Identifying Disconnected Agents in Multiagent Systems via External Estimators

This article addresses the problem of identifying disconnected agents in multiagent systems via external estimators. Specifically, we employ external estimators with an appropriately designed decision rule to identify the disconnectedness (i.e., the status of being disconnected) between two arbitrarily chosen agents in formation-control multiagent systems. The design of the decision rule is inspired by the unit-root testing problem of autoregressive time series. To make the best possible decision, a best-effort procedure is also proposed. Then, by introducing the concept of connected components (or just components) in graph theory, and using the methods of consensus analysis and time-series analysis, we develop an analytical framework to show the theoretical performance of the designed decision rule. A particularly important result shown by our analysis is that the miss probability of the decision rule can converge to 0 as the number of data samples increases. Finally, simulation results validate the performance of the decision rule and the best-effort procedure, showing that they can perform well even in small samples.

network is often modeled by a communication graph which is said to be connected if there is a path between every pair of vertices in the graph [18].With the aim of fully leveraging the benefits of multiagent systems, it is necessary to overcome many unprecedented issues arising from nonideal communication conditions [13], [17], [45].A differentiating factor in typical multiagent networks compared with the current majority of communication networks is the several-fold increase in the mobility of network nodes [28], [29].The high mobility of agents can cause multiagent networks to be easily disconnected, that is, some agents are disconnected from others [28], [29].

A. State of the Art and Prior Works
There has been significant research related to the estimation, maintenance (preservation), and control of the network connectivity for multiagent systems [1]- [23].Zhang et al. [1] recently proposed a decentralized method to estimate the algebraic connectivity of undirected graphs.Sabattini et al. [10] developed a decentralized estimation and control strategy to maintain the strong connectivity of directed communication graphs.Franceschelli et al. [14] presented a decentralized algorithm to estimate the eigenvalues of the Laplacian matrix that encodes the network topology of a multiagent system.Yang et al. [20] proposed estimating and controlling the global connectivity of a network using only decentralized computations and local communication via a power iteration algorithm.Dimarogonas and Kyriakopoulos [21] provided a distributed control strategy to preserve connectivity by using repulsive and attractive potential fields.De Gennaro and Jadbabaie [23] developed a decentralized algorithm for the connectivity control of a multiagent system.Zavlanos et al. [18] provided a theoretical framework for controlling network connectivity and discussed various distributed methods to maintain, control, and increase network connectivity.
Most literature on the network connectivity of multiagent systems employed the assumption that the communication graph is a connected graph [1]- [23].Nevertheless, in realistic situations, it is impossible for a multiagent system to keep its communication network connected all the time during the system evolution, that is, all-time connectivity may not be ensured [25].This has motivated further investigations on intermittent connectivity scenarios for multiagent networks [24]- [26].Kantaros and Zavlanos [24] designed a distributed controller for the agents so that the network connectivity is guaranteed over time, infinitely often.Khodayi-Mehr et al. [25] developed a distributed controller that designs sequences of communication events for the agents so that the network can be intermittently connected infinitely often.Aragues et al. [26] proposed an intermittent connectivity strategy that allows multiple agents to move on the 1-D cycle graph of the environment.
Very few studies have paid attention to disconnected network scenarios in which neither all-time connectivity nor intermittent connectivity is guaranteed.Being disconnected is the worst-case status of a multiagent network [28], [29], and it would be better to prevent it from happening [1]- [23], but there is no absolute guarantee that a practical multiagent network will never be disconnected.Multiagent systems need the ability to immediately reconnect after their communication networks are disconnected [28], [29], for which identifying the disconnected agents within the systems is a prerequisite.The difficulty in achieving this identification is caused by a lack of available modeling, analysis, and design methods.

