uses (56). Therefore, [ X i , x i ] contains both ˆ x i ( t ) and ˜ x i ( t + 1) . Then (70) and Lemma 13 yield | ˜ x i ( t + 1) − ˆ x i ( t ) | ≤ | x i − X i | = W ( L , D ( G − ) − L (cid:15)< D From (9-11), ˆ x i ( t + 1) = ˜ x i ( t + 1) . Am induction proves (88). Yuanqiu Mo was born in Yangzhou, China, in1991. He received the Ph.D. degree in electrical andcomputer engineering at the University of Iowa, in2019. He is currently a postdoc with the WestlakeUniversity. His research interests include distributedalgorithm design and stability theory. Dr. Mo hasbeen awarded the 2018 CDC Outstanding StudentPaper Award. He was a finalist of the Young AuthorAward in the IFAC World Congress 2020.
Soura Dasgupta, (M’87, SM’93, F’98) was bornin 1959 in Calcutta, India. He received the B.E.degree (Hons. I) in Electrical Engineering from theUniversity of Queensland (Australia) in 1980, andthe Ph.D. in Systems Engineering from the Aus-tralian National University, in 1985. He is currentlyF. Wendell Miller Distinguished Professor in theDepartment of Electrical and Computer Engineeringat the University of Iowa, U.S.A and holds a visitingappointment in the Shandong Academy of Sciences.In 1981, he was a Junior Research Fellow at theIndian Statistical Institute, Calcutta. He has held visiting appointments atthe University of Notre Dame, University of Iowa, Universite Catholiquede Louvain-La-Neuve, Belgium, Tata Consulting Services, Hyderabad, theAustralian National University and National ICT Australia.From 1988 to 1991, 1998 to 2009 and 2004 to 2007 he respectively servedas an Associate Editor of the IEEE TRANSACTIONS ON AUTOMATICCONTROL, IEEE Control Systems Society Conference Editorial Board, andthe IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS- II. He is aco-recipient of the Gullimen Cauer Award for the best paper published in theIEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS in the calendaryears of 1990 and 1991, a past Presidential Faculty Fellow, a past subjecteditor for the International Journal of Adaptive Control and Signal Processing,and a member of the editorial board of the EURASIP Journal of WirelessCommunications. In 2012 he was awarded the University Iowa CollegiateTeaching award. In the same year he was selected by the graduating class forexcellence in teaching and commitment to student success. From 2016-18 hewas a 1000 Talents Scholar in the People’s Republic of China.His research interests are in Controls, Signal Processing, Communicationsand Parkinson’s Disease. He was elected a Fellow of the IEEE in 1998.
Jacob Beal is a scientist at Raytheon BBN Tech-nologies in Cambridge, Massachusetts. His researchfocuses on the engineering of robust adaptive sys-tems, particularly on the problems of aggregate-level modeling and control for spatially distributedsystems like pervasive wireless networks, roboticswarms, and natural or engineered biological cells.Dr. Beal received a PhD in electrical engineeringand computer science from MIT. He is an associateeditor of ACM Transactions on Autonomous andAdaptive Systems, is on the steering committee ofthe IEEE International Conference on Self-Adapting and Self-OrganizingSystems (SASO), and is a founder of the Spatial Computing Workshop series.He is a Senior Member of IEEE. Contact him at [email protected] a scientist at Raytheon BBN Tech-nologies in Cambridge, Massachusetts. His researchfocuses on the engineering of robust adaptive sys-tems, particularly on the problems of aggregate-level modeling and control for spatially distributedsystems like pervasive wireless networks, roboticswarms, and natural or engineered biological cells.Dr. Beal received a PhD in electrical engineeringand computer science from MIT. He is an associateeditor of ACM Transactions on Autonomous andAdaptive Systems, is on the steering committee ofthe IEEE International Conference on Self-Adapting and Self-OrganizingSystems (SASO), and is a founder of the Spatial Computing Workshop series.He is a Senior Member of IEEE. Contact him at [email protected]