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.

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 Phase Changes in the Topological Indices of Scale-Free Trees

2013 ◽  
Vol 50 (2) ◽  
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 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 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.

Cascading failure model for improving the robustness of scale-free networks

2018 ◽  
Vol 29 (06) ◽  
pp. 1850044 ◽  
Author(s):  
Zhichao Ju ◽  
Jinlong Ma ◽  
Jianjun Xie ◽  
Zhaohui Qi
Keyword(s):  
Real Life ◽  
Cascading Failure ◽  
Edge Weight ◽  
Failure Model ◽  
Threshold Parameter ◽  
Connected Component ◽  
New Model ◽  
Scale Free ◽  
Small Load ◽  
Scale Free Networks

To control the spread of cascading failure on scale-free networks, we propose a new model with the betweenness centrality and the degrees of the nodes which are combined. The effects of the parameters of the edge weight on cascading dynamics are investigated. Five metrics to evaluate the robustness of the network are given: the threshold parameter ([Formula: see text]), the proportion of collapsed edges ([Formula: see text]), the proportion of collapsed nodes ([Formula: see text]), the number of nodes in the largest connected component ([Formula: see text]) and the number of the connected component ([Formula: see text]). Compared with the degrees of nodes’ model and the betweenness of the nodes’ model, the new model could control the spread of cascading failure more significantly. This work might be helpful for preventing and mitigating cascading failure in real life, especially for small load networks.

scholarly journals On the Zagreb Index of Random Recursive Trees

2011 ◽  
Vol 48 (04) ◽  
pp. 1189-1196 ◽  
Author(s):  
Qunqiang Feng ◽  
Zhishui Hu
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Normality ◽  
Random Process ◽  
Central Limit ◽  
Topological Indices ◽  
Zagreb Index ◽  
Martingale Central Limit Theorem ◽  
Recursive Tree ◽  
Recursive Trees

We investigate the Zagreb index, one of the topological indices, of random recursive trees in this paper. Through a recurrence equation, the first two moments ofZn, the Zagreb index of a random recursive tree of sizen, are obtained. We also show that the random process {Zn− E[Zn],n≥ 1} is a martingale. Then the asymptotic normality of the Zagreb index of a random recursive tree is given by an application of the martingale central limit theorem. Finally, two other topological indices are also discussed in passing.

THE TRAPPING PROBLEM OF WEIGHTED (2,2)-FLOWER NETWORKS WITH THE SAME WEIGHT SEQUENCE

Fractals ◽  
2019 ◽  
Vol 27 (07) ◽  
pp. 1950112
Author(s):  
CHANGMING XING
Keyword(s):  
Nearest Neighbor ◽  
Influence Factor ◽  
Network Models ◽  
Trapping Efficiency ◽  
Theoretical Research ◽  
Edge Weight ◽  
Weight Factor ◽  
Scale Free ◽  
Weight Sequence ◽  
Standard Weight

Intuitively, edge weight has an effect on the dynamical processes occurring on the networks. However, the theoretical research on the effects of edge weight on network dynamics is still rare. In this paper, we present two weighted network models by adjusting the matching relationship between weights and edges. Both network models are controlled by the weight factor [Formula: see text]. They have the same connection structure and weight sequence when [Formula: see text] is fixed. Based on their self-similar network structure, we study two types of biased walks with a trap. One is standard weight-dependent walk, while the other is mixed weight-dependent walk including both nearest-neighbor and next-nearest-neighbor jumps. For both weighted scale-free networks, we obtain exact solutions of the average trapping time (ATT) measuring the efficiency of the trapping process in both network models. Analyzing and comparing the obtained solutions, we find that the ATT is related to the walking rule and the spectral dimension of the fractal network, and not all ATT for the weighted networks are affected by the weight factor [Formula: see text]. In other words, not all weight adjustments can change the trapping efficiency in the network. We hope that the present findings could help us get deeper understanding about the influence factor of biased walk in complex systems.

On the size of paged recursive trees

2017 ◽  
Vol 09 (02) ◽  
pp. 1750021
Author(s):  
Mehri Javanian
Keyword(s):  
Contraction Method ◽  
Recursive Tree ◽  
Asymptotically Normal ◽  
Recursive Trees ◽  
Normal Formula

A paged recursive tree is constructed as a recursive tree except that it depends on an integer parameter [Formula: see text] representing a page capacity, small subtrees with size [Formula: see text]. We investigate the number of nodes [Formula: see text] (size of the tree) in paged recursive trees built from labels [Formula: see text]. The expectation and variance of [Formula: see text] are derived, and it is also shown that [Formula: see text] is asymptotically normal. [Formula: see text] as [Formula: see text] by applying the contraction method.

scholarly journals Exact Solutions of a Generalized Weighted Scale Free Network

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Li Tan ◽  
Dingyou Lei
Keyword(s):  
Random Processes ◽  
Analytic Solutions ◽  
Strength Distribution ◽  
Edge Weight ◽  
Scale Free Network ◽  
Scale Free ◽  
Scale Free Networks ◽  
Free Network ◽  
Vertex Degrees ◽  
Power Law Distributions

We investigate a class of generalized weighted scale-free networks, where the new vertex connects tompairs of vertices selected preferentially. The key contribution of this paper is that, from the standpoint of random processes, we provide rigorous analytic solutions for the steady state distributions, including the vertex degree distribution, the vertex strength distribution and the edge weight distribution. Numerical simulations indicate that this network model yields three power law distributions for the vertex degrees, vertex strengths and edge weights, respectively.

Sign in / Sign up

Export Citation Format

Share Document