B. Contributions
In this article, we consider multiagent systems whose communication networks are disconnected, meaning that alltime and intermittent connectivity cannot be guaranteed.We develop an approach to monitoring the connection status of two arbitrarily chosen agents via external estimators and to identify those that are disconnected, on the basis of the data samples.We show that an appropriately designed decision rule allows external estimators to accomplish this identification.In particular, we prove that the miss probability of the designed decision rule converges to 0 as the number of data samples increases.An interesting finding in the simulations shows that even with a small sample number, both miss and false alarm probabilities can be reduced to below 0.05 in many cases.
The main contributions and novelty of this article can be summarized as follows.
1) Prior works on the network connectivity of multiagent systems have mainly adopted the all-time or intermittent connectivity assumption [1]- [27].To the best of our knowledge, this article is one of the first to systematically study the disconnected network scenario (Assumption 1), with the aim of identifying disconnected agents.2) We build a model of external estimators that monitor the connection status of two arbitrarily chosen agents, and design a decision rule to judge whether these two agents are disconnected (Decision Rule 2).The design of the decision rule is inspired by the unit-root testing problem of autoregressive time series, in which the use of an ordinary least squares (OLS)-type estimate is a key ingredient.Moreover, we propose a best-effort procedure to make the best possible decision (Procedure 1).3) We develop an analytical framework to show the theoretical performance of the designed decision rule.To be specific, the concept of components in graph theory is first introduced to characterize disconnected communication graphs (Definition 1).Based on this concept, the basic equivalence conditions for two agents to be disconnected and connected are derived (Lemma 1).The methods of consensus analysis [31], [32] are then used to build a bridge between the dynamics model of multiagent systems and autoregressive time series (Lemma 5).Finally, the application of time-series analysis [40] reveals the theoretical performance of the decision rule (Theorems 1 and 2).Notations: Let R denote the real field.Vectors are set in boldface lowercase letters, and matrices in boldface capital letters.We write a i for the ith entry of the vector a, and a ij for the (i, j)th entry of the matrix A, that is,

II. PRELIMINARIES
In this section, we recall the basic definitions and early results of graph theory, algebraic graph theory, timeseries analysis, and estimation theory, those which will be used in this study.Please refer to [38]- [41] for more details.
First, consider an undirected graph G = (V G , E G ), where V G and E G denote the sets of vertices and edges, respectively.
Definition: The graph G is connected if each pair of vertices in G belongs to a path, and disconnected otherwise [38].

