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65
Upper bounds on algebraic connectivity via convex optimization
 Linear Algebra Appl
, 2006
"... The second smallest eigenvalue of the Laplacian matrix L of a graph is called its algebraic connectivity. We describe a method for obtaining an upper bound on the algebraic connectivity of a family of graphs G. Our method is to maximize the second smallest eigenvalue over the convex hull of the Lapl ..."
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The second smallest eigenvalue of the Laplacian matrix L of a graph is called its algebraic connectivity. We describe a method for obtaining an upper bound on the algebraic connectivity of a family of graphs G. Our method is to maximize the second smallest eigenvalue over the convex hull of the Laplacians of graphs in G, which is a convex optimization problem. By observing that it suffices to optimize over the subset of matrices invariant under the symmetry group of G, we can solve the optimization problem analytically for families of graphs with large enough symmetry groups. The same method can also be used to obtain upper bounds for other concave functions, and lower bounds for convex functions of L (such as the spectral radius). 1
A OneParameter Family of Distributed Consensus Algorithms with Boundary: From Shortest Paths to Mean Hitting Times
"... Abstract — We present a oneparameter family of consensus algorithms over a timevarying network of agents. The proposed family of algorithms contains the average and minimum consensus algorithms as two special cases. Furthermore, we investigate a closely related family of distributed algorithms whi ..."
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Abstract — We present a oneparameter family of consensus algorithms over a timevarying network of agents. The proposed family of algorithms contains the average and minimum consensus algorithms as two special cases. Furthermore, we investigate a closely related family of distributed algorithms which can be considered as a consensus scheme with fixed boundary conditions and constant inputs. The proposed algorithms recover both the BellmanFord iteration for finding shortest paths as well as the algorithm for calculating the mean hitting time of a random walk on a graph. Finally, we demonstrate the potential utility of these algorithms for routing in adhoc networks. I.
Algebraic Distance on Graphs
"... Measuring the connection strength between a pair of vertices in a graph is one of the most vital concerns in many graph applications. Simple measures such as edge weights may not be sufficient for capturing the local connectivity. In this paper, we consider a neighborhood of each graph vertex and pr ..."
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Measuring the connection strength between a pair of vertices in a graph is one of the most vital concerns in many graph applications. Simple measures such as edge weights may not be sufficient for capturing the local connectivity. In this paper, we consider a neighborhood of each graph vertex and propagate a certain property value through direct neighbors. We present a measure of the connection strength (called the algebraic distance, see [21]) defined from an iterative process based on this consideration. The proposed measure is attractive in that the process is simple, linear, and easily parallelized. A rigorous analysis of the convergence property of the process confirms the underlying intuition that vertices are mutually reinforced and that the local neighborhoods play an important role in influencing the vertex connectivity. We demonstrate the practical effectiveness of the proposed measure through several combinatorial optimization problems on graphs and hypergraphs. 1
Convergence and stochastic stability of continuous time consensus protocols
"... A unified approach to studying convergence and stochastic stability of continuous time consensus protocols (CPs) is presented in this work. Our method applies to networks with directed information flow; both cooperative and noncooperative interactions; networks under weak stochastic forcing; and tho ..."
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A unified approach to studying convergence and stochastic stability of continuous time consensus protocols (CPs) is presented in this work. Our method applies to networks with directed information flow; both cooperative and noncooperative interactions; networks under weak stochastic forcing; and those whose topology and strength of connections may vary in time. The graph theoretic interpretation of the analytical results is emphasized. We show how the spectral properties, such as algebraic connectivity and total effective resistance, as well as the geometric properties, such the dimension and the structure of the cycle subspace of the underlying graph, shape stability of the corresponding CPs. In addition, we explore certain implications of the spectral graph theory to CP design. In particular, we point out that expanders, sparse highly connected graphs, generate CPs whose performance remains uniformly high when the size of the network grows unboundedly. Similarly, we highlight the benefits of using random versus regular network topologies for CP design. We illustrate these observations with numerical examples and refer to the relevant graphtheoretic results.
A distributed control strategy for optimal reactive power flow with power and voltage constraints
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Improving Damping of Power Networks: Power Scheduling and Impedance Adaptation
"... Abstract — This paper studies the effect of power scheduling and line impedances on the damping of a power network. We relate the damping of a network with the algebraic connectivity of a state dependent Laplacian. Via implicit function theorem, we further characterize its dependence on network para ..."
