The Fréchet Mean of Inhomogeneous Random Graphs

We consider a queue to which only a finite pool of n customers can arrive, at times depending on their service requirement. A customer with stochastic service requirement S arrives to the queue after an exponentially distributed time with mean S-α for some [Formula: see text]; therefore, larger service requirements trigger customers to join earlier. This finite-pool queue interpolates between two previously studied cases: α = 0 gives the so-called [Formula: see text] queue and α = 1 is closely related to the exploration process for inhomogeneous random graphs. We consider the asymptotic regime in which the pool size n grows to infinity and establish that the scaled queue-length process converges to a diffusion process with a negative quadratic drift. We leverage this asymptotic result to characterize the head start that is needed to create a long period of activity. We also describe how this first busy period of the queue gives rise to a critically connected random forest.

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A Frechet Mean Approach for Compressive Sensing Date Acquisition and Reconstruction in Wireless Sensor Networks

IEEE Transactions on Wireless Communications ◽

10.1109/twc.2012.081612.111908 ◽

2012 ◽

Vol 11 (10) ◽

pp. 3598-3606 ◽

Cited By ~ 22

Author(s):

Wei Chen ◽

Miguel R. D. Rodrigues ◽

Ian J. Wassell

Keyword(s):

Wireless Sensor Networks ◽

Sensor Networks ◽

Compressive Sensing ◽

Wireless Sensor ◽

Fréchet Mean ◽

Frechet Mean

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On the consistency of procrustean mean shapes

Advances in Applied Probability ◽

10.1239/aap/1035227991 ◽

1998 ◽

Vol 30 (1) ◽

pp. 53-63 ◽

Cited By ~ 29

Author(s):

Huiling Le

Keyword(s):

Mild Condition ◽

Shape Space ◽

Entire Space ◽

Fréchet Mean ◽

Mean Shape ◽

Frechet Mean

We discuss the uniqueness of the Fréchet mean of a class of distributions on the shape space of k labelled points in ℝ2, the supports of which could be the entire space. From this it follows that the shape of the means is the unique Fréchet mean shape of the induced distribution with respect to an appropriate metric structure, provided the distribution of k labelled points in ℝ2 is isotropic and satisfies a further mild condition. This result implies that an increasing sequence of procrustean mean shapes defined in either of the two ways used in practice will tend almost surely to the shape of the means.

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Parameterized clique on inhomogeneous random graphs

Discrete Applied Mathematics ◽

10.1016/j.dam.2014.10.018 ◽

2015 ◽

Vol 184 ◽

pp. 130-138 ◽

Cited By ~ 3

Author(s):

Tobias Friedrich ◽

Anton Krohmer

Keyword(s):

Random Graphs ◽

Inhomogeneous Random Graphs

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On the consistency of procrustean mean shapes

Advances in Applied Probability ◽

10.1017/s0001867800008077 ◽

1998 ◽

Vol 30 (01) ◽

pp. 53-63 ◽

Cited By ~ 6

Author(s):

Huiling Le

Keyword(s):

Mild Condition ◽

Shape Space ◽

Entire Space ◽

Fréchet Mean ◽

Mean Shape ◽

Frechet Mean

We discuss the uniqueness of the Fréchet mean of a class of distributions on the shape space of k labelled points in ℝ2, the supports of which could be the entire space. From this it follows that the shape of the means is the unique Fréchet mean shape of the induced distribution with respect to an appropriate metric structure, provided the distribution of k labelled points in ℝ2 is isotropic and satisfies a further mild condition. This result implies that an increasing sequence of procrustean mean shapes defined in either of the two ways used in practice will tend almost surely to the shape of the means.

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Principal component analysis and the locus of the Fréchet mean in the space of phylogenetic trees

Biometrika ◽

10.1093/biomet/asx047 ◽

2017 ◽

Vol 104 (4) ◽

pp. 901-922 ◽

Cited By ~ 9

Author(s):

Tom M W Nye ◽

Xiaoxian Tang ◽

Grady Weyenberg ◽

Ruriko Yoshida

Keyword(s):

Principal Component Analysis ◽

Phylogenetic Trees ◽

Principal Component ◽

Component Analysis ◽

Fréchet Mean ◽

Frechet Mean

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The Largest Component in Subcritical Inhomogeneous Random Graphs

Combinatorics Probability Computing ◽

10.1017/s0963548310000180 ◽

2010 ◽

Vol 20 (1) ◽

pp. 131-154 ◽

Cited By ~ 8

Author(s):

TATYANA S. TUROVA

Keyword(s):

Random Graph ◽

Random Graphs ◽

Graph Model ◽

Exact Formula ◽

Connected Component ◽

Subcritical Case ◽

Random Graph Model ◽

Asymptotic Size ◽

Inhomogeneous Random Graphs ◽

Inhomogeneous Random Graph

We study the ‘rank 1 case’ of the inhomogeneous random graph model. In the subcritical case we derive an exact formula for the asymptotic size of the largest connected component scaled to log n. This result complements the corresponding known result in the supercritical case. We provide some examples of applications of the derived formula.

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