Definition 1 (Component):
The component of G is a maximal connected subgraph that is connected and is not contained in any other connected subgraph of G [38].
Note that to illustrate the concept of components, an example of G is shown in Fig. 1, where G is a disconnected graph that has three components.
Proposition 1 [38], [39]: Any two vertices belonging to the same component are connected by at least one path.
Proposition 2 [38], [39]: If G is a connected graph, it has exactly one component consisting of the whole graph.Otherwise G has at least two components, which are pairwise disjoint.
Definition (Algebraic Connectivity): Denote the Laplacian matrix of G by L. Let λ 1 , . . ., λ N be the eigenvalues of L, where λ i ≥ 0 is the ith smallest eigenvalue and λ 2 is the algebraic connectivity that indicates the global connectivity of G [39].
Proposition 3 [13], [31], [39]: Since G is an undirected graph, L is symmetric positive semidefinite and there exists an orthogonal matrix Next, let us review some details of the time-series analysis [40].A time series is a sequence of values of a variable at successive equally spaced points in time, denoted by {X k } K k=1 .The time-series analysis consists of methods for analyzing time-series data to extract meaningful features of the data, such as the convergence behavior below.
Definition (Convergence in Probability [40]): Let {X k } be a time series, which is said to converge in probability to c if for all ε > 0, lim k→∞ P(|X k − c| > ε) = 0.This is indicated as Definition (Convergence in Distribution [40]): For a time series {X k }, let F X k (x) denote the cumulative distribution function of X k .If there exists a cumulative distribution function F X (x) such that lim k→∞ F X k (x) = F X (x) for every number x at which F X (•) is continuous, then X K is said to converge in distribution to X, denoted as ( In deriving the main results of this article, we will be faced with a specific type of time series and its associated testing problem, which are defined as follows.
Definition: A time series {Y k } K k=0 is called a first-order autoregressive time series [40], if it satisfies where 0 < ρ ≤ 1, Y 0 is a constant, and {ε k } K k=1 is an i.i.d.white noise sequence with zero mean and variance σ 2 .This time series is said to be unit-root autoregressive (or called a unit-root autoregression) if ρ = 1 [40].
Definition (Unit-Root Testing Problem [40]): To test whether the time series (3) is unit-root autoregressive (i.e., ρ = 1) based on the data {Y k } K k=0 , is the so-called unit-root testing problem.
This problem is the starting point of many empirical timeseries studies, which can be resolved by using the OLS estimation. 1 The OLS estimation is to find an estimate of the parameter ρ from the sampled data {Y k } K k=0 of the time series (3).For any real valued 0 < ρ ≤ 1, the OLS estimate of ρ can be written as which is the maximum-likelihood estimate of ρ conditioned on {Y k } K k=0 .We note that there are several useful properties of OLS estimation.
Proposition 4 [40,Proposition 17.1,Example 7.15]: Consider the time series (3), denoted by and ).Based on the OLS estimation, the following decision rule can be used to judge whether the time series {Y k } K k=0 is unitroot autoregressive.
Decision Rule 1: After performing the OLS estimate as (4), the first-order autoregressive time series {Y k } K k=0 is judged to be unit-root autoregressive, if the following condition is satisfied: where C ∈ (0, 1) is a real number.The unit-root testing problem can be treated as a binary hypothesis-testing problem [41], by assuming that there are two hypotheses, that is, null hypothesis H 0 and alternative hypothesis H 1 , which are written as respectively.Therefore, we are able to implement the unit-root testing by using Decision Rule 1 to make decisions, that is, accepting or rejecting H 1 , based on the estimated ρK from the data {Y k } K k=0 .The relations between truth and falseness of the decision are illustrated in Table I.
A basic measure for the performance evaluation of Decision Rule 1 is the probability of making a wrong decision, called error probability and defined as follows.
Definition (Error Probability [41]): A decision rule for binary hypothesis testing has two types of error probability.
1) The first type is called miss probability, which is the probability that the rule makes a wrong decision, that is, rejecting H 1 when H 1 is true (H 0 is false).
2) The second type is called false alarm probability, which is the probability that the rule makes a wrong decision, that is, accepting H 1 when H 1 is false (H 0 is true).Remark: The miss and false alarm probabilities can be denoted as P(|ρ K ı, − 1| > C|H 1 ) and P(|ρ K ı, − 1| ≤ C|H 0 ), respectively.

A. Multiagent System
Consider a multiagent system consisting of N a agents with the second-order dynamics [33]- [35] where p i (t) ∈ R M and v i (t) ∈ R M are the position and velocity of agent i, respectively, and u i (t) ∈ R M denotes the control input.Assume that u i (t) uses the formation-control law as where α and β are positive constants satisfying αβ > 1. v * ∈ R M is the desired velocity.g ij denotes the interagent communication between two agents i and j, that is, g ij = 1 if there is a communication link between them and g ij = 0 otherwise; while g ii = 0 for all i ∈ {1, . . ., N a }.The communication is bidirectional, that is, g ij = g ji for i = j.In addition where δ i ∈ R M is the position offset of agent i in the formation, while ε i,j,p ∈ R M and ε i,j,v ∈ R M denote the random errors.Define The system (8) can thus be remodeled as [27], [31] ẋi where ε i,j (t) := ε i,j,p + βε i,j,v .Assume that all elements of ε i,j (t) are independent white noises [32], [35]; that is, by letting ε i,j,l (t) be the lth entry of ε i,j (t), one can have that ε i,j,l (t) ∼ N(0, σ2 ) for every i, j, l, and any t, and that {ε i,j,l (t)} are stationary processes, where ε ı,j 1 ,l 1 (t 1 ) and ε ,j 2 ,l 2 (t 2 ) are independent for ı = , Remark 1: For clarity of presentation, we focus on investigating multiagent systems using the formation-control law described by ( 9) and (10).In fact, the derived results should apply to multiagent systems with various control laws as long as the systems can be remodeled by ( 12) and (13), where (12) is a classic dynamics model for multiagent systems and ( 13) is a typical consensus-control input with noise [27], [45].
We now specify the communication graph of the system as follows [13], [34], [38].Let V a = {1, . . ., N a } be the set of N a agents with i ∈ V a representing agent i.The graph G a = (V a , E a ) is used to model the interagent communications among the agents, where G a is an undirected graph and E a ∈ V a × V a is the edge set of paired agents.An edge (j, i) ∈ E a implies that agent i can straightforwardly communicate with agent j, that is, g ij = 1.The Laplacian matrix of G a can be written as L a = [l ij ], where l ii = j =i g ij and l ij = −g ij for i = j.Since this study considers the disconnected network scenario, we make the following assumption.
Assumption 1: The communication graph G a is a disconnected graph.
This assumption means that G a has more than one component, under which we are interested in monitoring the connectivity between two arbitrarily chosen agents.This type of connectivity is especially useful for showing precise system status and helping to remerge the multiagent system when G a is a disconnected graph.
Without loss of generality, suppose that agents ı and are the two agents of interest.The formal definition of the connectivity between two agents is given as follows.
Definition 2 (Connected and Disconnected Agents): Agents ı and are said to be connected if there is a path between vertices ı and in graph G a , and disconnected otherwise.