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Abstract — This paper studies the effect of power scheduling and line impedances on the damping of a power network. We relate the damping of a network with the algebraic connectivity of a state dependent Laplacian. Via implicit function theorem, we further characterize its dependence on network parameters. This allows us to derive several updating directions that can locally improve the damping. The analysis also provides some interesting insight. For example, improving connectivity, by adding lines for instance, may not be beneficial in terms of damping. I.
On the optimal synchronization of oscillator networks via sparse interconnection graphs
 in Proceedings of the 2012 American Control Conference, 2012
"... Abstract — We consider the problem of synchronizing a given set of oscillators through the design of a conductance network, where the conductance connecting two oscillators models the amount of communication between them. Using optimal control theory, we formulate an optimization problem that addres ..."
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Abstract — We consider the problem of synchronizing a given set of oscillators through the design of a conductance network, where the conductance connecting two oscillators models the amount of communication between them. Using optimal control theory, we formulate an optimization problem that addresses the tradeoff between synchronization performance and conductance usage. Additionally, we promote the sparsity of the network by penalizing the number of interconnection links. We demonstrate that in the case of identical oscillators the optimization problem is convex and admits formulation as a semidefinite program. For nonidentical oscillators that can be considered as perturbations around a central oscillator, we show that it is meaningful to design an optimal conductance network by assuming that all oscillators are identical to the central (average) oscillator. Finally, we derive explicit formulas for the optimal conductance values for some special problems. Index Terms — Convex relaxation, optimization, oscillator synchronization, reweighted `1 minimization, semidefinite pro
Graph realizations associated with minimizing the maximum eigenvalue of the Laplacian.
, 2009
"... Abstract In analogy to the absolute algebraic connectivity of Fiedler, we study the problem of minimizing the maximum eigenvalue of the Laplacian of a graph by redistributing the edge weights. Via semidefinite duality this leads to a graph realization problem in which nodes should be placed as clos ..."
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Abstract In analogy to the absolute algebraic connectivity of Fiedler, we study the problem of minimizing the maximum eigenvalue of the Laplacian of a graph by redistributing the edge weights. Via semidefinite duality this leads to a graph realization problem in which nodes should be placed as close as possible to the origin while adjacent nodes must keep a distance of at least one. We prove three main results for a slightly generalized form of this embedding problem. First, given a set of vertices partitioning the graph into several or just one part, the barycenter of each part is embedded on the same side of the affine hull of the set as the origin. Second, there is an optimal realization of dimension at most the treewidth of the graph plus one and this bound is best possible in general. Finally, bipartite graphs possess a one dimensional optimal embedding.
Designing Node and Edge Weights of a Graph to Meet Laplacian Eigenvalue Constraints
"... Abstract—We consider agents connected over a network, and propose a method to design an optimal interconnection such that the gap between the largest and smallest Laplacian eigenvalues of the graph representing the network is minimized. We study ways of imposing constraints that may arise in physica ..."
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Abstract—We consider agents connected over a network, and propose a method to design an optimal interconnection such that the gap between the largest and smallest Laplacian eigenvalues of the graph representing the network is minimized. We study ways of imposing constraints that may arise in physical systems, such as enforcing lower bounds on connectivity and upper bounds on gain as well as network cost. In particular, we show that node and edge weights of a given graph can be simultaneously adjusted via convex optimization to achieve improvements in its Laplacian spectrum. I.
RESISTANCEBASED PERFORMANCE ANALYSIS OF THE CONSENSUS ALGORITHM OVER GEOMETRIC GRAPHS∗
"... Abstract. The performance of the linear consensus algorithm is studied by using a Linear Quadratic (LQ) cost. The objective is to understand how the communication topology influences this algorithm. This is achieved by exploiting an analogy between Markov Chains and electrical resistive networks. In ..."
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Abstract. The performance of the linear consensus algorithm is studied by using a Linear Quadratic (LQ) cost. The objective is to understand how the communication topology influences this algorithm. This is achieved by exploiting an analogy between Markov Chains and electrical resistive networks. Indeed, this permits to uncover the relation between the LQ performance cost and the average effective resistance of a suitable electrical network and, moreover, to show that, if the communication graph fulfils some local properties, then its behavior can be approximated by that of a grid, which is a graph whose associated LQ cost is wellknown. Key words. Multiagent systems, consensus algorithm, distributed averaging, largescale graphs AMS subject classifications. 68R10, 90B10, 94C15, 90B18, 05C50 1. Introduction. The