Muhammad U. Javed, Jorge I. Poveda and Xudong Chen M. U. Javed, J. I. Poveda, and X. Chen are with the Department of Electrical, Computer and Energy Engineering, at the University of Colorado, Boulder, USA. Emails: {muhammad.javed, jorge.poveda, xudong.chen}@colorado.edu.

Global Synchronization of Clocks in Directed Rooted Acyclic Graphs: A Hybrid Systems Approach

Muhammad U. Javed, Jorge I. Poveda and Xudong Chen M. U. Javed, J. I. Poveda, and X. Chen are with the Department of Electrical, Computer and Energy Engineering, at the University of Colorado, Boulder, USA. Emails: {muhammad.javed, jorge.poveda, xudong.chen}@colorado.edu.
Abstract

In this paper, we study the problem of robust global synchronization of resetting clocks in multi-agent networked systems, where by robust global synchronization we mean synchronization that is insensitive to arbitrarily small disturbances, and which is achieved from all initial conditions. In particular, we aim to address the following question: Given a set of homogeneous agents with periodic clocks sharing the same parameters, what kind of information flow topologies will guarantee that the resulting networked systems can achieve robust global synchronization? To address this question, we rely on the framework of robust hybrid dynamical systems and a class of distributed hybrid resetting algorithms. Using the hybrid-system approach, we provide a partial solution to the question: Specifically, we show that one can achieve robust global synchronization with no purely discrete-time solutions in any networked system whose underlying information flow topology is a rooted acyclic digraph. Such a result is complementary to the existing result [1] in which strongly connected digraphs are considered as the underlying information flow topologies of the networked systems. We have further computed in the paper the convergence time for a networked system to reach global synchronization. In particular, the computation reveals the relationship between convergence time and the structure of the underlying digraph. We illustrate our theoretical findings via numerical simulations toward the end of the paper.

I Introduction

In recent years, the problem of coordination and control of networked multi-agent systems (MAS) has been a major research area in control theory. For instance, analytical tools for MAS were studied in [2], [3], [4], [5], [6] in the context of networked systems, event-triggered control, and synchronization of clocks. Multi-agent networked systems have also been analyzed using graph-theoretic methods in [7], and geometric approaches in [8, 9]. In this work, we focus on one particular control problem that emerges in MAS, namely the global synchronization of a collection of homogeneous agents (i.e., agents with identical structures and parameters) with periodic behaviors. We assume that these agents can only communicate with their local neighbors, for which we will use a directed graph to describe the information flow topology. Such a problem finds applications in many areas where a globally synchronized periodic behavior is needed, but only local interactions between agents are allowed, i.e., there does not exist a centralized controller (or network manager) that can coordinate all the agents in the network. These applications include power systems, biological systems, and sampled-data systems where the synchronization problems emerges in a natural way. This has motivated the development of several deterministic and stochastic synchronization algorithms in [6, 10, 11] and [12], to just name a few.

In this paper, we focus on a particular phase synchronization problem, where a group of agents with periodic clocks aims to eventually aligned their phase position for all time in a distributed way. This problem is equivalent to the synchronization of identical oscillators flowing on the unit circle , a problem that is known for its infeasibility of achieving global synchronization that is robust to arbitrarily small disturbances by using smooth feedback control laws [13], [14]. Such infeasibility has motivated the development of algorithms that relax the global requirement and instead focus on achieving almost global synchronization, i.e., synchronization from all initial conditions except for a set of measure zero. Some examples of these algorithms can be found in [11] and [12]. On the other hand, under certain assumptions, robust global synchronization can be achieved if one implements a hybrid controller in a cyclic graph [15], or, alternatively, if there exists a global cue in the network [16]. More recently, it was shown in [17, 1] that a hybrid set-valued resetting algorithm (HSRA) achieves robust global synchronization in any network whose underlying information flow topology is characterized by a directed strongly connected graph, provided that some mild assumptions of tunable parameters of the vertices are satisfied (these tunable parameters determine the jump rules for the vertices).

While these results have been instrumental for the development of robust distributed asynchronous control and optimization feedback mechanisms [17, 18] for directed graphs (digraphs) other than strongly connected ones, it is still an open question whether or not robust global clock synchronization could also be achieved by using a hybrid-systems approach. Specifically, we aim to address the following question: Given a set of homogeneous agents with periodic clocks sharing the same frequency, what kind of information flow topologies will guarantee that the resulting networked system can achieve robust global synchronization?