B. External Estimator
We are concerned with identifying the disconnectedness between two arbitrarily chosen agents (denoted by agents ı and ) in multiagent systems via external estimators. 2 Assume that there is an external estimator monitoring the connection status of agents ı and in the multiagent system (8).The agents and external estimator have their own tasks to perform.
1) Agents ı and take the samples of x ı (t) and x (t) at t = kτ , respectively, with a sampling interval of τ , and then transmit the sampled data x ı (kτ ) and x (kτ ) to the external estimator via long-range communication links, 3where k = 0, 1, . . ., K and K is the number of data samples.2) After receiving the sampled data x ı (kτ ) and x (kτ ), k = 0, 1, . . ., K, from agents ı and , respectively, the external estimator first computes Fig. 2. Multiagent system (i.e., a multi-UAV system) with control stations that have external estimators.Control stations can be ground control centers and manned vehicles.Note that the long-range communications are independent with the interagent communications.
and then estimates Note that ρ K ı, is an OLS-type estimate that requires inner product •, • since d ı, [k] and d ı, [k − 1] are vectors, while the classical OLS estimate in time-series theory involves only the product of two real numbers [40] [see (4)].
Then, identifying the disconnectedness between agents ı and can be treated as binary hypothesis testing.Specifically, we assume that there are two hypotheses H 0 : Agents ı and are connected, (16) H 1 : Agents ı and are disconnected (17) and design the decision rule as follows.
Decision Rule 2: Agents ı and are judged to be disconnected (i.e., accepting H 1 ), if the following condition is satisfied: where K is the number of data samples, and C ∈ (0, 1).The formulation of this rule is inspired by Decision Rule 1 for the unit-root testing problem of the first-order autoregressive time series.Obviously both the estimate and decision have low complexity, because ( 14) and ( 15) require MK subtractions, 2MK multiplications, 2MK additions, and 1 division.We propose to set C = 0.5 since 0.5 is the median of 0 and 1, noting that this setting will be checked in the simulation.
The exact expressions for τ and K are not yet available due to the absence of a prior information about the communication graph G a at the external estimator.Therefore, we propose a best-effort procedure to identify disconnected agents, which is described by Procedure 1.The procedure uses progressively increasing τ as long as there is plenty of time.That is why it is called the best-effort procedure.In line 3 of Procedure 1, f (τ 0 , T) is a positive monotonically increasing function of T. For example, we will set f (τ 0 , T) = τ 0 T in the simulation.Besides, we propose to select K as large as possible while the simulation results (Figs. 8-10) will show that setting 10 ≤ K ≤ 20 might be large enough for many cases.
To evaluate the performance of Decision Rule 2, we will also use the measure of error probability in our analysis.
Remark 2: The miss probability of Decision rule 2 is the probability of event |ρ K ı, − 1| > C under the hypothesis that agents ı and are disconnected (i.e., H 1 ), which is written as and the false alarm probability is the probability of event |ρ K ı, − 1| ≤ C under the hypothesis that agents ı and are connected (i.e., H 0 ), which is written as ) Note that the definitions of H 1 and H 0 can be found in (17) and (16), respectively.Obviously, the miss and false alarm probabilities are the probabilities of Decision Rule 2 making a wrong decision under the conditions that agents ı and are disconnected and connected, respectively.

