The impact of multi-group multi-layer network structure on the performance of distributed consensus building strategies

Transcription

1 INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL Int. J. Robust. Nonlinear Control 203; 23: Published online 20 February 202 in Wiley Online Library (wileyonlinelibrary.com). DOI: 0.002/rnc.2783 The impact of multi-group multi-layer network structure on the performance of distributed consensus building strategies Yan Wan *,, Kamesh Namuduri, Swathik Akula and Murali Varanasi Electrical Engineering, University of North Texas, USA SUMMARY We consider a structural approach to the consensus building problem in multi-group multi-layer (MGML) distributed sensor networks (DSNs) common in many natural and engineering applications. From among the possible network structures, we focus on bipartite graph structure as it represents a typical MGML structure and has a wide applicability in the real world. We establish exact conditions for consensus and derive a precise relationship between the consensus value and the degree distribution of nodes in a bipartite MGML DSN. We also demonstrate that for subclasses of connectivity patterns, convergence time and simple characteristics of network topology can be captured by explicit algebra. Direct inference of the convergence behavior of consensus strategies from MGML DSN structure is the main contribution of this paper. The insights gained from our analysis facilitate the design and development of large-scale DSNs that meet specific performance criteria. Copyright 202 John Wiley & Sons, Ltd. Received 0 May 200; Revised 29 November 20; Accepted 9 December 20 KEY WORDS: consensus building; multi-group multi-layer networks; network structure. INTRODUCTION Consensus problem widely appears in natural and engineering applications and has received extensive study in a multitude of fields including sensor networking, distributed computing, and decentralized control [ 5] among others. However, it has not been investigated extensively in networks with multi-group multi-layer (MGML) communication structures; that is, networks composed of multiple groups, in each of which some nodes are designated as leaders and given the responsibility of communicating with other groups through certain layered communication schemes. Such communication structure is commonly observed in nature and has tremendous potential for engineered systems. For instance, this structure is typically observed in bird flocking as studied in, for example, [6, 7]. Similarly, in distributed sensor networks (DSNs), properly designed MGML communication structure is envisioned to be more secure and energy efficient [8, 9]. Network topology has been known to be crucial to the performance of consensus strategies, particularly to convergence time (see e.g., [0, ]), which impacts communication overhead and network lifetime in DSN applications. In designing MGML network structures for consensus building, one question is whether there exists a quantitative relationship between the network structure and its performance. This question is particularly important in large-scale networks, where scalability is of critical concern. We address this issue by studying the network design problem for some common MGML structures. Examples of such structures are illustrated in Figure. Bipartite graph structure (Figure (a)), which captures a wide arrange of 2-layer lead-follower communication topology, is widely used in coding theory [2, 3] and has been recently adopted *Correspondence to: Yan Wan, Electrical Engineering, University of North Texas, USA. ywan@live.com Copyright 202 John Wiley & Sons, Ltd.

2 654 Y. WAN ET AL. (a) Bipartite Graph Model (b) Hierarchical Model (c) Mesh Model (d) Semi - regular Graph Model Figure. Four sample networks, each consisting of 6 nodes (2 member nodes V to V2,and four leader nodes, C to C4) and four groups. In each group, one node serves as the leader. Information flows in both directions, that is, from the agents to the leaders and from the leaders to the agents. for sensor network applications [4]. Hierarchical graph structure (Figure (b)), a generalization of the bipartite graph structure, can capture more complex MGML communication schemes. In this paper, we focus first on the bipartite graph structure, and analyze this structure in detail. Then, we briefly discuss the hierarchical graph structure, which is a simple generalization of the bipartite structure. Our objective is to investigate the effect of network structure on the convergence behavior of consensus building strategies for MGML DSNs. Specifically, we take a structural approach to study the research questions of interest in consensus building, including