On the definiteness of the weighted Laplacian and its connection to effective resistance

Zelazo, Daniel; Bürger, Mathias

doi:10.1109/cdc.2014.7039834

Cited by 65 publications

(78 citation statements)

References 20 publications

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“…Remark 5: Theorem 6 gives two necessary and sufficient conditions for L G being PSD with only one zero eigenvalue. It generalizes the results in [20][21][22] that apply to those graphs with a special location distribution of negative weighted edges. Note that the sequential inclusion describes an expansion from a single node to the whole concerned set (V k+1 − \V k − refers to the node to be included in one step).…”

Section: Interpreting Graph Laplacian Definiteness By Effective Rsupporting

confidence: 68%

“…So constraint (20f) is equivalent to (19f) following the same idea as in (11). Then, according to [44], problem (20) has the same feasible set as (19), and relaxing the rank constraint (20h) gives a convex problem. The convex relaxation of OPF has been well studied, which is shown to be exact in most cases [45].…”

Section: B Transient Stability Enhancementmentioning

confidence: 99%

“…In this case, the coherent groups of generators are V a = {1} and V b = {2, 3}, and the corresponding effective conductance at equilibrium point A is 7.58. Then, we solve the relaxed convex problem of (20) with the settings f 1 = 0.01P 2 1 + 10P 1 ,…”

Section: An Illustrative Case Studymentioning

confidence: 99%

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On Extension of Effective Resistance With Application to Graph Laplacian Definiteness and Power Network Stability

Song

Hill

2019

IEEE Trans. Circuits Syst. I

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This paper extends the definitions of effective resistance and effective conductance to characterize the overall relation (positive coupling or antagonism) between any two disjoint sets of nodes in a signed graph. It generalizes the traditional definitions that only apply to a pair of nodes. The monotonicity and convexity properties are preserved by the extended definitions. The extended definitions provide new insights into graph Laplacian definiteness and power network stability. It is proved that the Laplacian matrix of a signed graph is positive semi-definite with only one zero eigenvalue if and only if the effective conductances between some specific pairs of node sets are positive. Also the number of Laplacian negative eigenvalues is upper bounded by the number of negative weighted edges. In addition, new conditions for the small-disturbance angle stability, hyperbolicity and type of power system equilibria are established, which intuitively interpret angle instability as the electrical antagonism between certain two sets of nodes in the defined active power flow graph. Moreover, a novel optimal power flow (OPF) model with effective conductance constraints is formulated, which significantly enhances power system transient stability. By the properties of extended effective conductance, the proposed OPF model admits a convex relaxation representation that achieves global optimality.

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Section: Interpreting Graph Laplacian Definiteness By Effective Rsupporting

confidence: 68%

Section: B Transient Stability Enhancementmentioning

confidence: 99%

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On Extension of Effective Resistance With Application to Graph Laplacian Definiteness and Power Network Stability

Song

Hill

2019

IEEE Trans. Circuits Syst. I

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“…In fact, it is possible that relative outputs ζ(t) do not converge to a point in I even when all the nodes are in their steady states with the equilibrium input u(t) = 0. Some typical examples are given in [6], [9], where the networks of single integrators with edge functions represented by (6) We will further discuss these conditions in Section IV. Before we proceed, we provide the following proposition showing an important property of the nodes only incident to strictly positive edges.…”

Section: B Signed Networkmentioning

confidence: 99%

Convergence Analysis of Signed Nonlinear Networks

Chen

Zelazo

Wang

2020

IEEE Trans. Control Netw. Syst.

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This work analyzes the convergence properties of signed networks with nonlinear edge functions. We consider diffusively coupled networks comprised of maximal equilibriumindependent passive (MEIP) dynamics on the nodes, and a general class of nonlinear coupling functions on the edges. The first contribution of this work is to generalize the classical notion of signed networks for graphs with scalar weights to graphs with nonlinear edge functions using notions from passivity theory. We show that the output of the network can finally form one or several steady-state clusters if all edges are positive, and in particular, all nodes can reach an output agreement if there is a connected subnetwork spanning all nodes and strictly positive edges. When there are non-positive edges added to the network, we show that the tension of the network still converges to the equilibria of the edge functions if the relative outputs of the nodes connected by non-positive edges converge to their equilibria. Furthermore, we establish the equivalent circuit models for signed nonlinear networks, and define the concept of equivalent edge functions which is a generalization of the notion of effective resistance. We finally characterize the relationship between the convergence property and the equivalent edge function, when a non-positive edge is added to a strictly positive network comprised of nonlinear integrators. We show that the convergence of the network is always guaranteed, if the sum of the equivalent edge function of the previous network and the new edge function is passive.

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“…where L e (G) ∈ R m×m represents the edge-Laplacian of the graph G and can be expressed as, Zelazo and Burger (2014). Also, using the fact that the underlying graph contains a spanning-tree G τ we partition the edge states after a suitable permutation as follows,…”

Section: Single Integratormentioning

confidence: 99%

Consensus analysis of systems with time-varying interactions : An event-triggered approach

2017

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We present consensus analysis of systems with single integrator dynamics interacting via time-varying graphs under the event-triggered control paradigm. Event-triggered control sparsifies the control applied, thus reducing the control effort expended. Initially, we consider a multi-agent system with persistently exciting interactions and study the behaviour under the application of event-triggered control with two types of trigger functions-static and dynamic trigger. We show that while in the case of static trigger, the edge-states converge to a ball around the origin, the dynamic trigger function forces the states to reach consensus exponentially. Finally, we extend these results to a more general setting where we consider switching topologies. We show that similar results can be obtained for agents interacting via switching topologies and validate our results by means of simulations.

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On the definiteness of the weighted Laplacian and its connection to effective resistance

Cited by 65 publications

References 20 publications

On Extension of Effective Resistance With Application to Graph Laplacian Definiteness and Power Network Stability

On Extension of Effective Resistance With Application to Graph Laplacian Definiteness and Power Network Stability

Convergence Analysis of Signed Nonlinear Networks

Consensus analysis of systems with time-varying interactions : An event-triggered approach

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