In this paper, we build on our previous results of [1] and we provide a partial solution to the above question by characterizing a different type of digraphs for which robust global synchronization can be achieved by using the HSRA. Moreover, we provide variations of an existing algorithm [17, 1] that can yield a better performance in terms of convergence time in some specific cases. In particular, the technical contributions of the paper are as follows:

  1. We provide a negative result which says that if the underlying network topology is not a rooted digraph, then the entire network system cannot achieve global synchronization under the HSRA in any case.

  2. We show that the HSRA can be used to achieve robust global synchronization in any rooted acyclic digraph. Because rooted acyclic digraphs are mutually exclusive from strongly connected ones, the proof of the above result relies on different techniques compared to our previous results established in [17]. We have also computed the corresponding convergence time for the entire networked system to reach synchronization: In the worst case, the convergence time is bounded above by the multiplication of the depth of the rooted acyclic digraph (see Def. II.2 in the next section) and (where is the oscillating frequency of the clock).

  3. We further show that in a special case where the common parameter shared by all agents is set to be certain extreme value, global synchronization can be achieved if and only if the digraph is rooted acyclic. Moreover, we show that in this special case, global synchronization can be achieved in at most units of time, which is the smallest upper bound on the convergence time in any case.

The rest of the paper is organized as follows: In Section II, we present preliminaries and definitions related to graph theory and hybrid dynamical systems. In Section III, we introduce the mathematical model for a network of resetting clocks and we detail our problem formulation. We also present our main result in this section. Section IV will then be devoted to the proof of the main result. Next we manifest the simulation results for the network of resetting clocks in Section V. Finally, we provide conclusions and further discussions in Section VI.

Ii Preliminaries

Ii-a Notation

The set of (nonnegative) real numbers is denoted by . The set of (nonnegative) integers is denoted by . A set-valued mapping is outer semi-continuous (OSC) at if for all sequences and such that we have . A set-valued mapping is said to be locally bounded (LB) at if there exists a neighborhood of such that is bounded. Given a set , the mapping is OSC and LB relative to if the set-valued mapping from to defined by for , and by for , is OSC and LB at each . We use to denote the closed convex hull of . The outer semi-continuous (OSC) hull of is the unique set-valued mapping satisfying , where . Given a compact set and a vector , we define , and we use to denote the standard Euclidean norm. Also, we use to denote the vector in with all entries equal to and to denote the unit vector of appropriate dimension with -th entry equal to 1. We denote a vector of ones by . For a couple of sets , we define the indicator function to be non-empty only on , and to satisfy if , and if . Further, the cardinality of a finite set is denoted as . denotes the unit circle in a plane.

Ii-B Graph Theory:

We characterize a directed graph (or simply a digraph) by a vertex set and an edge set . We denote by an edge from to in or equivalently, . We say that is an in-neighbor of and is an out-neighbor of . The sets of in-neighbors and out-neighbors of vertex are denoted by and , respectively. The in-degree and out-degree of vertex are defined as and , respectively.

Remark II.1

Throughout this paper, we adopt a convention that information flow is from vertex to a vertex if is an edge i.e. from to .

Let and be two vertices of . A walk from to , denoted by , is a sequence (with and ) in which is an edge of for all . A walk is said to be a path if all the vertices in the walk are pairwise distinct. A walk is said to be a cycle if there is no repetition of vertices in the walk other than the repetition of the starting and ending vertex. The length of a path/cycle/walk is defined to be the number of edges in that path/cycle/walk.

We next introduce the following definition:

Definition II.1

Let be a digraph. A vertex of is said to be a root if for any other vertex , there exists a path from to . Correspondingly, the digraph is called a rooted digraph. If, further, does not contain any cycle in it, then we call a rooted acyclic digraph.

We note here that if is rooted acyclic, then there is a unique root . We also need the following definition:

Definition II.2

Let be rooted and acyclic and be the root. The depth of a vertex , denoted by , is the minimum length of a path from to . The depth of is by default . Further, we define the depth of as .

With the above definition, we can decompose the vertex set of as follows:

(1)

where is comprised of all vertices of depth .

Figure 1 provides an example to illustrate the depths of rooted acyclic graphs.

(a)

(b)

(c)
Fig. 1: Rooted Acyclic Graphs with depths a) one b) one c) two. (Black indicates the root vertex)

A strongly connected component (SCC) is a maximal subset of vertices such that there exists a path from any one vertex to the other. Based on the definition of SCC, a condensation graph is defined as a graph obtained with every SCC ‘condensed’ as one vertex. The condensation of a graph is denoted as .

