On the definiteness of graph Laplacians with negative weights: Geometrical and passivity-based approaches

Signed graphs have appeared in a broad variety of applications, ranging from social networks to biological networks, from distributed control and computation to power systems. In this paper, we investigate spectral properties of signed Laplacians for undirected signed graphs. We find conditions on the negative weights under which a signed Laplacian is positive semidefinite via the Kron reduction and multiport network theory. For signed Laplacians that are indefinite, we characterize their inertias with the same framework. Furthermore, we build connections between signed Laplacians, generalized M-matrices, and eventually exponentially positive matrices.

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“…Corollary 1 is consistent with the statements in [12,Theorem III.4] and [13,Theorem 3.2]. Example 2:…”

Section: B Semidefiniteness and Conductance Matrixsupporting

confidence: 83%

On Spectral Properties of Signed Laplacians With Connections to Eventual Positivity

Chen

Wang

et al. 2021

IEEE Trans. Automat. Contr.

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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: 66%

“…By (6c) and (6d), the first term of right-hand-side of (27) is zero, and the second term is r eff (L G , V a , V b ). Therefore, r eff (L G , V a , V b ) is the optimal value of problem (22) as well as the original problem in (9). Moreover, we have r eff (L G , V a , V b ) > 0 since the objective v T L G v is nonnegative and the only case that makes it zero (v = 1 n ) is excluded by (22b).…”

Section: Proof Of Lemma 4: Formentioning

confidence: 93%

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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“…It was shown that the absolute value of the negative edge weight must be larger than the inverse of the effective resistance between the two nodes connected by the edge. The same result was rederived in [10], where two alternative proofs were provided based on geometrical and passivitybased approaches. The definiteness of the signed Laplacian matrix for the signed directed networks has also been studied in [11], [12].…”

Section: Introductionmentioning

confidence: 84%

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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On the definiteness of graph Laplacians with negative weights: Geometrical and passivity-based approaches

Cited by 39 publications

References 10 publications

On Spectral Properties of Signed Laplacians With Connections to Eventual Positivity

On Spectral Properties of Signed Laplacians With Connections to Eventual Positivity

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

Convergence Analysis of Signed Nonlinear Networks

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