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A family of random trees with random edge lengths

David Aldous, Jim Pitman
1999 Random structures & algorithms (Print)  
We i n troduce a family of probability distributions on the space of trees with I labeled vertices and possibly extra unlabeled vertices of degree 3, whose edges have positive real lengths.  ...  An interpretation is given in terms of sampling from the inhomogeneous continuum random tree of Aldous and Pitman 1998.  ...  The purpose of this paper is to describe a new model of random trees with edge lengths.  ... 
doi:10.1002/(sici)1098-2418(199909)15:2<176::aid-rsa4>3.0.co;2-4 fatcat:yju3chr5tfgv7jl44jk63luyzi

The Continuum Random Tree III

David Aldous
1993 Annals of Probability  
Then under an extra condition, this family determines a random continuum tree ?/, which it is convenient to represent as a random subset of 11.  ...  Let (W(k), k 2 1) be random trees with k leaves, satisfying a consistency condition: Removing a random leaf from R(k) gives R(k -1).  ...  Formally, a tree with n vertices (and hence n -1 edges) could be represented as a point (1) t = (F; x1,. ., xn-1) E Tn x Rn-l where ti is the corresponding tree without edge lengths, and t edge lengths  ... 
doi:10.1214/aop/1176989404 fatcat:4oyqnmkw5nfbvivyfuipdchaie

Unidentifiable divergence times in rates-across-sites models

S.N. Evans, T. Warnow
2004 IEEE/ACM Transactions on Computational Biology & Bioinformatics  
We consider an example of a clock-like tree with three taxa, one unknown edge length, a known root state, and a parametric family of scale factor distributions that contain the gamma family.  ...  This model has the property that, for a generic choice of unknown edge length and scale factor distribution, there is another edge length and scale factor distribution which generates data with exactly  ...  Acknowledgments The authors thank the anonymous referees for a number of very helpful suggestions and observations.  ... 
doi:10.1109/tcbb.2004.34 pmid:17048388 fatcat:3pjbnutje5hjbiy473xwefbiyi

Percolation-like scaling exponents for minimal paths and trees in the stochastic mean field model

D. J. Aldous
2005 Proceedings of the Royal Society A  
A parallel study with trees instead of paths gives scaling exponent β = 2.  ...  For 0 < ℓ < ∞ and in the n →∞ limit, let δ(ℓ) = 1/n × (maximum number of steps in a path whose average step-length is ≤ℓ).  ...  For any tree t in the graph, with edges e 1 , . . . , e m , write size(t) for the number of edges of t and A(t) for the average edge-length: size(t) = m A(t) = m −1 e∈t d(e).  ... 
doi:10.1098/rspa.2004.1388 fatcat:t4cw7sgu3jb2zp5qbe6rfwwxze

Recovering a tree from the lengths of subtrees spanned by a randomly chosen sequence of leaves

Steven N. Evans, Daniel Lanoue
2018 Advances in Applied Mathematics  
We show that if T is known a priori to belong to one of various families of edge-weighted trees, then the answer is, "Yes."  ...  Given an edge-weighted tree T with n leaves, sample the leaves uniformly at random without replacement and let W k , 2 ≤ k ≤ n, be the length of the subtree spanned by the first k leaves.  ...  The last family of edge-weighted trees with general edge-weights whose elements we can identify up to isomorphism from the joint probability distributions of their random length sequences is the class  ... 
doi:10.1016/j.aam.2018.01.001 pmid:30220760 pmcid:PMC6135540 fatcat:c5x5ymk5z5bspcwpnhyzjckfvm

Unidentifiable divergence times in rates-across-sites models [article]

Steven N. Evans, Tandy Warnow
2004 arXiv   pre-print
We consider an example of a clock-like tree with three taxa, one unknown edge length, and a parametric family of scale factor distributions that contain the gamma family.  ...  This model has the property that, for a generic choice of unknown edge length and scale factor distribution, there is another edge length and scale factor distribution which generates data with exactly  ...  Acknowledgments The authors thank the anonymous referees for a number of very helpful suggestions and observations.  ... 
arXiv:q-bio/0408011v2 fatcat:azedgv6ykjfelehsgckxatqbka

First Passage Percolation on Random Geometric Graphs and an Application to Shortest-Path Trees

C. Hirsch, D. Neuhäuser, C. Gloaguen, V. Schmidt
2015 Advances in Applied Probability  
length of the longest branch in the shortest-path tree extracted from a typical segment system if the intensity of network stations converges to 0.  ...  We consider Euclidean first passage percolation on a large family of connected random geometric graphs in the d-dimensional Euclidean space encompassing various well-known models from stochastic geometry  ...  above by total edge length inside certain bad clusters G = random geometric graph in R 2 as above G * = Palm version of G informally: shifting o to random location on the edge set of G Description of  ... 
doi:10.1239/aap/1435236978 fatcat:xwno776zqjbe3jp44o7jyv3d54

Recovering a tree from the lengths of subtrees spanned by a randomly chosen sequence of leaves [article]