Ii-C Hybrid Dynamical Systems:

In this paper, we will model the clocks and the synchronization algorithms as hybrid dynamical systems (HDS) that combine continuous-time dynamics and discrete-time dynamics [19]. A HDS with state can be described by the following dynamics:

(2a)
(2b)

where is a continuous function describing the continuous-time dynamics (or flows), is an outer-semicontinuous and locally bounded set-valued mapping describing the discrete-time dynamics (or jumps), is a closed set describing the points in the space where the system can flow, and is a closed set describing the points in the space where the system can jump. Under this definition of the data the HDS is said be well-posed. Solutions of (2) are defined on hybrid time domains111We refer the reader to [19, Ch. 2] for a comprehensive introduction to hybrid time domains., i.e., depends on a continuous-time index , and a discrete-time index . Solutions with unbounded time domain are said to be complete.

We will use the following stability notion to characterize the synchronization problem considered in this paper.

Definition II.3

[19] Consider a hybrid system with state . A compact set is said to be uniformly globally asymptotically stable (UGAS) if there exists a function such that all solutions of system (2) satisfy the bound

(3)

for all .

After introducing the preliminaries, now we are in a position to discuss the main contribution of this paper.

Iii Problem Formulation and Main Result

In this section, we will first introduce the network of resetting clocks and a class of hybrid synchronization dynamics. Then, we will formally state our problem formulation to finally, present Theorem 3.3 which manifests the main result of this paper over rooted acyclic digraphs.

Iii-a System Model

Consider a network of clocks with same resetting period and individual flow dynamics given as:

(4)

Whenever the clock of each agent satisfies , agent resets its own clock according to

(5)

and signals its out-neighbours to update their clock as follows:

(6)

where is a homogenous parameter of the network that partitions the decision rule on the unit interval . On the other hand, agents that are not out-neighbours of agent keep their state constant i.e.

Since the dynamics of the clocks combine continuous-time updates and discrete-time updates, the overall system is a hybrid dynamical system of the form (2). However, finding the data such that the overall system is well-posed is not trivial. Indeed, for a HDS to be well-posed in the sense of [19], we expect that the behavior of a sequence of solutions from a sequence of initial conditions should approach the behavior of the limiting solution obtained from the limiting initial condition , where the limit should be understood in the graphical sense [19, Ch. 5]. For the clock synchronization problem, this means that if a sequence of solutions with initial conditions satisfying , for all , and , generates solutions with sequential jumps with smaller and smaller times between jumps, then the limiting behavior from should generate also sequential jumps, in this case with no time between jumps. This implies that whenever two or more agents satisfy the condition , a well-posed model of the hybrid synchronization mechanism should generate non-unique solutions, since the system should capture all the possible combinations of sequential jumps that could emerge from this point.

Remark III.1

Well-posed HDS capture not only the behavior of the nominal system, but also the emerging behavior under arbitrarily small disturbances acting on the states and dynamics. Therefore, by designing a clock synchronization algorithm that is modeled by a well-posed HDS we can simultaneously study the robustness properties of the algorithm.

In order to obtain a well-posed hybrid model for the synchronization algorithm, let be a set-valued mapping that is nonempty only when for some and for . This set-valued mapping is defined as

where the set-valued map is given by

Note that captures the simultaneous jumps described by equations (5) and (6). Finally, to capture all the possible jumps that could emerge in the network whenever more than one agent satisfies , we define the overall jump map to be the outer semi-continuous hull of the mapping . Using this definition and combining all clocks, we obtain a HDS with state and dynamics

(7a)
(7b)

Based on this, the robust global clock synchronization problem can be equivalently cast as studying the asymptotic stability properties of system (7) with respect to the following compact set:

(8)

In particular, one is interested in HDS that render the set UGAS and which generate no purely discrete-time solution that jumps forever.

We summarize the synchronization algorithm realized from the hybrid dynamical system (7) as Algorithm 1 below.

1:procedure Synchronization
2:     
3:     
4:     
5:     if  then      
6:     for  do
7:         if  then
8:              for  do
9:                  if  then
10:                       if  then                        
11:                       if  then                        
12:                       if  then                        
13:                  else
14:                                                                      
Algorithm 1 Distributed Clock Synchronization

Iii-B Problem Formulation

Given the homogeneous parameter , we let be the collection of digraphs for which the hybrid system (7) satisfies the following two properties: 1) It renders UGAS the compact set (8). 2) It does not generate solutions that jump for ever. We aim to characterize such a set for any . We also aim to compute the convergence time for any given , i.e., we compute the least upper bound of the time for the entire networked system to reach synchronization starting with any initial condition. We denote such an upper bound by (or simply if there is no ambiguity).