whether consensus can be reached by the network, what is the final consensus value, and how many iterations are needed to reach the consensus. These results allow us to explicitly design DSNs with connectivity patterns that meet the desirable performance specifications without using complicated numerical computations. Next, we describe the main contributions of this paper and provide a brief review of the most relevant literature in the field with the aim of connecting with and distinguishing our work from the existing literature. () Systematic approach to study consensus building algorithms in MGML DSNs. Consensus building studies are mostly concerned with single-group single-layer networks consisting of egalitarian sensor nodes [, 5, 6]. We focus on MGML consensus building dynamics in this paper. We will show that through a simple manipulation, the dynamics can be transformed to a simple first-order dynamics captured by a stochastic (or equivalently Laplacian) matrix widely studied in the literature (see a tutorial paper [4]). We then use algebraic graph theory to connect the network structure with its dynamics and to provide useful insights into network design [7]. (2) Direct inference of convergence behavior of consensus strategies from simple features of MGML DSN structure. Complementary to the existing studies (e.g., [3]), which provide general results relating network structure with its performance, we derive explicit closedform relationship between the two. Some efforts on directly relating network structure and the performance can be found in [5, 6]). We further extend these explicit structure-dependent performance analysis to more complex graph structures. (3) Effective design of communication structure for a desired consensus value and convergence time. Our analytical results linking MGML communication structure and consensus performance lead to efficient and scalable strategies for designing DSNs (e.g., through the selection of the number of leaders and network connections) to meet desirable performance requirements. This is different from many numerical network design studies such as [], in which a semi-definite programming approach is used to find optimal network design for fast convergence. Some limited studies on systematic network designs can also be found in [8 20]. (4) Novel way to investigate algebraic graph theory-related aspects from indirect graphs rather than graphs directly associated with network dynamics. In this paper, the network structures directly associated with network dynamics are much more complicated than the commonly studied structures in the literature [5,6]. However, by working with indirect/hidden graphs Copyright 202 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control 203; 23: DOI: 0.002/rnc

3 THE IMPACT OF MGML STRUCTURE ON CONSENSUS BUILDING 655 (e.g., the graphs capturing MGML routing structures), many structure-related results can be obtained. Investigating indirect graphs provides a new way to classifying network structures and finding the connections between network structures and the associated dynamics. The remainder of this paper is organized in the following. In Section 2, we formulate the research problem, and Section 3 includes results on consensus building for bipartite MGML structures. Discussions and future directions are provided in Section PROBLEM FORMULATION We begin with a functional description of the network. Each sensor node in the network has the same functionality in the sense of sensing and information processing. However, in each group, some sensor nodes are designated as leaders or fusion centers (see Figure 2(a)). To facilitate our analysis, we remove the sensing functionality from fusion centers, and introduce the concept of virtual fusion centers (VFCs), whose responsibilities are solely on establishing connections within and between the groups (as shown in Figure 2(b)). The VFCs are represented by square nodes, the sensors are represented by circles, and the communication links are represented by edges. The functionality of a VFC can be implemented at any node. Our focus in this paper is on networks whose structure can be mapped to a bipartite graph as shown in Figure (a) (see [4] for a detailed description). Bipartite graphs, are also known as Tanner graphs [3], in coding theory. The networking structure of a bipartite graph is represented by its parity matrix, also referred to as routing matrix in the context of a DSN. Assuming that a DSN has m VFCs and n sensor nodes, its structure can be captured by a mn routing matrix H,inwhich the entry in.i, j/ is set to if there is communication link between the i th VFC and thej th 0 sensor node. For instance, the routing matrix