scholarly journals The $k$-Cut Model in Deterministic and Random Trees

10.37236/9486 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Gabriel Berzunza ◽  
Xing Shi Cai ◽  
Cecilia Holmgren
Keyword(s):  
Random Trees ◽  
Convergence In Distribution ◽  
Scale Free ◽  
Offspring Distribution ◽  
Logarithmic Height ◽  
Split Trees ◽  
Recursive Trees ◽  
Cutting Model

The \(k\)-cut number of rooted graphs was introduced by Cai et al. as a generalization of the classical cutting model by Meir and Moon. In this paper, we show that all moments of the \(k\)-cut number of conditioned Galton-Watson trees converge after proper rescaling, which implies convergence in distribution to the same limit law regardless of the offspring distribution of the trees. This extends the result of Janson. Using the same method, we also show that the \(k\)-cut number of various random or deterministic trees of logarithmic height converges in probability to a constant after rescaling, such as random split-trees, uniform random recursive trees, and scale-free random trees. 

scholarly journals Deterministic Edge Weights in Increasing Tree Families

2009 ◽  
Vol 19 (1) ◽  
pp. 99-119
Author(s):  
MARKUS KUBA ◽  
STEPHAN WAGNER
Keyword(s):  
Normal Limit ◽  
Random Trees ◽  
Edge Weight ◽  
Combinatorial Approach ◽  
Related Parameter ◽  
Scale Free ◽  
Recursive Tree ◽  
Combinatorial Model ◽  
Recursive Trees

In this work we study edge weights for two specific families of increasing trees, which include binary increasing trees and plane-oriented recursive trees as special instances, where plane-oriented recursive trees serve as a combinatorial model of scale-free random trees given by the m = 1 case of the Barabási–Albert model. An edge e = (k, l), connecting the nodes labelled k and l, respectively, in an increasing tree, is associated with the weight we = |k − l|. We are interested in the distribution of the number of edges with a fixed edge weight j in a random generalized plane-oriented recursive tree or random d-ary increasing tree. We provide exact formulas for expectation and variance and prove a normal limit law for this quantity. A combinatorial approach is also presented and applied to a related parameter, the maximum edge weight.

scholarly journals Embedding Small Digraphs and Permutations in Binary Trees and Split Trees

Algorithmica ◽  
2020 ◽  
Vol 82 (3) ◽  
pp. 589-615
Author(s):  
Michael Albert ◽  
Cecilia Holmgren ◽  
Tony Johansson ◽  
Fiona Skerman
Keyword(s):  
Large Class ◽  
Random Permutation ◽  
Random Trees ◽  
Binary Trees ◽  
Independent Interest ◽  
Explicit Formulas ◽  
Random Node ◽  
Labelled Trees ◽  
Logarithmic Height ◽  
Split Trees

AbstractWe investigate the number of permutations that occur in random labellings of trees. This is a generalisation of the number of subpermutations occurring in a random permutation. It also generalises some recent results on the number of inversions in randomly labelled trees (Cai et al. in Combin Probab Comput 28(3):335–364, 2019). We consider complete binary trees as well as random split trees a large class of random trees of logarithmic height introduced by Devroye (SIAM J Comput 28(2):409–432, 1998. 10.1137/s0097539795283954). Split trees consist of nodes (bags) which can contain balls and are generated by a random trickle down process of balls through the nodes. For complete binary trees we show that asymptotically the cumulants of the number of occurrences of a fixed permutation in the random node labelling have explicit formulas. Our other main theorem is to show that for a random split tree, with probability tending to one as the number of balls increases, the cumulants of the number of occurrences are asymptotically an explicit parameter of the split tree. For the proof of the second theorem we show some results on the number of embeddings of digraphs into split trees which may be of independent interest.

scholarly journals Tree limits and limits of random trees

2021 ◽  
pp. 1-45
Author(s):  
Svante Janson
Keyword(s):  
Continuous Time ◽  
Branching Process ◽  
Preferential Attachment ◽  
Binary Search ◽  
Random Trees ◽  
Ordered Trees ◽  
Labelled Trees ◽  
Split Trees ◽  
Recursive Trees ◽  
Simply Generated Trees

Abstract We explore the tree limits recently defined by Elek and Tardos. In particular, we find tree limits for many classes of random trees. We give general theorems for three classes of conditional Galton–Watson trees and simply generated trees, for split trees and generalized split trees (as defined here), and for trees defined by a continuous-time branching process. These general results include, for example, random labelled trees, ordered trees, random recursive trees, preferential attachment trees, and binary search trees.

scholarly journals Profiles of random trees: plane-oriented recursive trees

2005 ◽  
Vol DMTCS Proceedings vol. AD,... (Proceedings) ◽  
Author(s):  
Hsien-Kuei Hwang
Keyword(s):  
Correlation Coefficients ◽  
Random Trees ◽  
Binary Trees ◽  
Asymptotic Approximations ◽  
Square Root ◽  
International Audience ◽  
Expected Width ◽  
Random Binary Trees ◽  
Logarithmic Height ◽  
Recursive Trees

International audience We summarize several limit results for the profile of random plane-oriented recursive trees. These include the limit distribution of the normalized profile, asymptotic bimodality of the variance, asymptotic approximations of the expected width and the correlation coefficients of two level sizes. We also unveil an unexpected connection between the profile of plane-oriented recursive trees (with logarithmic height) and that of random binary trees (with height proportional to the square root of tree size).