IV. ANALYSIS
This section is dedicated to the analysis of identifying disconnected agents with Decision Rule 2 and Procedure 1.We will first give useful lemmas and then present the derived results, for which detailed proofs will be provided in an independent section at the end.

A. Useful Lemmas
For the multiagent system (8), we adopt the concept of components in graph theory to describe and analyze the communication graph G a that is a disconnected graph (see Assumption 1).
Definition 3: Suppose that agent ı belongs to component G ı of the communication graph G a and agent belongs to Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
That is, there are N agents, denoted by i 1 , . . . ,i N , belonging to G ı , where N ≤ N a .
Remark 3: By definition, it is known that G ı has at most two components.That is, when G ı and G are disjoint (i.e., G ı ∩ G = ∅), G ı consists of two components, implying that agents ı and are disconnected.
Under Assumption 1, the communication graph G a is disconnected, while agents ı and are either disconnected or connected.
Lemma 1: The following statements are equivalent.1) Agents ı and are disconnected.
2) G ı consists of only one component, such that agents ı and belong to the same component (i.e., G ı = G ). 3) λ 1 = 0, and λ i > 0 for all i = 2, . . ., N. According to this lemma, it is always true that λ 1 = 0 and λ i > 0 with i = 3, . . ., N. However, the value of λ 2 is different in two cases, that is, λ 2 > 0 if agents ı and are connected, and λ 2 = 0 if they are disconnected.Note that Fig. 3 Lemma 2: With A, B, and K being defined in (11), if α > 0, β > 0, and αβ > 1, then for each λ i ≥ 0, we have where We now focus on i , especially on 2 , recalling that the value of λ 2 can be used to distinguish whether agents ı and are connected or not (see Lemma 1).
Lemma 6: Suppose that Assumption 1 holds.Let ).If agents ı and are disconnected, then ( 30)-( 33) hold, as shown at the bottom of the page.This lemma is useful in the computation of ρ K ı, , since which can further make use of ( 30)- (33).
By Theorem 1, we can explain why Decision Rule 2 can effectively identify the disconnectedness between agents ı and ; that is, if two agents are disconnected, for sufficiently large K, ρ K ı, will be close enough to 1 such that (18) will certainly be satisfied.This means that the miss probability can be kept arbitrarily small, as precisely pointed out in the following theorem.
Theorem 2: Let Assumption 1 hold.If agents ı and are disconnected, then for any C ∈ (0, 1), the miss probability converges to 0 as K increases, that is lim This theorem describes the convergence behavior of the miss probability.Elegant theoretical results for the false alarm probability of Decision Rule 2 are, however, still beyond analytical reach because analyzing ρ K ı, is complicated when agents ı and are connected.To gain further insights into the false alarm probability, simulations will be carried out in Section V. Before that, let us provide some comments helpful to understanding the behavior of the false alarm probability.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Remark 4: Under the condition that agents ı and are connected, it can be known that: 1) the convergence as in ( 37) is generally not valid in this case.Because the expressions in the right-hand sides of equalities in ( 32) and ( 33) only differ by the additional term I 2M ⊗ e n 2 +1 τ in (32) while this term causes ρ K ı, to deviate from 1.Note that I 2M ⊗ e n 2 +1 τ is a diagonal matrix whose diagonal entries consist of e φ n+1,m τ , n ∈ {1, . . ., N − 1}, m ∈ {1, . . ., 2M}.For all n and all m, we have e φ n+1,m τ < 1 such that e φ n+1,m τ = 1; 2) as τ → ∞, each diagonal entry of I 2M ⊗ e n 2 +1 τ decreases exponentially to 0, since φ n+1,m < 0. Thus, with ever-increasing τ , ρ K ı, tends to 0, making the false alarm probability, that is, P(|ρ K ı, − 1| ≤ C|Connected), to decrease.This means that Procedure 1 should be able to substantially reduce the false alarm probability, which will be shown in the simulation (Section V).

