On a Conjecture of Godsil Concerning Controllable Random Graphs

O’Rourke, Sean; Touri, Behrouz

doi:10.1137/15m1049622

Cited by 36 publications

(29 citation statements)

References 28 publications

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“…In view of Theorem 3.7, though, this conjecture would imply Conjecture 3.8. Specifically, Theorem 3.7 hints at another approach to prove Conjecture 3.8, which would be entirely different from the proofs given in [39,40]. While the proofs in these previous works focused on the eigenvector structure, this new method only requires working with the eigenvalues (in particular, the characteristic polynomial) of A n .…”

Section: 3mentioning

confidence: 96%

“…One can view Conjecture 3.8 as stating that controllability (alternatively, minimal controllability) is a universal property of graphs. Conjecture 3.8 was recently proven in [39,40]. The proof relies on Kalman's rank condition (Theorem 3.3) and one of its corollaries known as the Popov-Belevitch-Hautus (PBH) test (see [23,Section 12.2] for details).…”

Section: 3mentioning

confidence: 99%

“…The proof relies on Kalman's rank condition (Theorem 3.3) and one of its corollaries known as the Popov-Belevitch-Hautus (PBH) test (see [23,Section 12.2] for details). In particular, the proof given in [39,40] involves studying the additive structure of the eigenvectors of the random adjacency matrix A n .…”

Section: 3mentioning

confidence: 99%

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Low-Degree Factors of Random Polynomials

O’Rourke

Wood

2018

J Theor Probab

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We study the probability that a monic polynomial with integer coefficients has a low-degree factor over the integers, which is equivalent to having a low-degree algebraic root. It is known in certain cases that random polynomials with integer coefficients are very likely to be irreducible, and our project can be viewed as part of a general program of testing whether this is a universal behavior exhibited by many random polynomial models. Our main result shows that pointwise delocalization of the roots of a random polynomial can be used to imply that the polynomial is unlikely to have a low-degree factor over the integers. We apply our main result to a number of models of random polynomials, including characteristic polynomials of random matrices, where strong delocalization results are known.

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Section: 3mentioning

confidence: 96%

Section: 3mentioning

confidence: 99%

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Low-Degree Factors of Random Polynomials

O’Rourke

Wood

2018

J Theor Probab

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“…This property can also be characterized by the main eigenvalues and main eigenvectors of the graph, see [7]. The relevance of controllability becomes clear from the recent work of O'Rourke and B. Touri [14] who proved Godsil's conjecture [7] that asymptotically all graphs are controllable. The theorem above also implies the following well-known fact concerning controllable graphs:…”

Section: The Determinant Of the Walk Matrixmentioning

confidence: 99%

New families of graphs determined by their generalized spectrum

Liu

Siemons

Wang

2019

Discrete Mathematics

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We construct infinite families of graphs that are determined by their generalized spectrum. This construction is based on new formulae for the determinant of the walk matrix of a graph. The graphs constructed here all satisfy a lower divisibility for the determinant of their walk matrix. 1 spectrum, or a DS graph for short. In [4,8,9] it is conjectured that almost all graphs are DS, more recent surveys can be found in [5].A variant of this problem concerns the generalized spectrum of G which is given by the pair spec(G), spec(G) . In this situation G is said to be determined by its generalized spectrum, or a DGS graph for short, if spec(H), spec(H) = spec(G), spec(G) implies that H is isomorphic to G. For the most recent results on DGS graphs see [13] and [20] in particular. To date only a few families of graphs are known to be DS or DGS. This includes some almost complete graphs [1], pineapple graphs [10] and kite graphs [11]. These are all known to be DS, and in particular DGS. In addition all rose graphs are determined by their Laplacian spectrum except for two specific examples, see [12].In this paper we construct infinite sequences of DGS graphs from certain small starter graphs. This construction involves the walk matrix of a graph and recent results in [20]. Let n be the number of vertices of G and let e be the column vector of height n with all entries equal to 1. Then the walk matrix of G is the n × n matrix W = e, Ae, A 2 e, · · · , A n−1 e

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“…Further studies in this direction include enumerating and counting strong structurally controllable graphs for a given set of network parameters (leaders and nodes) [22], [23], leader selection to achieve desired structural controllability (e.g., [24], [4], [25], [5], [7], [6], [26]), and network topology design for a desired control performance (e.g., [27], [28], [29], [30], [31]).…”

Section: A Related Workmentioning

confidence: 99%

On the Computation of the Distance-based Lower Bound on Strong Structural Controllability in Networks

Shabbir

Abbas

Yazıcıoğlu

2019

2019 IEEE 58th Conference on Decision and Control (CDC)

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A network of agents with linear dynamics is strong structurally controllable if agents can be maneuvered from any initial state to any final state independently of the coupling strengths between agents. If a network is not strong structurally controllable with a given set of input nodes (leaders), then the dimension of strong structurally controllable subspace quantifies the extent to which a network can be controlled by the same inputs. Computing this dimension exactly is computationally challenging. In this paper, we study the problem of computing a sharp lower bound on the dimension of strong structurally controllable subspace in networks with Laplacian dynamics. The bound is based on a sequence of vectors containing distances between leaders and the remaining nodes in the underlying network graph. Such vectors are referred to as the distanceto-leader vectors. We provide a polynomial time algorithm to compute a particular sequence of distance-to-leader vectors with a fixed set of leaders, which directly provides a lower bound on the dimension of strong structurally controllable subspace. We also present a linearithmic approximation algorithm to compute such a sequence, which provides near optimal solutions in practice. Using these results, we also explore connections between graph-theoretic properties and length of longest such sequences in path and cycle graphs. Finally, we numerically evaluate and compare our bound with other bounds in the literature on various networks. arXiv:1909.03565v1 [eess.SY]

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On a Conjecture of Godsil Concerning Controllable Random Graphs

Cited by 36 publications

References 28 publications

Low-Degree Factors of Random Polynomials

Low-Degree Factors of Random Polynomials

New families of graphs determined by their generalized spectrum

On the Computation of the Distance-based Lower Bound on Strong Structural Controllability in Networks

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