scholarly journals Asymptotic Properties of a Random Graph with Duplications

2015 ◽  
Vol 52 (2) ◽  
pp. 375-390 ◽  
Author(s):  
Ágnes Backhausz ◽  
Tamás F. Móri
Keyword(s):  
Random Graph ◽  
Discrete Time ◽  
Exponential Decay ◽  
Degree Distribution ◽  
Asymptotic Properties ◽  
Graph Model ◽  
Random Graph Model ◽  
Stretched Exponential ◽  
Stretched Exponential Decay

We deal with a random graph model evolving in discrete time steps by duplicating and deleting the edges of randomly chosen vertices. We prove the existence of an almost surely asymptotic degree distribution, with stretched exponential decay; more precisely, the proportion of vertices of degree d tends to some positive number cd > 0 almost surely as the number of steps goes to ∞, and cd ~ (eπ)1/2d1/4e-2√d holds as d → ∞.

scholarly journals Asymptotic Properties of a Random Graph with Duplications

2015 ◽  
Vol 52 (02) ◽  
pp. 375-390 ◽  
Author(s):  
Ágnes Backhausz ◽  
Tamás F. Móri
Keyword(s):  
Random Graph ◽  
Discrete Time ◽  
Exponential Decay ◽  
Degree Distribution ◽  
Asymptotic Properties ◽  
Graph Model ◽  
Random Graph Model ◽  
Stretched Exponential ◽  
Stretched Exponential Decay

We deal with a random graph model evolving in discrete time steps by duplicating and deleting the edges of randomly chosen vertices. We prove the existence of an almost surely asymptotic degree distribution, with stretched exponential decay; more precisely, the proportion of vertices of degreedtends to some positive numbercd> 0 almost surely as the number of steps goes to ∞, andcd~ (eπ)1/2d1/4e-2√dholds asd→ ∞.

scholarly journals Scale-Free Property for Degrees and Weights in a Preferential Attachment Random Graph Model

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
István Fazekas ◽  
Bettina Porvázsnyik
Keyword(s):  
Asymptotic Behaviour ◽  
Random Graph ◽  
Power Law ◽  
Degree Distribution ◽  
Preferential Attachment ◽  
Graph Model ◽  
Random Graph Model ◽  
Scale Free ◽  
Attachment Model ◽  
Law Degree

A random graph evolution mechanism is defined. The evolution studied is a combination of the preferential attachment model and the interaction of four vertices. The asymptotic behaviour of the graph is described. It is proved that the graph exhibits a power law degree distribution; in other words, it is scale-free. It turns out that any exponent in(2,∞)can be achieved. The proofs are based on martingale methods.

scholarly journals Analysis of E-Commerce Product Graphs

2020 ◽  
Author(s):  
Shalin Shah
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Power Law ◽  
Real World ◽  
Degree Distribution ◽  
Graph Model ◽  
Graph Analysis ◽  
Random Graph Model ◽  
Product Graphs ◽  
Clustering Coefficients

<p>Consumer behavior in retail stores gives rise to product graphs based on copurchasing</p><p>or co-viewing behavior. These product graphs can be analyzed using</p><p>the known methods of graph analysis. In this paper, we analyze the product graph</p><p>at Target Corporation based on the Erd˝os-Renyi random graph model. In particular,</p><p>we compute clustering coefficients of actual and random graphs, and we find that</p><p>the clustering coefficients of actual graphs are much higher than random graphs.</p><p>We conduct the analysis on the entire set of products and also on a per category</p><p>basis and find interesting results. We also compute the degree distribution and</p><p>we find that the degree distribution is a power law as expected from real world</p><p>networks, contrasting with the ER random graph.</p>

scholarly journals Analysis of E-Commerce Product Graphs

2020 ◽  
Author(s):  
Shalin Shah
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Power Law ◽  
Real World ◽  
Degree Distribution ◽  
Graph Model ◽  
Graph Analysis ◽  
Random Graph Model ◽  
Product Graphs ◽  
Clustering Coefficients

<p>Consumer behavior in retail stores gives rise to product graphs based on copurchasing</p><p>or co-viewing behavior. These product graphs can be analyzed using</p><p>the known methods of graph analysis. In this paper, we analyze the product graph</p><p>at Target Corporation based on the Erd˝os-Renyi random graph model. In particular,</p><p>we compute clustering coefficients of actual and random graphs, and we find that</p><p>the clustering coefficients of actual graphs are much higher than random graphs.</p><p>We conduct the analysis on the entire set of products and also on a per category</p><p>basis and find interesting results. We also compute the degree distribution and</p><p>we find that the degree distribution is a power law as expected from real world</p><p>networks, contrasting with the ER random graph.</p>

Degree distribution of a scale-free random graph model

2012 ◽  
Vol 28 (3) ◽  
pp. 587-598 ◽  
Author(s):  
Li Tan ◽  
Zhen Ting Hou ◽  
Xin Ru Liu
Keyword(s):  
Random Graph ◽  
Degree Distribution ◽  
Graph Model ◽  
Random Graph Model ◽  
Scale Free

Random graphs

2018 ◽  
Author(s):  
Mark Newman
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Large Scale ◽  
Random Network ◽  
Graph Model ◽  
Clustering Coefficient ◽  
Giant Component ◽  
Random Graph Model ◽  
Definition Of ◽  
Average Size

An introduction to the mathematics of the Poisson random graph, the simplest model of a random network. The chapter starts with a definition of the model, followed by derivations of basic properties like the mean degree, degree distribution, and clustering coefficient. This is followed with a detailed derivation of the large-scale structural properties of random graphs, including the position of the phase transition at which a giant component appears, the size of the giant component, the average size of the small components, and the expected diameter of the network. The chapter ends with a discussion of some of the shortcomings of the random graph model.

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