Before stating the main result of the paper (Theorem III.3), we first introduce a known positive result and a negative result. We start with the following fact established in [17]:

Lemma III.1

If is strongly connected then for any , where is the number of agents in the network.

We next have the following result:

Lemma III.2

If a graph is not rooted, then for any .

{proof}

Suppose that the graph is not rooted, then there exist at least two distinct vertices and a positive integer such that i.e. . Therefore, corresponding to these graph components and an arbitrary initial condition, there exist at least two isolated components of the graph (may not be disjoint) that do not share any information. Hence, the dynamics of the components evolve independently and the agents in the graph will not achieve synchronization. Thus, a rooted graph is a necessary condition for system (7) to be UGAS with respect to (8).

We note here that the converse of Lemma 3.2 is not true i.e. a rooted digraph does not necessarily belong to for any We provide below an example to illustrate such a fact:

Example III.1

Consider the HDS in Figure 2, where the dynamics of the agents are given by (7). Due to the periodic nature of the problem, agents are characterized on a unit circle rotating at a frequency with and identified as the same point. First, note that if , then as soon as any vertex hits one we have that vertices and will keep jumping forever while vertex undergoes a cyclic motion. Thus, in this case, the network cannot reach synchronization. We next consider the case where . We let the initial positions of the three vertices be such that and . Then, the following events will occur subsequently:

  1. hits 1 and , so and jump to zero.

  2. hits 1 and , so and jump to zero.

  3. The clocks flow for less than seconds such that hits 1 and , so and jump to zero.

Since all the clocks have the same resetting frequencies, from this point forward events 2) and 3) will repeat forever and will oscillate between the positions of and . Hence, , and cannot achieve synchronization.

0

1

(a)

(b)
Fig. 2: Three resetting clocks with initial condition, and

A key feature that prevents the hybrid system (7) in the above example to render UGAS the compact set (8) is that vertices other than the root vertex form a strongly connected component. The existence of the counter-example indicates that the problem we posed at the beginning of the section is nontrivial in a sense that one cannot simply reach the conclusion that is the class of rooted digraphs as a straightforward extension of Lemma III.1.

Iii-C Main result and sketch of proof

We now take in the paper the first step to characterize : We focus on a relatively simple class of rooted digraphs, namely, the class of rooted acyclic digraphs. We will now state the main result of the paper:

Theorem III.3

The following statements hold:

  1. If is a rooted acyclic digraph, then for any . Moreover, for any , the convergence time of is .

  2. If , then is exactly the class of rooted acyclic digraphs. Moreover, the convergence time of is given by .

We provide below an example to illustrate global synchronization of resetting clocks over rooted acyclic digraphs.

Example III.2

Consider Figure 3 where there are four agents in the network and the depth of the underlying rooted acyclic graph is two. The set of all agents is further partitioned into three disjoint subsets based on their depths as follows: . Note that , and . Now following Algorithm 1, as hits 1, it jumps to 0 and a subset of its out-neighbour agents jumps to 0 while the other jumps to 1. Indeed, after this first step, the agents and are synchronized together. Next, the agent hits 1 and jumps to zero without affecting any other agent in or . Finally, the agents , which are already synchronized, will hit 1 and trigger the event that agent jumps to either or . Thus, all four agents flow and jump in a synchronized manner.

A complete proof of Theorem III.3 will be given in the next Section. An outline of the proof is given below:

Sketch of Proof of Theorem III.3: We follow a Lyapunov-based approach for hybrid systems. The proof consists of three main arguments:

  1. We define the Lyapunov function such that is the infimum of the lengths of all arcs touching all agents , where the points and on the interval are identified to be the same, to form a circle .

  2. We show that the Lyapunov function remains constant during the flows and does not increase during the jumps.

  3. We prove by induction the following fact: When an agent of depth  reaches , it will force the agents on the next level, i.e., agents of depth , to be synchronized, this is also illustrated in Example 3.2. These jumps will necessarily decrease the Lyapunov function unless all the agents in depths and are already synchronized. By moving forward on we will eventually exhaust the depths of the graph, implying that the Lyapunov function converges to zero in finite time, by construction this implies synchronization.