C. Proofs
The proofs of the derived results are provided as follows.Proof of Lemma 1: Equivalence of 1) and 2) follows from Propositions 1 and 2. On the other hand, 3) is equivalent to 1) and 2), because the multiplicity of the eigenvalue 0 of L equals the number of components [37].To be specific, if G ı has only one component, then the multiplicity is 1 such that λ 1 = 0 and λ i > 0 for all i = 2, . . . ,N; if G ı has two components, then the multiplicity is 2 such that λ 1 = λ 2 = 0 and λ i > 0 for all i = 3, . . ., N.
As shown in Appendix A, there exists a link between the Ornstein-Uhlenbeck process and autoregressive time series.We can apply this link to obtain (28), where by letting ε n,m,q,k = σ τ 0 e φ n+1,m (τ −s ) dB q ((k − 1)τ + s ).It is easy to check that ε n,m,q,k is Gaussian with zero mean and variance −(σ 2 /2φ n+1,m )(1 − e 2φ n+1,m τ ), noting that the variance will be σ 2 τ if φ n+1,m = 0.This means that ε n,m,k is also Gaussian with zero mean, and the sequence { ε n,m,k } K k=1 has independent elements.
Proof of Lemma 5: By using the definitions of d ı, [k], ξ l (t), η n (t), ζ n (t), and η n [k], together with the relation we can obtain (29) and thus prove the result.
Proof of Lemma 6: First, ( 30) and ( 33) can be derived by using the following properties of the Euclidean inner product, trace, and vectorization: x, y = x T y = tr(x T y) for vectors x ∈ R m and y ∈ R m ; tr(XY) = tr(YX) and tr(X The next step of the proof is to show (31).We see that the term 30) is a matrix whose (i, j)th , that is where i, j ∈ {1, . . ., 2M}.Using (28), we can show that where the convergence holds true since ε n 2 ,i,k and η n 1 ,j,k−1 are independent.Implicit in (42) is that the (i, j)th entry of , where e n 2 +1 τ = ∞ q=1 (τ q /q!) q n 2 +1 is a diagonal matrix since n 2 +1 is diagonal.At this point, applying (41) and ( 42) to (30) results in (31).
Finally, using a fact about two matrices X ∈ R k×l and Y ∈ R l×m that vec(XY) = (I m ⊗ X)vec(Y), for (31), we have Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Hence, (32) and ( 33) hold, which completes the proof.Proof of Theorem 1: It is important to recall at this point that if agents ı and are disconnected, then it follows from Lemma 3 that φ based on the expression in (31).On the one hand, if The convergence in (43) follows from ( 42) and Proposition 4. That is, for each i ∈ {M + 1, . . ., 2M}, p −→ 0; however, for each i ∈ {1, . . ., M}, {η 1,i,k−1 } K k=0 is unit-root autoregressive because e φ 2,i τ = 1.It follows from Lemma 4 and Proposition 4 that: On the other hand, if n 1 = 1 or n 2 = 1, it is easy to see that by Hölder's inequality [44].Without loss of generality, we assume that n 1 = 1, and obtain p −→ 0. This further allows us to have Substituting ( 43)-( 46) into (31) and letting where Similar to the derivations of (35), we can also evaluate ) and obtain (36).After obtaining (35) and (36), we finally complete the proof by applying [40, 4. Two cases considered in the simulations, which have the same desired formation shape but different communication graphs G a .Note that the formation shape is specified by parameters δ i .(a) Disconnected case (i.e., agents ı = 1 and = 9 are disconnected), which implies that the hypothesis H 1 as in ( 17) is true.(b) Connected case (i.e., agents ı = 1 and = 9 are connected), which implies that the hypothesis H 0 as in ( 16) is true.definition of convergence in probability to show that for all C ∈ (0, 1), the miss probability satisfies lim K→∞ P(|ρ K ı, −1| ≥ C|Disconnected) = 0.

