Spectral Theory for Dynamics on Graphs Containing Attractive and Repulsive Interactions

Bronski, Jared C.; DeVille, Lee

doi:10.1137/130913973

Cited by 64 publications

(83 citation statements)

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“…Necessary and sufficient conditions are established in [13,56] on when the repelling Laplacian L rw G is positive semidefinite from linear matrix inequalities, which can be utilized to establish deeper results compared to Theorem 2. Also see [11] for a much more detailed spectrum analysis of repelling Laplacians.…”

Section: Repelling Negative Dynamicsmentioning

confidence: 99%

Dynamics over Signed Networks

Shi¹,

Altafini²,

Baras³

2019

SIAM Rev.

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A signed network is a network with each link associated with a positive or negative sign. Models for nodes interacting over such signed networks, where two different types of interactions take place along the positive and negative links, respectively, arise from various biological, social, political, and economic systems. As modifications to the conventional DeGroot dynamics for positive links, two basic types of negative interactions along negative links, namely the opposing rule and the repelling rule, have been proposed and studied in the literature. This paper reviews a few fundamental convergence results for such dynamics over deterministic or random signed networks under a unified algebraic-graphical method.We show that a systematic tool of studying node state evolution over signed networks can be obtained utilizing generalized Perron-Frobenius theory, graph theory, and elementary algebraic recursions.j∈V 2 x j (0) /n, i ∈ V 1 , and lim t→∞ x i (t) = − j∈V 1 x j (0) − j∈V 2 x j (0) /n, i ∈ V 2 .

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Section: Repelling Negative Dynamicsmentioning

confidence: 99%

Dynamics over Signed Networks

Shi¹,

Altafini²,

Baras³

2019

SIAM Rev.

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“…Ever since then Laplacians have been studied and applied in various fields. For an example of studying the applications of Laplacians to spectral theory, we refer the interested reader to Bronski and DeVille (2014) in which they study the class of Signed graph Laplacians (a symmetric matrix, which is special case of above defined Laplacian).…”

Section: Introductionmentioning

confidence: 99%

Laplacian Dynamics with Synthesis and Degradation

Mirzaev

Bortz

2015

Bull Math Biol

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Analyzing qualitative behaviors of biochemical reactions using its associated network structure has proven useful in diverse branches of biology. As an extension of our previous work, we introduce a graph-based framework to calculate steady state solutions of biochemical reaction networks with synthesis and degradation. Our approach is based on a labeled directed graph G and the associated system of linear non-homogeneous differential equations with first order degradation and zeroth order synthesis. We also present a theorem which provides necessary and sufficient conditions for the dynamics to engender a unique stable steady state. Although the dynamics are linear, one can apply this framework to nonlinear systems by encoding nonlinearity into the edge labels. We answer an open question from our previous work concerning the non-positiveness of the elements in the inverse of a perturbed Laplacian matrix. Moreover, we provide a graph theoretical framework for the computation of the inverse of such a matrix. This also completes our previous framework and makes it purely graph theoretical. Lastly, we demonstrate the utility of this framework by applying it to a mathematical model of insulin secretion through ion channels in pancreatic β-cells.

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“…Most of the remaining definitions and notation of this section are similar to those of [6], but are stated for signed graphs that may be disconnected.…”

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confidence: 99%

“…Applying Theorem 2.10 of [6] to each connected component of a weighted signed graph gives the following result. Theorem 1.3.…”

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confidence: 99%

“…Note that the proof of the above theorem in [6] uses a 1-parameter family Γ(t) introduced below, but it is clear that the results hold for any consistent weighting of the edges of G. Let Γ = (G, γ) be a weighted signed graph. We denote by Γ + the weighted signed graph that consists of G + and the restriction of γ to E + , and by Γ − the weighted signed graph that consists of G − and the restriction of γ to E − .…”

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Inertias of Laplacian matrices of weighted signed graphs

et al. 2019

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We study the sets of inertias achieved by Laplacian matrices of weighted signed graphs. First we characterize signed graphs with a unique Laplacian inertia. Then we show that there is a sufficiently small perturbation of the nonzero weights on the edges of any connected weighted signed graph so that all eigenvalues of its Laplacian matrix are simple. Next, we give upper bounds on the number of possible Laplacian inertias for signed graphs with a fixed flexibility τ (a combinatorial parameter of signed graphs), and show that these bounds are sharp for an infinite family of signed graphs. Finally, we provide upper bounds for the number of possible Laplacian inertias of signed graphs in terms of the number of vertices.The signs of the real parts of the eigenvalues of a coefficient matrix in a system of linear ordinary differential equations determine the stability of the dynamical system that it is describing. In this paper, we focus on systems with a coefficient matrix that is a symmetric Laplacian matrix, for which all eigenvalues are real. In order to understand the behaviour of such a dynamical system, the number of positive, negative, and zero eigenvalues of the coefficient matrix, together known as the inertia of the matrix, are studied. The sign pattern of the Laplacian matrix is described by a graph with positive and negative edges. The spectrum of the Laplacian of a signed graph reveals important information about the underlying dynamics, for example see [1,2,3].A signed graph is a simple graph G = (V (G), E(G)) together with an assignment of the signs + or − to its edges. An edge is positive if it is assigned the sign +, and negative if it is assigned the sign −. We assume throughout that a signed graph G has vertex set V (G) = {1, 2, . . . , n}, and a total of m edges of which m + are positive and m − are negative. The number of components of *

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Spectral Theory for Dynamics on Graphs Containing Attractive and Repulsive Interactions

Cited by 64 publications

References 65 publications

Dynamics over Signed Networks

Dynamics over Signed Networks

Laplacian Dynamics with Synthesis and Degradation

Inertias of Laplacian matrices of weighted signed graphs

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