We then appeal to the Hybrid Invariance Principle (see [20]) to establish that system (7) renders UGAS the compact set (8). For an arbitrary initial condition, the worst-case convergence time corresponds to the case when every agent at each depth jumps to zero. Hence, by repeatedly applying the above arguments (starting with ), we will show that the entire networked system will be synchronized in at most seconds.

Iv Proof of Theorem 3.3

This section is dedicated to the proof of Theorem 3.3. We will provide the proof statement-wise.

Iv-a Proof of item (1) of Theorem iii.3

{proof}

We consider the Lyapunov function as introduced in Sketch of Proof. By [17], this Lyapunov function satisfies the following properties: (i) It is positive definite with respect to compact set (2). (ii) It remains constant during flows because all the clocks have the same frequency . (iii) It does not increase at jumps since jumps never increase the number of distinct points occupied by the agents. (iv) The Lyapunov function is upper bounded by considering the max-min problem of the arc evolving on i.e. .

We claim that every solution must satisfy that goes to zero in finite time. Indeed, suppose by contradiction that there exists a solution such that for all . Further recall from (1) that defines all vertices/agents of depth  . Since the graph is rooted acyclic no agent influences the unique root agent, and without loss of generality we can assume that corresponds to that unique root agent .

Based on this, we show that agents synchronize in at most seconds. We do this by performing an induction on the depth of the vertices of the rooted acyclic digraph.

Base Case: In at most seconds of flow, will reach a point such that . When , it will trigger all the to either go to 0 or 1. Thus, based on , there will exist a partition of that is defined by the index sets () such that:

(i) for all , (the agents will jump to 1 and trigger ), (ii) for all , (the agents will jump to 0 and flow for at most seconds to trigger ), (iii) for all , (the agents will have a set-valued jump . If the agent jumps to 1, it will follow (i), otherwise, it will follow (ii)).

Note that after the first jump, synchronize with within at most seconds and remain synchronized since does not influence by the acyclic property of the graph.

Induction Step: Suppose that agents synchronize in at most seconds, where . Since the graph does not have a cycle, and the root vertex/agent has a path to all the agents, we have that agents only influence agents and never influence already synchronized agents to go out of synchronization. Thus, agents synchronize in at most seconds. Therefore, the agents synchronize in at most seconds and remain synchronized after that i.e., they occupy the same position on the unit circle for all . This contradicts our assumption that Hence, every solution synchronizes in at most seconds. Now, UGAS of (7) with respect to (8) follows directly by the Hybrid Invariance Principle in [20], since no complete solution can stay in a positive level set of for all time. Absence of purely or eventually discrete-time solutions follows by noting that once agents are synchronized they will eventually have to flow since the root agent is not forced to jump by any neighboring agent. This completes the proof of item 1).

0

1

(a)

(b)
Fig. 3: Example 3.2: Synchronization of resetting clocks over rooted acyclic digraph

Iv-B Proof of item (2) of Theorem iii.3

{proof}

Let and suppose that the graph is rooted acyclic. When the root agent satisfies , all the agents in the graph will undergo at most consecutive jumps, after which all agents will satisfy . Indeed, note that the only case where out-neighbors of the root agent will not jump to correspond to the case where and , i.e., agents are already synchronized, and since without loss of generality we can assume that the root agent satisfied after flowing for at least seconds of time, then it must also be true that agent satisfied before resetting its state to zero. Therefore agent already triggered all its out-neighbors. This argument can be repeated for all agents that are out-neighbors of agent , until all agents have already jump to zero in at most seconds. Thus, the agents synchronize in at most seconds of flow. By using the same Lyapunov function as in the proof of item (1), and using again the hybrid invariance principle, we conclude that when having a rooted acyclic graph is a sufficient condition to obtain UGAS of the set (8).

Next we show that when , having a rooted acyclic graph is also a necessary condition in order to have UGAS with no purely or eventually discrete-time solutions. Suppose by contradiction that there exists a cycle in the rooted graph and that the set is UGAS under the hybrid dynamics (7) with no purely discrete-time solution. Let be the same starting and ending vertex/agent of a cycle. By the definition of cycles, such always exists. Thus, there exists a positive integer such that . Moreover, when after flowing for at least seconds, all agents in the set will sequentially update their clocks to 1. However, since and , there exists solutions where the agents in the set keep updating their clock state to without flowing, i.e., these solutions will jump for ever. This establishes the result.