associated with Figure 2 is H D.If 0 the degrees (i.e., the number of communication links) of all VFC nodes are the same, we say that the DSN is regular. Now let us describe the consensus building process. The motivation for the consensus building strategies that we study comes from the belief propagation concepts developed by Gallager in 963 [2] with applications to channel coding and Pearl s algorithm developed in the 980s in artificial intelligence community. It begins with the sensors observing a common phenomenon by taking their own measurements. The iterative consensus building algorithm is consisted of two operations in each iteration: () Forward operation each sensor sends its own value to the VFC to which it is connected; and (2) Backward operation each VFC updates the sensor nodes with its new result according to some averaging algorithm. A VFC may employ a linear or a nonlinear process to aggregate its input dependent upon the type of application. In this paper, it is assumed that the collected values from all connected sensor nodes are aggregated by a simple averaging operation based on the number of sensors that the VFC receives information from. In other words, each VFC calculates the average of all observations it receives and sends this new estimate to each connected sensor node. Each sensor node then updates its value by taking the average of new estimates it receives from the connected VFCs. We refer to this typical consensus building algorithm as the iterative averaging algorithm. (a) (b) (c) VFC VFC2 S S2 FC S3 S4 FC2 S S3 S4 S2 S3 S4 Figure 2. (a) An example of a regular DSN, (b) its bipartite graph representation, and (c) the graph capturing the direct network structure associated with the consensus building dynamics. Copyright 202 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control 203; 23: DOI: 0.002/rnc

4 656 Y. WAN ET AL. 3. INFERRING CONSENSUS FROM NETWORK STRUCTURE In this section, we analyze the impact of network structure on consensus condition and convergence time, using the MGML DSN model and the iterative averaging algorithm formulated in Section 2. We first present the general conditions for convergence in consensus building strategies and the consensus estimate from structure in Section 3.. In Section 3.2, we discuss an important performance measure the time needed to reach consensus. We provide results that directly infer convergence time from some very simple structural characteristics of a DSN, such as the number and degrees of sensor nodes and VFCs, and so on. 3.. Condition for convergence and estimate of the consensus value in consensus building strategies In order to analyze the convergence behavior of consensus strategies, let us first obtain the dynamics of sensor values. We introduce a vector xœk 2 R n to hold the values of all sensor nodes at cycle k. According to the iterative averaging algorithm, after the forward operation, the values held at the VFCs are yœk C D K 2 H xœk, where the m m diagonal matrix K 2 D Œdiag.H n /. Here, diag() represents the operation that places the vector in the parenthesis onto the main diagonal. Similarly, the values at the sensors after the backward operation can be computed using xœk C D K H T yœk C, where the n n diagonal matrix K D Œdiag.H T m /. The effect of the two operations in one cycle can be captured by the following LTI difference equation: xœk C D AxŒk D K H T K 2 H xœk, () where A is the system matrix, capturing the direct network structure associated with the consensus building dynamics (see Figure 2(c) for an example). We note that even though the routing structure associated with H is bipartite, the direct network structure can be very complicated. The eigenvalues of A provide rich information into the system dynamics, and in turn on the convergence behavior of the consensus strategy. We note that () maps the consensus problem in bipartite networks to a typical discrete-time first-order consensus problem studied in the literature, and hence general results presented in, for example, [,3] can be adapted for our analysis. Different from many of these studies, the system matrix in () is not necessarily symmetric. Here in Theorem, let us first describe the properties of eigenvalues associated with A, through the introduction of a new symmetric matrix A. O We denote the normalized left and right eigenvectors associated with eigenvalue i of A as w T i and v i, respectively. Theorem The eigenvalues of the system matrix A described in () are real, simple, and reside between 0 and. At least one eigenvalue is equal to. Moreover, the multiplicity of the eigenvalue with a value 0 is greater than or equal to n m. Because A is a stochastic matrix, it has at least one