V. SIMULATIONS
In this section, we present simulation results of the proposed Decision Rule 2 and Procedure 1 available to external estimators for a multiagent system (8) with 16 agents.The simulations are performed in R 2 space with initial conditions p i (0) = δ i + w i and v i (0) = 0 2 for all i ∈ {1, . . ., 16}, where w i = [w i,1 w i,2 ] T ∈ R 2 whose entries satisfy w i,m ∼ N(0, 40 2 ).As such, we can write δ i = [δ i,1 δ i,2 ] T and p i (t) = [p i,1 (t) p i,2 (t)] T .For the control law (9), we let α = 1.0 and β = 2.9.Without loss of generality, the disconnectedness between agents ı = 1 and = 9 are of interest, and we set τ = 3.
To provide comparative performance plots, let us consider two cases as illustrated in Fig. 4.More specifically, one is the disconnected case, that is, the case in which agents ı = 1 and = 9 are disconnected, and the other is the connected case, that is, the case in which two agents are connected.These two cases are related to H 1 and H 0 hypotheses specified by ( 17) and ( 16), respectively.In Fig. 5(a) and (b), we show position trajectories and achieved formation shapes in the disconnected and connected cases with one trial, respectively.It can be seen from Fig. 5(a) that the desired formation shape cannot be achieved in the disconnected case since the communication graph is disconnected.
Numerical examples of ρ K ı, are provided in Fig. 6, where we show the behavior of ρ K ı, as K increases in both disconnected and connected cases with five trials.The results show us an obvious contrast between the disconnected and connected cases with respect to the behavior of ρ K ı, .That is, in the disconnected case, ρ K ı, approaches 1 as K increases, while in the connected case, it appears that ρ K ı, is evidently less than 1.In Fig. 6.Evolution of ρ K ı, as K increases in both disconnected and connected cases.Fig. 7. Error probabilities of Decision Rule 2 as K increases, that is, miss and false alarm probabilities for disconnected and connected cases, respectively.Different values of C are used, including C = 0.1 and C = 0.5.The inset plots the data of error probabilities as logarithmic scale for the y-axis.addition, Fig. 6 shows that ρ K ı, has already become close to 1 when K ≤ 200 in the disconnected case.
The error probabilities of Decision Rule 2 in both disconnected and connected cases are plotted in Fig. 7. Simulation results of the miss and false alarm probabilities are provided for C = 0.1 and C = 0.5.As expected by Theorem 1, the miss probability P(|ρ K ı, − 1| > C|Disconnected) decays with growing K in the disconnected case.Also it can be seen that the false alarm probability P(|ρ K ı, − 1| ≤ C|Connected) quickly decreases as K grows larger in the connected case.Perhaps the most inspiring finding is that with C = 0.5 both the miss and false alarm probabilities can be reduced to less than 0.05 with K = 30.By comparing the performances with C = 0.1 and with C = 0.5, we see that there is no significant difference in the rates of decrease of miss and false alarm probabilities with C = 0.5, while the miss probability decreases much more slowly than the false alarm probability with C = 0.1.Thus, as mentioned in Section III, we propose to set C = 0.5, because Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.it turns out numerically that C = 0.5 (i.e., the median of 0 and 1) is an appropriate threshold for Decision Rule 2 and makes it simple to apply the decision rule in practice.
In Fig. 8, the miss probability of Decision Rule 2 is depicted for different values of τ in the disconnected case.We see that the miss probability can quickly decay to 0 in the disconnected case with τ = 2, 3, 4, 6.It seems from the inset of Fig. 8 that the value of τ does not have an apparent effect on the decaying rate of the miss probability.This observation is consistent with Theorems 1 and 2 that do not make an additional assumption about τ .In other words, Theorems 1 and 2 always hold true regardless of the value of τ , meaning that the miss probability can always converge to 0 in the disconnected case.
For comparison, the false alarm probability of Decision Rule 2 with different values of τ in the connected case is illustrated by Fig. 9.It is shown that the false alarm probability behaves differently as τ varies.We can see that in the connected case with τ = 3, 4, 6, the false alarm probability can decrease to be less than 0.05 when K reaches 20; however, with τ = 2, the false alarm probability is relatively large and cannot decrease to the small value for K ≤ 20.These results tell us that the value of τ can affect the false alarm probability, that is, the larger τ is, the smaller the false alarm probability is likely to be.This can be explained by Remark 4.
To illustrate the effectiveness of Procedure 1, we perform simulations by setting τ 0 = 1, K = 10, and considering four agent pairs, that is, (ı = 1, = 11), (ı = 3, = 12), (ı = 6, = 16), and (ı = 8, = 15).Fig. 10 plots the error probabilities of the decision made in Procedure 1, from which two important trends can be observed.First, the miss probability can always be small and does not significantly decrease as τ increases (τ = τ 0 T), as we have already observed from Fig. 8. Second, the false alarm probability will become considerably small as τ grows, even if this probability is initially large (close to 1).This implies that using large τ in Procedure 1 (if allowed) can effectively improve the false alarm performance of the decision, as we can expect from Remark 4.

