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Network Coherence in Fractal Graphs 2011 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC) Orlando, FL, USA, December 12-15, 2011 Network Coherence in Fractal Graphs Stacy Patterson and Bassam Bamieh Abstract— We study distributed consensus algorithms in frac- with first-order dynamics, the per-node variance of the de- tal networks where agents are subject to external disturbances. viation from consensus scales linearly in the number nodes, We characterize the coherence of these networks in terms of an while in the two-dimensional torus, the per-node variance H2 norm of the system that captures how closely agents track the consensus value. We show that, in first-order systems, the scales logarithmically in the number of nodes. coherence measure is closely related to the global mean first In this work, we explore the robustness of consensus al- passage time of a simple random walk. We can therefore draw gorithms in networks with dimensions between one and two. directly from the literature on random walks in fractal graphs Namely, we study the coherence of consensus algorithms in to derive asymptotic expressions for the coherence in terms of self-similar, fractal graphs. For first-order systems, we are the network size and dimension. We then show how techniques employed in the random walks setting can be extended to able to draw directly from literature on random walks on analyze the coherence of second-order consensus algorithms fractal networks to show that, in a network with N nodes, 1/df in fractal graphs with tree-like structures, and we present the per-node variance scales as N where df is the fractal asymptotic results for these second-order systems. dimension of the network. For networks with second-order I. INTRODUCTION consensus dynamics, we are unaware of any such analysis for general fractal graphs. For fractals with tree-like structures, Distributed consensus algorithms are important tools in we show how techniques used for the analysis of random the domains of multi-agent systems and the vehicle pla- walks [12], [13] can be extended to analyze the coherence of tooning problem [1], [2], [3] as a means by which agents second-order consensus systems, and we present asymptotic can reach and maintain agreement on quantities such as results for the per-node variance in terms of the network size velocity, heading, and inter-vehicle spacing using only local and spectral dimension. communication. In these settings, in addition to verifying The remainder of this paper is organized as follows. In the correctness of distributed consensus algorithms, it is also Section II, we present the system models for first-order important to consider how robust these algorithms are to and second-order consensus dynamics and give a formal external disturbances. definition of network coherence for each setting. We also Several recent works have studied the robustness of dis- present several properties from other domains that are math- tributed consensus algorithms for systems with first-order ematically similar to network coherence. In Section III, we and second-order dynamics in terms of an H2 norm. This describe the fractal graph models, and in Section IV we norm is a quantification of the network coherence; it captures present analytical results on the scalings of the first-order and how well a network can maintain its formation in the face second-order coherence measures in fractal graphs. Finally, of stochastic external disturbances. For the first-order case, we conclude in Section V. it has been shown that the H2 norm can be characterized by the trace of the pseudo-inverse of the Laplacian matrix [4], II. NETWORK COHERENCE [5], [6], [7]. This value has important meaning not just in We consider simple, first-order and second-order con- consensus systems, but in electrical networks [8], [9], random sensus algorithms over an undirected, connected network walks [10], and molecular connectivity [11]. For systems modeled by a graph G with N nodes and M edges. Our with second-order dynamics, the H2 norm is also determined objective is to quantify the robustness of these algorithms to by the spectrum of the Laplacian. However, we are unaware stochastic perturbations at the nodes using a quantity that we of any analogous concepts in other fields. call network coherence. In the first-order setting, each node Our recent work [5], [7] gave scalings for the H2 norm has a single state, and in the second order setting, each node of first and second order consensus algorithms in torus has two states corresponding to position and velocity. This and lattice networks in terms of the number of nodes and difference leads to different expressions for coherence and the network dimension. These results showed that there is therefore different asymptotic scalings, as we show in the a marked difference in coherence between first-order and sequel. second-order systems and also between networks of different dimensions. For example, in a one-dimensional ring network A. Coherence in Networks with First-Order Dynamics This research is partially supported through AFOSR grant FA9550-10-1- In the first-order consensus problem, the state of the 0143, and NSF grants ECCS-0802008 and CMMI-0626170. system, the value at each node, is a vector x RN . Each 2 S. Patterson and B. Bamieh are with the Dept. of Mechanical node state is subject to stochastic disturbances, and the Engineering, University of California, Santa Barbara, Santa Barbara, California 93106, USA [email protected], objective is for nodes to maintain consensus at the average [email protected] of their states. 978-1-61284-799-3/11/$26.00 ©2011 IEEE 6445 Let L be the Laplacian of the graph, defined by L := Definition 2.2: The second-order network coherence is D A, where D is diagonal matrix of node degrees and A the mean, steady-state variance of the deviation of each − is the adjacency matrix of G. The dynamics of the system vehicle’s position error from the average of all vehicle are defined as follows, position errors. Let the output for the system (4) be x˙ = Lx + w, (1) − x y = P 0 , (5) where w is an N-vector of zero-mean, white noise processes. v In the absence of the noise processes, the system converges ⇥ ⇤ asymptotically to consensus at the the average of the initial As in the consensus case, the variance is given by the H2 states. With the additive noise term, the nodes do not con- norm of the system defined by (4) and (5). This value is also verge to consensus, but instead, node values fluctuate around completely determined by the eigenvalues of the Laplacian the average of the current node states. Network coherence matrix [5], [7], and is given by captures the variance of these fluctuations. 1 N 1 Definition 2.1: The first-order network coherence is de- H(2) = . fined as the mean, steady-state variance of the deviation from 2N λ2 i=2 i the average of all node values, X C. Related Concepts N N (1) 1 1 The eigenvalues of the Laplacian are linked to the topology H := lim E xi(t) xj(t) . N t 8 − N 9 of the network, and therefore, it is not surprising that these i=1 !1 < j=1 = X X 1 eigenvalues play a role in many graph problems. In fact, the Let P be the projection operator P = I 11⇤, where − N 1 is the N-vector of all ones.: We define the output; of the sum N 1 system (1) to be S := λ y = Px. (2) i=2 i X (1) H relates to the H2 norm of the system defined by (1) that appears in the expression for first order coherence (3) and (2) as follows, is an important quantity, not just in the study of consensus algorithms, but in several other fields. We can leverage work (1) 1 1 L⇤t Lt H = tr e− Pe− dt . in these fields to develop analytic expressions for network co- N ✓Z0 ◆ herence for different graph topologies. We first review these It has been shown that H(1) is s completely determined by related properties, and in the following sections, we present the spectrum of L [4], [5], [7], [6]. Let the eigenvalues of L analysis for the asymptotic behavior of network coherence be 0=λ <λ ... λ . Then, the network coherence measures in several classes of self-similar networks. 1 2 N of the first-order system is given by 1) Effective Resistance in an Electrical Network: The graph G represents an electrical network where each edge N 1 1 is a unit resistor. The resistance distance r between two H(1) = . (3) ij 2N λ nodes i and j is the potential distance between them when a i=2 i X one ampere current source is connected from node j to node B. Coherence in Networks with Second-Order Dynamics i. The total effective resistance of the network, also called the In the vehicle formation problem, there are N vehicles, Kirchoff index [8], [9], is the sum of the resistance distances each with a position and a velocity. The objective is for over all pairs of nodes in the graph, each vehicle to travel at a constant target velocity while maintaining a fixed, pre-specified distance between itself and R := rij. i,j V each of its neighbors. The state of the second-order system Xi=2j consists of a position vector x and a velocity vector v. The 6 states are measured relative to the target velocity v and It has been shown that the total effective resistance depends position x(t).
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