A Primer on Laplacian Dynamics in Directed Graphs

Veerman, J. J. P.; Lyons, Ronan A.

doi:10.48550/arxiv.2002.02605

Cited by 4 publications

(4 citation statements)

References 8 publications

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“…which is essentially a type of generalized Laplacian dynamics with more complicated details (see [37][38][39] for the elementary Laplacian dynamics). See figure 1(c) for illustrations.…”

Section: Joint Global Dynamics Of Convergent and Divergent Evolutionmentioning

confidence: 99%

Laplacian dynamics of convergent and divergent collective behaviors

Tian

Sun

2023

J. Phys. Complex.

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Collective dynamics is ubiquitous in various physical, biological, and social systems, where simple local interactions between individual units lead to complex global patterns. A common feature of diverse collective behaviours is that the units exhibit either convergent or divergent evolution in their behaviors, i.e., becoming increasingly similar or distinct, respectively. The associated dynamics changes across time, leading to complex consequences on a global scale. In this study, we propose a generalized Laplacian dynamics model to describe both convergent and divergent collective behaviors, where the trends of convergence and divergence compete with each other and jointly determine the evolution of global patterns. We empirically observe non-trivial phase-transition-like phenomena between the convergent and divergent evolution phases, which are controlled by local interaction properties. We also propose a conjecture regarding the underlying phase transition mechanisms and outline the main theoretical difficulties for testing this conjecture. Overall, our framework may serve as a minimal model of collective behaviors and their intricate dynamics.

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“…which is essentially a type of generalized Laplacian dynamics with more complicated details (see [37][38][39] for the elementary Laplacian dynamics). See figure 1(c) for illustrations.…”

Section: Joint Global Dynamics Of Convergent and Divergent Evolutionmentioning

confidence: 99%

Laplacian dynamics of convergent and divergent collective behaviors

Tian

Sun

2023

J. Phys. Complex.

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“…The operator L θ is reduced to the weighted symmetric graph Laplacian if k + = k − and also to the conventional graph Laplacian if k + = k − = 1 [91,92]. Equation 6can also cover linear transport on graphs, a class of linear electric circuits [93], consensus dynamics on graphs [94], and other linear dynamics on graphs [86,95]. 11…”

Section: Definition 3 (Weighted Asymmetric Graph Laplacianmentioning

confidence: 99%

Information geometry of dynamics on graphs and hypergraphs

Kobayashi,

Loutchko,

Kamimura

et al. 2023

Info. Geo.

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We introduce a new information-geometric structure associated with the dynamics on discrete objects such as graphs and hypergraphs. The presented setup consists of two dually flat structures built on the vertex and edge spaces, respectively. The former is the conventional duality between density and potential, e.g., the probability density and its logarithmic form induced by a convex thermodynamic function. The latter is the duality between flux and force induced by a convex and symmetric dissipation function, which drives the dynamics of the density. These two are connected topologically by the homological algebraic relation induced by the underlying discrete objects. The generalized gradient flow in this doubly dual flat structure is an extension of the gradient flows on Riemannian manifolds, which include Markov jump processes and nonlinear chemical reaction dynamics as well as the natural gradient. The information-geometric projections on this doubly dual flat structure lead to information-geometric extensions of the Helmholtz–Hodge decomposition and the Otto structure in $$L^{2}$$ L 2 -Wasserstein geometry. The structure can be extended to non-gradient nonequilibrium flows, from which we also obtain the induced dually flat structure on cycle spaces. This abstract but general framework can broaden the applicability of information geometry to various problems of linear and nonlinear dynamics.

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“…where deg(i) is the in-degree of node i. According to [22] the multiplicity of the eigenvalue 0 of L equals to the number of maximal reachable vertex sets. In other words, multiplicity of zero eigenvalues is the number of trees needed to cover G. Therefore, matrix L has m b + m f eigenvalues equal to 0.…”

Section: High-level Planning: Recovery Trajectory Planningmentioning

confidence: 99%

A Physics-Based Safety Recovery Approach for Fault-Resilient Multi-Quadcopter Coordination

Emadi¹,

Harshvardhan²,

Rastgoftar³

2021

Preprint

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This paper develops a novel physics-based approach for fault-resilient multi-quadcopter coordination in the presence of abrupt quadcopter failure. Our approach consists of two main layers: (i) high-level physics-based guidance to safely plan the desired recovery trajectory for every healthy quadcopter and (ii) low-level trajectory control design by choosing an admissible control for every healthy quadcopter to safely recover from the anomalous situation, arisen from quadcopter failure, as quickly as possible. For the high-level trajectory planning, first, we consider healthy quadcopters as particles of an irrotational fluid flow sliding along streamline paths wrapping failed quadcopters in the shared motion space. We then obtain the desired recovery trajectories by maximizing the sliding speeds along the streamline paths such that the rotor angular speeds of healthy quadcopters do not exceed certain upper bounds at all times during the safety recovery. In the low level, a feedback linearization control is designed for every healthy quadcopter such that quadcopter rotor angular speeds remain bounded and satisfy the corresponding safety constraints. Simulation results are given to illustrate the efficacy of the proposed method.

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A Primer on Laplacian Dynamics in Directed Graphs

Cited by 4 publications

References 8 publications

Laplacian dynamics of convergent and divergent collective behaviors

Laplacian dynamics of convergent and divergent collective behaviors

Information geometry of dynamics on graphs and hypergraphs

A Physics-Based Safety Recovery Approach for Fault-Resilient Multi-Quadcopter Coordination

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