This finishes the proof for Theorem 3.3. For the next section, we will now implement the hybrid system (7) for different examples of network topologies to illustrate our main result.

V Simulation Results

Consider the hybrid system (7) for five resetting clocks. The tunable parameter for all agents is chosen as and the frequency of the clocks is selected as . We implement this system with an underlying rooted acyclic graph. We provide three examples of such graphs as shown in Figure 4. For each example, we validate Theorem 3.3 by showing that: (i) the hybrid system (7) synchronizes, (ii) the convergence time is related to the depth of graph, (iii) no solution remains in the jump set for ever, i.e., all solutions eventually have to flow.

For digraphs (a) and (b) in Figure 4, we used the same initial conditions to implement Algorithm 1. The simulation results over these networks are then illustrated in Figure 5. Observe that for the network topology (a), as soon as hits 1, it will synchronize its out-neighbors and , after which will force to synchronized. However, remains unaffected. Soon after that, hits 1 and synchronizes with it in second. After that, all the clocks remain synchronized and there is no solution that stays in the jump set. Note that in this case, the agents synchronize in 1 second which is less that seconds. Similarly, for the network topology (b), as soon as hits 1, it synchronizes with itself. Soon after that hits 1 and synchronize together. Finally, as hits 1, it synchronizes all the agents in the network. Again, note that all the clocks still synchronize in 1 second which is less that seconds. Observe that even though the network synchronization behavior of case (a) is different from case (b), overall network converges to the set in 1 second. This is due to the fact that in both graphs the number of edges are different but the depths are the same.

Now, consider the network topology (c) as shown in Figure 4. We set the initial conditions of each clock very close to zero and we apply Algorithm 1 to obtain the plots shown in Figure 6. For this case, observe that when hits 1, it synchronizes with itself. Then both hit 1 and synchronize with them and so on. Note that when the network converges in 3.6 seconds which is less than the theoretical worst-case convergence time of seconds. Similarly, if we homogeneously set the value of , the network will synchronize in seconds which is less than the theoretical worst-case convergence time of second.

Hence, the simulation results validate the following aspects: (i) For rooted acyclic graphs, the hybrid system (7) synchronizes, (ii) the convergence time is related to the depth of the digraph by Theorem 3.3, and (iii) the hybrid dynamics generate no purely discrete-time solutions, i.e., solutions that remain in the jump set.

(a)

(b)

(c)
Fig. 4: Rooted Acyclic Graphs with depths a) two b) two c) three. (Black indicates the root vertex)
Fig. 5: Resetting Clocks Synchronization over graphs (a)-(b) in Figure 4.
Fig. 6: Resetting Clocks Synchronization over graph (c) in Figure 4.

Vi Conclusions and Further Discussions

We have introduced in the paper the problem of characterizing the set which is comprised all the information flow topologies that will guarantee robust global synchronization of the resulting network of resetting clocks. Our investigation has led us to the partial solution (Theorem III.3) to the above problem: Specifically, we have shown that rooted acyclic digraphs always belong to for any . Moreover, if , then is exactly the class of rooted acyclic digraphs. We have further computed in the paper the convergence time for a networked system to reach global synchronization. In particular, we have related the convergence time to the depth of the underlying rooted acyclic digraph. Finally, we have provided numerical results that illustrate the main points of this paper.

We note here that the class of rooted acyclic digraphs can be further extended to a broader class of digraphs: One can replace the unique root in a rooted acyclic digraph with a strongly connected component. We refer the reader to Figure 7-a) for an illustration. We can show that with such a replacement, the resulting digraph still satisfies item (1) of Theorem III.3. Due to space constraints, we omit the proof here and we will present it in another occasion.

Amongst other potential extensions of the work, we first mention the problem of synchronization of logic-states and sampled-data systems (see [1]) as a straightforward generalization of the framework considered in the paper. Indeed, by combining the synchronization results of this paper with the coordination results of [17] and [1], we can now consider the design of several distributed asynchronous control and optimization algorithms that do not require anymore of strongly connected graphs.

Finally, a future direction of this work will introduce random graph theory to the existing framework. The randomness can be either on the connecting edges or on the vertices (for which, vertices jump or not will be at random). We believe that such a stochastic framework may expand significantly the class of digraphs that can guarantee global synchronization almost surely.

(a)

(b)
Fig. 7: a) Root set forms a strongly connected component b) Condensation Graph is Rooted Acyclic

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