VI. CONCLUSION
This article provides an approach to monitoring the connection status of two arbitrarily chosen agents via external estimators and to identifying those that are disconnected, based on the sampled data.We design a decision rule to realize this kind of identification for formation-control multiagent systems and prove that the miss probability of this rule converges to 0 as the number of data samples increases.Interestingly, both miss and false alarm probabilities can be reduced to below 0.05 in many simulated cases with only a small sample number.The ability to identify disconnected agents in multiagent systems is important for practical purposes, as it allows actions to be taken to reconnect the disconnected agents.This study is only a first step in this direction, and considerable effort is needed to improve the decision rule, develop more powerful analysis methods, and devise reconnection strategies.Extending the derived results to multiagent systems using the control laws with random errors other than white noise is also meaningful for future work.
being its diagonal entries.Let I m be the m × m identity matrix and O m be the m × m zero matrix.We use 1 n ∈ R n and 0 n ∈ R n to denote the all-one and all-zero vectors, respectively.The superscript T is the transpose, and ⊗ represents the Kronecker product.•, • denotes the inner product.tr(•) and vec(•) are the trace and vectorization operators in linear algebra, respectively.Denote the zero-mean real Gaussian distribution with variance σ 2 by N(0, σ 2 ).

Fig. 3 .
Fig. 3. Two examples of G ı = G ı ∪ G , when the communication graph G a has three components (i.e., G c1 , G c2 , andG c3 ) such that G a = G c1 ∪ G c2 ∪ G c3 .It is easy to see that G ı = G ı ∪ G ⊂ G a .For clarity, G ı Fig. 3. Two examples of G ı = G ı ∪ G , when the communication graph G a has three components (i.e., G c1 , G c2 , andG c3 ) such that G a = G c1 ∪ G c2 ∪ G c3 .It is easy to see that G ı = G ı ∪ G ⊂ G a .For clarity, G ıis shown in blue.(a) If agents ı and belong to the same component (i.e., ı = i ∈ V G c2 and = m ∈ V G c2 ), then G ı = G ı = G = G c2 has only one component and thus two agents are connected.(b) If agents ı and belong to two disjoint components (i.e., ı = l ∈ V G c2 and = r ∈ V G c3 ), then G ı = G ı ∪ G = G c2 ∪ G c3 consists of two components (i.e., G c2 and G c3 ), and thus two agents are disconnected.
Fig. 4. Two cases considered in the simulations, which have the same desired

Fig. 5 .
Fig. 5. Position trajectories and achieved formation shapes in the disconnected and connected cases.(a) Disconnected case (one trial).(b) Connected case (one trial).

Fig. 8 .
Fig. 8. Miss probability of Decision Rule 2 as K increases, for different τ in the disconnected case.The inset plots the data of miss probability as logarithmic scale for the y-axis.

Fig. 9 .
Fig. 9. False alarm probability of Decision Rule 2 as K increases, for different τ in the connected case.The inset plots the data of false alarm probability as logarithmic scale for the y-axis.

Fig. 10 .
Fig. 10.Error probabilities of the decision made in Procedure 1 with τ = f (τ 0 , T) = τ 0 T, for different agent pairs in both disconnected and connected cases.

TABLE I RELATIONS
BETWEEN TRUTH AND FALSENESS OF THE DECISION