scholarly journals Degree distribution of random Apollonian network structures and Boltzmann sampling

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Alexis Darrasse ◽  
Michèle Soria
Keyword(s):  
Power Law ◽  
Degree Distribution ◽  
Random Trees ◽  
Network Structures ◽  
Higher Dimension ◽  
Random Generation ◽  
International Audience ◽  
Boltzmann Model

International audience Random Apollonian networks have been recently introduced for representing real graphs. In this paper we study a modified version: random Apollonian network structures (RANS), which preserve the interesting properties of real graphs and can be handled with powerful tools of random generation. We exhibit a bijection between RANS and ternary trees, that transforms the degree of nodes in a RANS into the size of particular subtrees. The distribution of degrees in RANS can thus be analysed within a bivariate Boltzmann model for the generation of random trees, and we show that it has a Catalan form which reduces to a power law with an exponential cutoff: $α ^k k^{-3/2}$, with $α = 8/9$. We also show analogous distributions for the degree in RANS of higher dimension, related to trees of higher arity.

scholarly journals Traffic-driven epidemic spreading on scale-free networks with tunable degree distribution

2016 ◽  
Vol 27 (11) ◽  
pp. 1650125 ◽  
Author(s):  
Han-Xin Yang ◽  
Bing-Hong Wang
Keyword(s):  
Power Law ◽  
Degree Distribution ◽  
Network Structures ◽  
Epidemic Spreading ◽  
Epidemic Threshold ◽  
Scale Free ◽  
Scale Free Networks ◽  
Law Degree

We study the traffic-driven epidemic spreading on scale-free networks with tunable degree distribution. The heterogeneity of networks is controlled by the exponent [Formula: see text] of power-law degree distribution. It is found that the epidemic threshold is minimized at about [Formula: see text]. Moreover, we find that nodes with larger algorithmic betweenness are more likely to be infected. We expect our work to provide new insights in to the effect of network structures on traffic-driven epidemic spreading.

scholarly journals The Degree Distribution of Thickened Trees

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Michael Drmota ◽  
Bernhard Gittenberger ◽  
Alois Panholzer
Keyword(s):  
Power Law ◽  
Real World ◽  
Degree Distribution ◽  
Combinatorial Structure ◽  
Random Real ◽  
Asymptotic Power ◽  
Scale Free ◽  
International Audience ◽  
Recursive Trees

International audience We develop a combinatorial structure to serve as model of random real world networks. Starting with plane oriented recursive trees we substitute the nodes by more complex graphs. In such a way we obtain graphs having a global tree-like structure while locally looking clustered. This fits with observations obtained from real-world networks. In particular we show that the resulting graphs are scale-free, that is, the degree distribution has an asymptotic power law.

scholarly journals The structure of co-publications multilayer network

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Ghislain Romaric Meleu ◽  
Paulin Yonta Melatagia
Keyword(s):  
Power Law ◽  
Degree Distribution ◽  
Preferential Attachment ◽  
Small World ◽  
Generation Model ◽  
Small World Network ◽  
Power Law Distribution ◽  
Multilayer Networks ◽  
Multilayer Network ◽  
Scientific Papers

AbstractUsing the headers of scientific papers, we have built multilayer networks of entities involved in research namely: authors, laboratories, and institutions. We have analyzed some properties of such networks built from data extracted from the HAL archives and found that the network at each layer is a small-world network with power law distribution. In order to simulate such co-publication network, we propose a multilayer network generation model based on the formation of cliques at each layer and the affiliation of each new node to the higher layers. The clique is built from new and existing nodes selected using preferential attachment. We also show that, the degree distribution of generated layers follows a power law. From the simulations of our model, we show that the generated multilayer networks reproduce the studied properties of co-publication networks.

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 The profile of unlabeled trees

2005 ◽  
Vol DMTCS Proceedings vol. AD,... (Proceedings) ◽  
Author(s):  
Bernhard Gittenberger
Keyword(s):  
Local Time ◽  
Joint Distribution ◽  
Random Trees ◽  
Rooted Trees ◽  
Brownian Excursion ◽  
Level Number ◽  
International Audience ◽  
The Times ◽  
Simply Generated Trees

International audience We consider the number of nodes in the levels of unlabeled rooted random trees and show that the joint distribution of several level sizes (where the level number is scaled by $\sqrt{n}$) weakly converges to the distribution of the local time of a Brownian excursion evaluated at the times corresponding to the level numbers. This extends existing results for simply generated trees and forests to the case of unlabeled rooted trees.

scholarly journals Models And Mining Of On-Line Social Networks

2021 ◽  
Author(s):  
Yanhua Tian
Keyword(s):  
Social Networks ◽  
Power Law ◽  
Degree Distribution ◽  
Small World ◽  
Spectral Expansion ◽  
On Line ◽  
Simulation Results ◽  
Small World Property ◽  
P Type ◽  
Law Degree

Power law degree distribution, the small world property, and bad spectral expansion are three of the most important properties of On-line Social Networks (OSNs). We sampled YouTube and Wikipedia to investigate OSNs. Our simulation and computational results support the conclusion that OSNs follow a power law degree distribution, have the small world property, and bad spectral expansion. We calculated the diameters and spectral gaps of OSNs samples, and compared these to graphs generated by the GEO-P model. Our simulation results support the Logarithmic Dimension Hypothesis, which conjectures that the dimension of OSNs is m = [log N]. We introduced six GEO-P type models. We ran simulations of these GEO-P-type models, and compared the simulated graphs with real OSN data. Our simulation results suggest that, except for the GEO-P (GnpDeg) model, all our models generate graphs with power law degree distributions, the small world property, and bad spectral expansion.

scholarly journals RANDOM WALKS ON DIRECTED NETWORKS: THE CASE OF PAGERANK

2007 ◽  
Vol 17 (07) ◽  
pp. 2343-2353 ◽  
Author(s):  
SANTO FORTUNATO ◽  
ALESSANDRO FLAMMINI
Keyword(s):  
Random Walk ◽  
Power Law ◽  
Degree Distribution ◽  
Directed Graphs ◽  
Damping Factor ◽  
Web Pages ◽  
Stationary Probability ◽  
Exact Results ◽  
Limit Values ◽  
Underlying Graph

PageRank, the prestige measure for Web pages used by Google, is the stationary probability of a peculiar random walk on directed graphs, which interpolates between a pure random walk and a process where all nodes have the same probability of being visited. We give some exact results on the distribution of PageRank in the cases in which the damping factor q approaches the two limit values 0 and 1. When q → 0 and for several classes of graphs the distribution is a power law with exponent 2, regardless of the in-degree distribution. When q → 1 it can always be derived from the in-degree distribution of the underlying graph, if the out-degree is the same for all nodes.

scholarly journals The use of the power law for designing wireless networks

2018 ◽  
Vol 21 ◽  
pp. 00012
Author(s):  
Andrzej Paszkiewicz
Keyword(s):  
Wireless Networks ◽  
Power Law ◽  
Degree Distribution ◽  
Limited Resources ◽  
Random Networks ◽  
New Approach ◽  
Available Bandwidth ◽  
Scale Free ◽  
Scale Free Networks ◽  
Node Removal

The paper concerns the use of the scale-free networks theory and the power law in designing wireless networks. An approach based on generating random networks as well as on the classic Barabási-Albert algorithm were presented. The paper presents a new approach taking the limited resources for wireless networks into account, such as available bandwidth. In addition, thanks to the introduction of opportunities for dynamic node removal it was possible to realign processes occurring in wireless networks. After introduction of these modifications, the obtained results were analyzed in terms of a power law and the degree distribution of each node.

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