eigenvalue at with associated right eigenvector n. Now let us show that the eigenvalues of matrix A are real, simple, and nonnegative. To do this, let us show that the eigenvalues of A are the same as the eigenvalues of the symmetric matrix AO D K 2 H T K 2 HK 2. Because w T i K H T K 2 H D w T i i,wehavew T i K 2 K 2 H T K 2 HK 2 D w T i K 2 i by multiplying K 2 on both sides. Therefore, for any eigenvalue i and left eigenvector w T i associated with A, AO D K 2 H T K 2 HK 2 has the same eigenvalue i with the left eigenvector w T i K 2. Because eigenvalues associated with a symmetric and positive semi-definite matrix AO are all real, simple, and nonnegative, the eigenvalues of A have the same properties. What is left to show is that the multiplicity of the eigenvalue at 0 is greater than or equal to n m. This is straightforward because the multiplicity of the eigenvalue 0 equals n rank.a/ and the rank of A is less than or equal to the rank of H, which is at most m. Copyright 202 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control 203; 23: DOI: 0.002/rnc

5 THE IMPACT OF MGML STRUCTURE ON CONSENSUS BUILDING 657 As shown in Theorem, because the eigenvalues are real and reside between 0 and, we can denote the range of eigenvalues of the system matrix A as D > 2 > ::: > n > 0. In Theorem 2, we will give the necessary and sufficient condition for consensus, and present a quantitative description of the dependence of consensus value on network structure captured by the routing matrix H. We note that general results on consensus condition and consensus value in terms of system matrix A are well known [, 3, 4]. Theorem 2 describes the results in terms of the routing matrix H instead of the system matrix A. Theorem 2 Consider a MGML DSN represented by a bipartite graph with routing matrix H. The consensus of the DSN is guaranteed if and only if there is at least a path between any pair of sensor nodes in the graph. The final consensus value that the DSN converges to is m H n m H xœ0, where xœ0 2 R n contains initial values of all sensor nodes in the DSN. Let us decompose the system matrix A and rewrite () as xœk D P n id v i k i wt i xœ0. Only the terms having eigenvalue contribute to the final values at sensor nodes. First, let us prove the if condition. Because all the sensor nodes are connected, we know that the system matrix A is irreducible. Moreover, because the underlining Markov chain for A is ergodic, from the Perron Frobenius theorem [7], we know the following facts: () the dominant eigenvalue of A,,isand strictly larger than the magnitudes of all other eigenvalues; and (2) the right eigenvector associated with it, v,is n. The two facts guarantee that the sensor nodes will reach the same value asymptotically. To prove the only if condition, let us note the fact that if sensor nodes are not connected, matrix A will have at least two eigenvalues at. The eigenvectors associated with all eigenvalues of will contribute to the final consensus. Hence, the sensor nodes cannot reach consensus. In the case that the consensus condition is satisfied, the final value of xœk can be calculated as w T xœ0. It can be easily checked that w T D m H n m H, because this expression satisfies w T A D wt. Theorem 2 gives the condition for convergence of the consensus strategy and the dependence of consensus value on initial values of all sensor nodes. We note that in works such as [], the final consensus value is an average of all sensor nodes initial values. In reality, sensor nodes may contribute unequally to the final consensus value; hence, we allow the contribution of each sensor to the consensus value as a free design variable. It is also interesting to notice that the degrees of VFCs do not impact the final consensus value. In fact, the degrees of sensor nodes (the column sum of H matrix, as indicated by m H ) are the sole factors in determining the final consensus value. The final consensus value is the average of the initial conditions weighted by the corresponding sensor nodes degrees. This simple relationship allows us to quickly design routing structure (captured by the routing matrix H ) to achieve certain consensus value. In particular, we can achieve the desired consensus value by assigning important sensor nodes with higher degrees. Theorem 2 naturally leads to the following corollary. Corollary Consider a MGML DSN represented by a bipartite graph with routing matrix H. Given that the DSN can reach consensus, the final consensus value is the average of all initial values if the degrees of all sensor nodes are the same Dependence of convergence time on network structure As is well known and can be found in, for example, [, 3], the convergence time (i.e., the number of iterations such that the difference between sensor value and final consensus value is within ı Ergodic implies both recurrent and aperiodic. Copyright 202 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control 203; 23: DOI: 0.002/rnc