Steven N. Evans, Daniel Lanoue
2015 arXiv   pre-print
We show that if T is known a priori to belong to one of various families of edge-weighted trees, then the answer is, "Yes."  ...  Given an edge-weighted tree T with n leaves, sample the leaves uniformly at random without replacement and let W_k, 2 < k < n, be the length of the subtree spanned by the first k leaves.  ...  The last family of edge-weighted trees with general edge-weights whose elements we can identify up to isomorphism from the joint probability distributions of their random length sequences is the class  ... 
arXiv:1506.01091v1 fatcat:x7mh2kw6p5harcyikhxgucufae

Color-coding

Noga Alon, Raphy Yuster, Uri Zwick
1994 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing - STOC '94  
We describe a novel randomized method, the method of color-coding for finding simple paths and cycles of a specified length k, and other small subgraphs, within a given graph G = (V, E).  ...  The randomized algorithms obtained using this method can be derandomized using families of perfect hash functions.  ...  Definition 6.2 A tree-decomposition of a graph G = (V, E) is a pair (X, T ) where T = (I, F ) is a tree and X = {X i : i ∈ I} is a family of subsets of V such that (i) ∪ i∈I X i = V ; (ii) for every edge  ... 
doi:10.1145/195058.195179 dblp:conf/stoc/AlonYZ94 fatcat:7escc6ce2nfmtowkqhrauzn3ae

Right-angled hexagon tilings of the hyperbolic plane [article]

Richard Kenyon
2015 arXiv   pre-print
We prove that, for each measure μ in a certain natural family of measures on right-angled hexagons, there is an isometry-invariant measure on Ω whose marginal distribution on tiles is μ.  ...  We study isometry-invariant probability measures on the space Ω of tilings of the hyperbolic plane with right-angled hexagons of varying shapes.  ...  Then, for each vertex of the tree, use the three adjacent random variables as lengths of edges 1, 3, 5 of a RAH.  ... 
arXiv:1503.05510v1 fatcat:or3spjzauvbjpn6varq2htb5em

On the Hyperbolicity of Small-World and Tree-Like Random Graphs [chapter]

Wei Chen, Wenjie Fang, Guangda Hu, Michael W. Mahoney
2012 Lecture Notes in Computer Science  
may easily destroy the hyperbolicity of the graphs, except for a class of random edges added using an exponentially decaying probability function based on the ring distance among the leaves.  ...  On the other hand, for a class of tree-like graphs called ringed trees that have constant hyperbolicity, adding random links among the leaves in a manner similar to the small-world graph constructions  ...  The second family of graphs is random ringed trees. A ringed tree is a binary tree with nodes in each level of the tree connected by a ring (Figure 1(d) ).  ... 
doi:10.1007/978-3-642-35261-4_31 fatcat:v4o3zxazprd6dgc3tmmkcvyjum

On the Hyperbolicity of Small-World and Treelike Random Graphs

Wei Chen, Wenjie Fang, Guangda Hu, Michael W. Mahoney
2013 Internet Mathematics  
may easily destroy the hyperbolicity of the graphs, except for a class of random edges added using an exponentially decaying probability function based on the ring distance among the leaves.  ...  On the other hand, for a class of tree-like graphs called ringed trees that have constant hyperbolicity, adding random links among the leaves in a manner similar to the small-world graph constructions  ...  The second family of graphs is random ringed trees. A ringed tree is a binary tree with nodes in each level of the tree connected by a ring (Figure 1(d) ).  ... 
doi:10.1080/15427951.2013.828336 fatcat:nulyjmpdnfcn7iu5bdqbpxpqtm

Color-coding

Noga Alon, Raphael Yuster, Uri Zwick
1995 Journal of the ACM  
We describe a novel randomized method, the method of color-coding for finding simple paths and cycles of a specified length k, and other small subgraphs, within a given graph G = (V, E).  ...  The randomized algorithms obtained using this method can be derandomized using families of perfect hash functions.  ...  Definition 6.2 A tree-decomposition of a graph G = (V, E) is a pair (X, T ) where T = (I, F ) is a tree and X = {X i : i ∈ I} is a family of subsets of V such that (i) ∪ i∈I X i = V ; (ii) for every edge  ... 
doi:10.1145/210332.210337 fatcat:xcxlixjrcbbrxkgn2lamsijyke

Embedding spanning trees in random graphs [article]

Michael Krivelevich
2010 arXiv   pre-print
We prove that if T is a tree on n vertices wih maximum degree D and the edge probability p(n) satisfies: np>c*maxD*logn,n^ϵ for some constant ϵ>0, then with high probability the random graph G(n,p) contains  ...  a copy of T.  ...  Let G be a random bipartite graph with sides A and B, where each pair (a, b), aA, b ∈ B, is an edge of G with probability p, independently of other pairs.  ... 
arXiv:1007.2326v2 fatcat:vwinzoc4inbuxhorazvjifuvzq

Poisson point process limits in size-biased Galton-Watson trees

Jochen Geiger
2000 Electronic Journal of Probability  
Decompose the population at t according to the particles' degree of relationship with a distinguished particle picked purely at random from those alive at t.  ...  Keeping track of the times when the different families grow out of the distinguished line of descent and the related family sizes at t, we represent this relationship structure as a point process in a  ...  Let T be the random family tree obtained by having one edge for each particle produced with the length of an edge being the particle's lifetime.  ... 
doi:10.1214/ejp.v5-73 fatcat:xvluendy45d6bb34j3zdn32vya
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