scholarly journals Betweenness centrality profiles in trees

2017 ◽  
Vol 5 (5) ◽  
pp. 776-794
Author(s):  
Benjamin Fish ◽  
Rahul Kushwaha ◽  
György Turán
Keyword(s):  
Betweenness Centrality ◽  
Shortest Paths ◽  
Random Trees ◽  
Experimental Results ◽  
Basic Notion ◽  
Worst Case ◽  
Scale Free ◽  
Order Of Magnitude

Abstract Betweenness centrality of a vertex in a graph measures the fraction of shortest paths going through the vertex. This is a basic notion for determining the importance of a vertex in a network. The $k$-betweenness centrality of a vertex is defined similarly, but only considers shortest paths of length at most $k$. The sequence of $k$-betweenness centralities for all possible values of $k$ forms the betweenness centrality profile of a vertex. We study properties of betweenness centrality profiles in trees. We show that for scale-free random trees, for fixed $k$, the expectation of $k$-betweenness centrality strictly decreases as the index of the vertex increases. We also analyse worst-case properties of profiles in terms of the distance of profiles from being monotone, and the number of times pairs of profiles can cross. This is related to whether $k$-betweenness centrality, for small values of $k$, may be used instead of having to consider all shortest paths. Bounds are given that are optimal in order of magnitude. We also present some experimental results for scale-free random trees.

scholarly journals Concentration Properties of Extremal Parameters in Random Discrete Structures

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Michael Drmota
Keyword(s):  
Maximum Degree ◽  
Random Trees ◽  
Height Distribution ◽  
Exponential Tail ◽  
Scale Free ◽  
Main Emphasis ◽  
Discrete Structures ◽  
International Audience ◽  
Concentration Properties

International audience The purpose of this survey is to present recent results concerning concentration properties of extremal parameters of random discrete structures. A main emphasis is placed on the height and maximum degree of several kinds of random trees. We also provide exponential tail estimates for the height distribution of scale-free trees.

scholarly journals Almost Giant Clusters for Percolation on Large Trees with Logarithmic Heights

2013 ◽  
Vol 50 (3) ◽  
pp. 603-611 ◽  
Author(s):  
Jean Bertoin
Keyword(s):  
Random Trees ◽  
Applied Probability ◽  
Bond Percolation ◽  
General Criterion ◽  
Asymptotic Results ◽  
Percolation Clusters ◽  
Large Trees ◽  
Logarithmic Height

This paper is based on works presented at the 2012 Applied Probability Trust Lecture in Sheffield; its main purpose is to survey the recent asymptotic results of Bertoin (2012a) and Bertoin and Uribe Bravo (2012b) about Bernoulli bond percolation on certain large random trees with logarithmic height. We also provide a general criterion for the existence of giant percolation clusters in large trees, which answers a question raised by David Croydon.

Phase Changes in the Topological Indices of Scale-Free Trees

2013 ◽  
Vol 50 (02) ◽  
pp. 516-532 ◽  
Author(s):  
Qunqiang Feng ◽  
Zhishui Hu
Keyword(s):  
Nonnegative Integer ◽  
Topological Indices ◽  
Critical Values ◽  
Phase Changes ◽  
Zagreb Index ◽  
Asymptotic Behaviors ◽  
Scale Free ◽  
Recursive Tree ◽  
Free Tree ◽  
Recursive Trees

A scale-free tree with the parameter β is very close to a star if β is just a bit larger than −1, whereas it is close to a random recursive tree if β is very large. Through the Zagreb index, we consider the whole scene of the evolution of the scale-free trees model as β goes from −1 to + ∞. The critical values of β are shown to be the first several nonnegative integer numbers. We get the first two moments and the asymptotic behaviors of this index of a scale-free tree for all β. The generalized plane-oriented recursive trees model is also mentioned in passing, as well as the Gordon-Scantlebury and the Platt indices, which are closely related to the Zagreb index.

scholarly journals Minimax functions on Galton–Watson trees

2019 ◽  
Vol 29 (3) ◽  
pp. 455-484 ◽  
Author(s):  
James B. Martin ◽  
Roman Stasiński
Keyword(s):  
Branching Process ◽  
Continuous Distribution ◽  
Random Trees ◽  
Value Of The Game ◽  
Offspring Distribution ◽  
Common Distribution ◽  
Open Questions ◽  
Terminal Values ◽  
Level 2 ◽  
Distributional Limits

AbstractWe consider the behaviour of minimax recursions defined on random trees. Such recursions give the value of a general class of two-player combinatorial games. We examine in particular the case where the tree is given by a Galton–Watson branching process, truncated at some depth 2n, and the terminal values of the level 2n nodes are drawn independently from some common distribution. The case of a regular tree was previously considered by Pearl, who showed that as n → ∞ the value of the game converges to a constant, and by Ali Khan, Devroye and Neininger, who obtained a distributional limit under a suitable rescaling.For a general offspring distribution, there is a surprisingly rich variety of behaviour: the (unrescaled) value of the game may converge to a constant, or to a discrete limit with several atoms, or to a continuous distribution. We also give distributional limits under suitable rescalings in various cases.We also address questions of endogeny. Suppose the game is played on a tree with many levels, so that the terminal values are far from the root. To be confident of playing a good first move, do we need to see the whole tree and its terminal values, or can we play close to optimally by inspecting just the first few levels of the tree? The answers again depend in an interesting way on the offspring distribution.We also mention several open questions.

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