6 658 Y. WAN ET AL. of its initial value) is log 2 ı as ı! 0, where 2 0 is the second largest eigenvalue associated with the system matrix A. The case that 2 D 0 is trivial in that consensus can be reached in one iteration. Whereas this quantitative result is important and well known, it does not directly tell us the convergence time from DSN structures. In the rest of this section, we relate the structure of the routing matrix H to 2 0, and in turn to the convergence time. In Theorem 3 and Corollary 2, we consider a general DSN structure first and then present the results for a regular DSN. Theorem 3 The second largest eigenvalue of the system matrix A associated with a general DSN model represented by a bipartite graph can be found as the maximum of id.k 2 2 H y/ 2 i,wherey2rn is subject to the following constraints: () P n id K i y i D 0; and(2) P n id K i yi 2 D. Matrix AO is symmetric. The Courant Fischer theorem informs us that the second largest eigenvalue 2 associated with A (or A) O can be found as [7] 2 D max x?w T K 2,jjxjjD x T Ax O D max x?w T K 2,jjxjjD x T K 2 H T K 2 HK 2 x. (2) Let us introduce a new vector y D K 2 x. Because w T aligns with nk, y is subject to the constraint that y? K n. Moreover, because jjxjj D, we obtain jjk 2 yjj D. This result is then straightforward. Corollary 2 Considering a regular DSN, the second largest eigenvalue of the system matrix A can be found as the maximum of p id.h y/2 i,wherep is the regular degree of VFCs, and y 2 Rn is subject to the following constraints: () P n id K i y i D 0; and(2) P n id K i yi 2 D. This result can be derived directly from Theorem 2, with the observation that K 2 D p I. Theorem 3 and Corollary 2 provide an approach to obtain the second largest eigenvalue from the routing structure H, and hence the convergence time for various subclasses of DSNs. Complementing the results in the consensus building literature [], we show in Theorems 4, 5, and 6 that convergence time can be directly obtained from some simple structural characteristics for subclasses of MGML DSNs. Theorem 4 Consider a regular DSN that contains m groups. Each group has VFC and k>sensor nodes that communicate with it. In each group, there is one and only one node that communicates with all VFCs. No other cross-group communication exists. The consensus time for this regular DSN is log k kcm ı. The DSN considered in the theorem has m VFCs with regular degree p D k C m and n D mk sensor nodes. Without loss of generality, let us index the sensor node within group i and communicates with all VFCs with C k.i /. Moreover, we introduce set S containing the indices of such sensor nodes for all groups, and set G i containing the indices of all sensor nodes in group i. Copyright 202 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control 203; 23: DOI: 0.002/rnc

7 THE IMPACT OF MGML STRUCTURE ON CONSENSUS BUILDING 659 Let us define the Lagrange operator L as mx nx.h y/ 2 i C. K i y i / C ˇ. id id nx id K i y 2 i / D mx. X id j 2G i CS The optimal solution for (y l,, ˇ) is the one @y l D 0 leads to 2 mx. X id j 2G i l y j / 2 C. C ˇ. nx id nx id K i y i / K i y 2 i / (3) D 0 for D D 0. y j / C K l C 2ˇK l y l D 0 if l 2 G i (4) Equation (4) informs us that for each group i from to m, y l are equal for all l 2 G i S, because K l D for all l 2 G i S. Similarly, y l are equal for l 2 S as K l D m for l 2 S. Therefore, the constraint P n id K i y i D 0 becomes m 2 y C.k / id y k.i /C2 D 0. id.h y/ Finally, let us obtain the maximum of 2 P i nid. Using the constraint that m 2 y K y 2 C.k / i i id y k.i /C2 D 0 and the observations from (4), we obtain id.h y/2 i P n id K i y 2 i id D.P j 2G i CS y j / 2 m P i2s y2 i C P id D.P j 2S y j C P j 2G i S y j / 2 i S y2 i m 2 y 2 C.k / id y2 k.i /C2 id D.my C.k /y k.i /C2 / 2 m 2 y 2 C.k / id y2 k.i /C2 D m3 y 2 C.k /2 id y2 k.i /C2 C 2.k /my id y k.i /C2 m 2 y 2 C.k / id y2 k.i /C2 D m3 y 2 C.k /2 id y2 k.i /C2 m 2 y 2 C.k / id y2 k.i /C2 6.k P /2 m id y2 k.i /C2.k / D k id y2 k.i /C2 The case k D is trivial as 2 D 0 and consensus can be reached in one iteration. The equality holds when y i D 0 for all i 2 S. Hence, the maximum of D k. Because the second largest eigenvalue 2 is the maximum of p id.hy/ 2 i P nid K i y 2 i id.hy/ 2 P i nid, we obtain K y 2 2 D k kcm. i i Theorem 4 says that for a common class of DSN that contains multiple groups, and in each of the groups exists a leader that communicates to all the other leaders, the