Consensus in Self-Similar Hierarchical Graphs and Sierpiński Graphs: Convergence Speed, Delay Robustness, and Coherence

Qi, Yi; Zhang, Zhongzhi; Yi, Yuhao; Li, Huan

doi:10.1109/tcyb.2017.2781714

Cited by 39 publications

(22 citation statements)

References 79 publications

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“…Physical objects in IoT network should be robust with respect to different parameters, including hardware failure, environmental uncertainty and communication failures. Robustness to uncertainty and noise can be effectively measured by network coherence [30], [31]. Definition 2: Network coherence is also defined as robustness to noise, and it can be measured by the deviation of each node's state from the global average of all current states.…”

Section: Brief Review Of Average Consensus Algorithmmentioning

confidence: 99%

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Analysis of Distributed Average Consensus Algorithms for Robust IoT networks

Dhuli,

Atik

2021

Preprint

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Internet of Things(IoT) is a heterogeneous network consists of various physical objects such as large number of sensors, actuators, RFID tags, smart devices, and servers connected to the internet. IoT networks have potential applications in healthcare, transportation, smart home, and automotive industries. To realize the IoT applications, all these devices need to be dynamically cooperated and utilize their resources effectively in a distributed fashion. Consensus algorithms have attracted much research attention in recent years due to their simple execution, robustness to topology changes, and distributed philosophy. These algorithms are extensively utilized for synchronization, resource allocation, and security in IoT networks. Performance of the distributed consensus algorithms can be effectively quantified by the Convergence Time, Network Coherence, Maximum Communication Time-Delay. In this work, we model the IoT network as a q-triangular r-regular ring network as q-triangular topologies exhibit both small-world and scale-free features. Scale-free and small-world topologies widely applied for modelling IoT as these topologies are effectively resilient to random attacks. In this paper, we derive explicit expressions for all eigenvalues of Laplacian matrix for q-triangular r-regular networks. We then apply the obtained eigenvalues to determine the convergence time, network coherence, and maximum communication timedelay. Our analytical results indicate that the effects of noise and communication delay on the consensus process are negligible for q-triangular r-regular networks. We argue that q-triangulation operation is responsible for the strong robustness with respect to noise and communication time-delay in the proposed network topologies.

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Section: Brief Review Of Average Consensus Algorithmmentioning

confidence: 99%

“…Definition 4: Maximum Communication Time-Delay [31], [5] measures the ability of consensus algorithm resilient to maximum communication delay between nodes and it is expressed as…”

Section: Brief Review Of Average Consensus Algorithmmentioning

confidence: 99%

Analysis of Distributed Average Consensus Algorithms for Robust IoT networks

Dhuli,

Atik

2021

Preprint

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“…dynamics [2], [3], [4] and bond percolation [5] on a graph. Concerning the Laplacian matrix, its smallest and largest nonzero eigenvalues are closely related to the time of convergence and delay robustness of the consensus problem [6]; all the nonzero eigenvalues determine the number of spanning trees [7] and the sum of resistance distances over all node pairs [8], [9], with the latter encoding the performance of different dynamical processes, such as the total hitting times of random walks [10], [11], [12] and robustness to noise in consensus problem [13], [14], [15].…”

Section: Introductionmentioning

confidence: 99%

Maximizing the Smallest Eigenvalue of Grounded Laplacian Matrix

Wang¹,

Zhou²,

Li³

et al. 2021

Preprint

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For a connected graph G = (V, E) with n nodes, m edges, and Laplacian matrix L, a grounded Laplacian matrix L(S) of G is a (n − k) × (n − k) principal submatrix of L, obtained from L by deleting k rows and columns corresponding to k selected nodes forming a set S ⊆ V . The smallest eigenvalue λ(S) of L(S) plays a pivotal role in various dynamics defined on G. For example, λ(S) characterizes the convergence rate of leader-follower consensus, as well as the effectiveness of a pinning scheme for the pinning control problem, with larger λ(S) corresponding to smaller convergence time or better effectiveness of a pinning scheme. In this paper, we focus on the problem of optimally selecting a subset S of fixed k ≪ n nodes, in order to maximize the smallest eigenvalue λ(S) of the grounded Laplacian matrix L(S). We show that this optimization problem is NP-hard and that the objective function is non-submodular but monotone. Due to the difficulty to obtain the optimal solution, we first propose a naïve heuristic algorithm selecting one optimal node at each time for k iterations. Then we propose a fast heuristic scalable algorithm to approximately solve this problem, using derivative matrix, matrix perturbations, and Laplacian solvers as tools. Our naïve heuristic algorithm takes Õ(knm) time, while the fast greedy heuristic has a nearly linear time complexity of Õ(km), where Õ(•) notation suppresses the poly(log n) factors. We also conduct numerous experiments on different networks sized up to one million nodes, demonstrating the superiority of our algorithm in terms of efficiency and effectiveness compared to baseline methods.

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“…This concept of the network coherence helps to study the relationship between the Laplacian eigenvalues and network consistency. Great progress has been made for some special networks such as Vicsek fractals [10], tree-like networks [11], Sierpiński graphs [18] and weighted networks [19]. Many works have been devoted to studying the network coherence.…”

Section: Introductionmentioning

confidence: 99%

Network Coherence in a Family of Book Graphs

Chen¹,

Sun

2020

Front. Phys.

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In this paper, we study network coherence characterizing the consensus behaviors with additive noise in a family of book graphs. It is shown that the network coherence is determined by the eigenvalues of the Laplacian matrix. Using the topological structures of book graphs, we obtain recursive relationships for the Laplacian matrix and Laplacian eigenvalues and further derive exact expressions of the network coherence. Finally, we illustrate the robustness of network coherence under the graph parameters and show that the parameters have distinct effects on the coherence.

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Consensus in Self-Similar Hierarchical Graphs and Sierpiński Graphs: Convergence Speed, Delay Robustness, and Coherence

Cited by 39 publications

References 79 publications

Analysis of Distributed Average Consensus Algorithms for Robust IoT networks

Analysis of Distributed Average Consensus Algorithms for Robust IoT networks

Maximizing the Smallest Eigenvalue of Grounded Laplacian Matrix

Network Coherence in a Family of Book Graphs

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