A New Notion of Effective Resistance for Directed Graphs—Part I: Definition and Properties

Young, George Forrest; Scardovi, Luca; Leonard, Naomi Ehrich

doi:10.1109/tac.2015.2481978

Cited by 54 publications

(61 citation statements)

References 31 publications

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“…Note that according to (Young et al, 2016a, Section II) L(P ) is not unique and depends on the choice of Q, while R is independent of the choice of Q. To compare R with our proposed resistance distance Ω, it boils down to a comparison between Y and the fundamental matrix F or the group inverse D. While both R and Ω does not define a metric in general, in Proposition 1.2 below we show that Ω does define a metric when P is doubly stochastic while it is unclear whether R also defines a metric in this setting.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On resistance distance of Markov chain and its sum rules

Choi

2019

Linear Algebra and its Applications

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Motivated by the notion of resistance distance on graph, we define a new resistance distance between two states on a given finite ergodic Markov chain based on its fundamental matrix. We prove a few equivalent formulations and discuss its relation with other parameters of the Markov chain such as its group inverse, stationary distribution, eigenvalues or hitting time. In addition, building upon existing sum rules for the hitting time of Markov chain, we give sum rules of this new resistance distance of Markov chains that resembles the sum rules of the resistance distance on graph. This yields Markov chain counterparts of various classical formulae such as Foster's first formula or the Kirchhoff index formulae.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

On resistance distance of Markov chain and its sum rules

Choi

2019

Linear Algebra and its Applications

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“…However, this is not directly possible. Indeed, the notion of effective resistance has been recently introduced for both directed and undirected graphs as [30],…”

Section: Example 3 Consider the Graphsmentioning

confidence: 99%

“…We also highlight the case where the condition becomes necessary and sufficient. Furthermore, it is argued that for directed graphs, the recently proposed notion of effective resistance [30] is not applicable to interpret the obtained upper bound. By partitioning the nodes of the graph into some sets, we identify "sensitive pairs of nodes" with the following property: If there exists at least one edge with sufficiently small negative weight, the Laplacian matrix has at least one eigenvalue with negative real part.…”

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confidence: 96%

On eigenvalues of Laplacian matrix for a class of directed signed graphs

Ahmadizadeh

Shames

Martín

et al. 2017

Linear Algebra and its Applications

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The eigenvalues of the Laplacian matrix for a class of directed graphs with both positive and negative weights are studied. First, a class of directed signed graphs is investigated in which one pair of nodes (either connected or not) is perturbed with negative weights. A necessary condition is proposed to attain the following objective for the perturbed graph: the real parts of the non-zero eigenvalues of its Laplacian matrix are positive. A sufficient condition is also presented that ensures the aforementioned objective for unperturbed graph. It is then highlighted the case where the condition becomes necessary and sufficient. Secondly, for directed graphs, a subset of pairs of nodes are identified where if any of the pairs is connected by an edge with infinitesimal negative weight, the resulting Laplacian matrix will have at least one eigenvalue with negative real part. Illustrative examples are presented to show the applicability of our results.

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“…This therefore allows one to study connectivity of vertices without employing paths; which is potentially useful since defining a path, as mentioned above, is problematic for matrix-weighted graphs. One may ask why our formulation is in terms of effective conductance instead of the commoner effective resistance, e.g., [17]. The reason is that the conductances we work with are matrix-valued and not necessarily invertible.…”

Section: Introductionmentioning

confidence: 99%

Observability Through a Matrix-Weighted Graph

Tuna

2018

IEEE Trans. Automat. Contr.

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Observability of an array of identical LTI systems with incommensurable output matrices is studied, where an array is called observable when identically zero relative outputs imply synchronized solutions for the individual systems. It is shown that the observability of an array is equivalent to the connectivity of its interconnection graph, whose edges are assigned matrix weights. The interconnection graph is studied by means of a collection of simpler graphs, each of which is associated to an eigenvalue of the system matrix of individual dynamics. It is reported that the interconnection graph is connected if and only if no member of this collection is disconnected. Moreover, to better understand the relative behavior of distant units, pairwise observability which concerns with the synchronization of a certain pair of individual systems in the array is studied. This milder version of observability is shown to be closely related to certain connectivity properties of the interconnection graph as well. Pairwise observability is also analyzed using the circuit theoretic tool effective conductance. The observability of a certain pair of units is proved to be equivalent to the nonsingularity of the (matrix-valued) effective conductance between the associated pair of nodes of a resistive network (with matrix-valued parameters) whose node admittance matrix is the Laplacian of the array's interconnection graph.

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A New Notion of Effective Resistance for Directed Graphs—Part I: Definition and Properties

Cited by 54 publications

References 31 publications

On resistance distance of Markov chain and its sum rules

On resistance distance of Markov chain and its sum rules

On eigenvalues of Laplacian matrix for a class of directed signed graphs

Observability Through a Matrix-Weighted Graph

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