consensus time is only dependent on the number of groups and the number of nodes in each group. For example, for the DSN structure shown in Figure 3, according to Theorem 4, 2 can be directly obtained as 0.5, and the consensus time as log 0.5 ı. It is worthwhile to note that the value of the theorem resides in largescale networks, for which the consensus time can be inferred directly from simple characteristics of DSN structures, without any complicated numerical computation. Theorem 5 Consider a DSN containing two VFCs with regular degree p and n < 2p sensor nodes. The consensus time is logn p p ı. (5) The proof is omitted as it is similar to the proof of Theorem 4. Copyright 202 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control 203; 23: DOI: 0.002/rnc

8 660 Y. WAN ET AL. (a) (b) Figure 3. (a) A DSN consisting of three groups; each group leader communicates to the other two group leaders. (b) The corresponding bipartite graph representation with k D 4 and m D 3. In Theorem 5, we consider another class of regular DSN containing two VFCs. The consensus time can be directly inferred from the regular degree and the number of sensor nodes in the DSN. The theorem is exemplified in an example shown in Figure 4(a). According to Theorem 5, we can directly obtain the second largest eigenvalue as 2 5, and therefore the consensus time as log 0.4ı. In the next theorem, we consider an irregular DSN that has one and only one sensor node that communicates with all VFCs. Theorem 6 Consider a DSN that contains m>2groups (see Figure 4(b)). Each group has VFC and k> sensor nodes that communicate with it. There exists one and only one sensor node in the DSN that communicates with all VFCs. No other cross-group communication exists. The consensus time for this DSN is log k ı. kc The DSN considered in the theorem has m VFCs and n D mk sensor nodes. Let us use G i to indicate the set containing the indices of sensor nodes in group i. Without loss of generality, let us assume that the index of the sensor node communicating with all VFCs is, and the group containing this particular sensor node is group. Moreover, we assume that the sensor nodes in group i are indexed from k.i / C to ki. (a) (b) Figure 4. (a) The bipartite graph representation for a regular DSN that has two VFCs with regular degree 6 and 8 sensor nodes. (b) The bipartite graph representation for an irregular DSN studied in Theorem 6. Copyright 202 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control 203; 23: DOI: 0.002/rnc

9 THE IMPACT OF MGML STRUCTURE ON CONSENSUS BUILDING 66 P mid 2.K2 Let us use Theorem 3 to obtain 2, which is the maximum of P H y/2 i nid,wherey is subject K y 2 i i to the constraint: P n id K l y i D 0. Using Lagrange multipliers, we can show that for each group i from to m, all entries of y corresponding to the sensor node inside a group G i ¹º are the same. This result is discussed in the proof of Theorem 4. Further, similar to the proof of Theorem 4, we find when m 2, id.k 2 2 H y/ 2 i P n id K i y 2 i 6..k /2 /y 2 k 2 C. k2 kc /.k /y2 2 C id2 ky2 k.i /C id2 y2 k.i /C, (6) The equality holds when y D 0. Denoting y2 2 be X and equation becomes.k / 2 k X C k2.kc/ Y.k /X C ky D k k C k 2 Y.k /Y kc.k /X C ky 6 k C k id2 y2 k.i /C k 2 Y.k /Y kc ky be Y, the preceding D k k C (7) The equality holds when y i2g D 0. The proof is complete. 4. CONCLUDING REMARKS AND FUTURE WORK In this paper, we study bipartite structure as a typical MGML communication structure and show that consensus properties can be directly inferred from the characteristics of its routing structure. Theorems 4, 5, and 6 demonstrate three classes of DSNs, whose convergence time can be directly obtained from connectivity structures based upon the results in Theorem 3. Using this approach, we can obtain similar results for other classes of DSNs, including many irregular DSNs. We emphasize that by linking eigenvalues with the properties of the indirect routing matrix H rather than the system matrix A directly, rich structural information can be brought out to aid in the network analysis and design. The inference of eigenvalue from the structure of graphs (especially asymmetric graphs) is not a trivial task. This work provides an interesting approach to expose hidden graph structure that is not obvious from the graph directly associated with network dynamics; such hidden structure may help to address algebraic graph problems with rich insights. These results also provide a direct approach to DSN design. First, the degrees for sensor nodes can be selected to reach desired consensus value. Consensus time can also be achieved through the DSN design. For instance, for the class of DSNs considered in Theorem 4, we can easily choose the number of VFCs for any predefined convergence time by using the explicit relation between 2 and the number of VFCs. Considering a network containing 000 sensor nodes and the desired convergence time of 0 (associated with ı D 0.0), we can calculate that m D 25 by solving for log (according to the inequality log m 2 ı 6 k). The results are particularly Cm 000 m useful when large-scale networks are considered, as existing numerical solutions for computing convergence times can be very time-consuming []. The results presented so far are focused on bipartite graphical models. However, they can be easily extended to other graphical models and iterative consensus algorithms as well. For instance, let us consider a three-layer hierarchical tree structure. The system dynamics for this three-layer hierarchical structure (see Figure (b)) can be easily captured by xœkc D K H T.K 3H2 T K 4H 2 /K 2 H xœk, where H and H 2 are the routing matrices associated with the first and second layers, respectively, and K D Œdiag.H T /, K 2 D Œdiag.H /, K 3 D Œdiag.H2 T /,andk 4 D Œdiag.H 2 /. For the particular case in Figure (b), the system dynamics can be represented by xœk C D.I 33 4 / T. T /.I /xœk. Explicit results characterizing the convergence behavior, similar to those derived for the bipartite graphical models, can thus also be derived. We leave the full development of analysis on hierarchical graph models and other MGML structures in future work. Copyright 202 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control 203; 23: DOI: 0.002/rnc

10 662 Y. WAN ET AL. ACKNOWLEDGEMENTS The work is supported by the National Science Foundation under grant numbers and The authors would like to thank Dr Sandip Roy and Mr Rahul Dhal at Washington State University for illuminating discussions. We also thank the editor and reviewers for valuable suggestions. REFERENCES. Boyd S, Diaconis P, Xiao L. Fast mixing Markov chain on a graph. SIAM Review 2004; 46(4): Lynch NA. Distributed Algorithms. Springer: San Mateo, CA, Olfati-Saber AFR, Murry RM. Consensus and cooperation in networked multi-agent systems. Proceedings of the IEEE January 2007; 95(): Ren W, Beard RW, Atkins EA. Information consensus in multivehicle cooperative control. IEEE Control Systems Magazine April 2007: Roy S, Saberi A, Herlugson K. A control-theoretic perspective on the design of agreement protocol. International Journal of Robust and Nonlinear Control December 2006; 7(0 ): Nagy M, Ákos A, Biro D, Vicsek T. Hierarchical group dynamics in pigeon flocks. Nature April 200; 464: Quera V, Beltran FS, Dolado R. Flocking behaviour: agent-based simulation and hierarchical leadership. Journal of Artificial Societies and Social Simulation 200; 3(2):8. 8. Tarannum S, Suraiya S, Asha DS, Venugopal KR. Dynamic hierarchical communication paradigm for wireless sensor networks: a centralized, energy effient approach. Wireless Sensor Network November 2009; (4): Zhou B, Ngoh LH, Lee BS, Fu CP. HDA: a hierarchical data aggregation scheme for sensor networks. Computer Communications November 2006; 29: Kolmogorov V, Wainwright M. On the optimality of tree-reweighted max-product message passing. Proceedings of the 2st Conference on Uncertainty in Artificial Intelligence, Arlington, Virginia, July 2005; Kschischang F. Codes defined on graphs. IEEE Communications Magazine August 2003; 4(8): Gallager RG. Low-Density Parity-Check Codes. MIT Press: Cambridge, MA, Tanner RM. A recursive approach to low complexity codes. IEEE Transactions on Information Theory September 98; 27(5): Shukair M, Namuduri K. LDPC-like belief propagation algorithm for consensus building in wireless sensor network. 43rd Annual Conference on Information Sciences and Systems, Baltimore, MD, March 2009; Chen G, Duan Z. Network synchronizability analysis: a graph-theoretic approach. Chaos 2008; 8: 03702() 03702(0). 6. Duan Z, Chen G, Huang L. Complex network synchronizability: analysis and control. Physical Review E 2007; 76:05603() 05604(6). 7. Chung FRK. Spectral Graph Theory. American Mathematical Society: Providence, RI, Roy S, Saberi A, Petite P. Scaling: a canonical design problem for networks. Proceedings of the 2006 American Control Conference, Minneapolis, Minnesota, June 2006; Wan Y, Roy S, Saberi A. A new focus in the science of networks: towards methods for design. Proceedings of the Royal Society A March 2008; 464: Wan Y, Roy S, Wang X, Saberi A, Yang T, Xue M, Malek B. On the structure of graph edge designs that optimize the algebraic connectivity. Proceedings of 47th IEEE Conference on Decision and Control, Cancun, Maxico, December 2008; Copyright 202 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control 203; 23: DOI: 0.002/rnc