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 Width of a scale-free tree

2005 ◽  
Vol 42 (03) ◽  
pp. 839-850 ◽  
Author(s):  
Zsolt Katona
Keyword(s):  
Theoretical Result ◽  
Power Law ◽  
Real World ◽  
Degree Distribution ◽  
Graph Model ◽  
Random Graph Model ◽  
Scale Free ◽  
Slight Generalization ◽  
Free Tree ◽  
Law Degree

Consider the random graph model of Barabási and Albert, where we add a new vertex in every step and connect it to some old vertices with probabilities proportional to their degrees. If we connect it to only one of the old vertices then this will be a tree. These graphs have been shown to have a power-law degree distribution, the same as that observed in some large real-world networks. We are interested in the width of the tree and we show that it is at the nth step; this also holds for a slight generalization of the model with another constant. We then see how this theoretical result can be applied to directory trees.

scholarly journals Width of a scale-free tree

2005 ◽  
Vol 42 (3) ◽  
pp. 839-850 ◽  
Author(s):  
Zsolt Katona
Keyword(s):  
Theoretical Result ◽  
Power Law ◽  
Real World ◽  
Degree Distribution ◽  
Graph Model ◽  
Random Graph Model ◽  
Scale Free ◽  
Slight Generalization ◽  
Free Tree ◽  
Law Degree

Consider the random graph model of Barabási and Albert, where we add a new vertex in every step and connect it to some old vertices with probabilities proportional to their degrees. If we connect it to only one of the old vertices then this will be a tree. These graphs have been shown to have a power-law degree distribution, the same as that observed in some large real-world networks. We are interested in the width of the tree and we show that it is at the nth step; this also holds for a slight generalization of the model with another constant. We then see how this theoretical result can be applied to directory trees.

GROWING HIERARCHICAL SCALE-FREE NETWORKS BY MEANS OF NONHIERARCHICAL PROCESSES

2007 ◽  
Vol 17 (07) ◽  
pp. 2447-2452 ◽  
Author(s):  
S. BOCCALETTI ◽  
D.-U. HWANG ◽  
V. LATORA
Keyword(s):  
Power Law ◽  
Degree Distribution ◽  
Preferential Attachment ◽  
Rate Equations ◽  
Scale Free ◽  
Scale Free Networks ◽  
Degree Correlations ◽  
Law Degree ◽  
Hierarchical Scale

We introduce a fully nonhierarchical network growing mechanism, that furthermore does not impose explicit preferential attachment rules. The growing procedure produces a graph featuring power-law degree and clustering distributions, and manifesting slightly disassortative degree-degree correlations. The rigorous rate equations for the evolution of the degree distribution and for the conditional degree-degree probability are derived.

scholarly journals Sublinear Random Access Generators for Preferential Attachment Graphs

2021 ◽  
Vol 17 (4) ◽  
pp. 1-26
Author(s):  
Guy Even ◽  
Reut Levi ◽  
Moti Medina ◽  
Adi Rosén
Keyword(s):  
Random Graph ◽  
Preferential Attachment ◽  
Random Access ◽  
Graph Model ◽  
Efficient Estimation ◽  
Sublinear Algorithms ◽  
Tree Model ◽  
Random Graph Model ◽  
Time Space ◽  
Attachment Model

We consider the problem of sampling from a distribution on graphs, specifically when the distribution is defined by an evolving graph model, and consider the time, space, and randomness complexities of such samplers. In the standard approach, the whole graph is chosen randomly according to the randomized evolving process, stored in full, and then queries on the sampled graph are answered by simply accessing the stored graph. This may require prohibitive amounts of time, space, and random bits, especially when only a small number of queries are actually issued. Instead, we propose a setting where one generates parts of the sampled graph on-the-fly, in response to queries, and therefore requires amounts of time, space, and random bits that are a function of the actual number of queries. Yet, the responses to the queries correspond to a graph sampled from the distribution in question. Within this framework, we focus on two random graph models: the Barabási-Albert Preferential Attachment model (BA-graphs) ( Science , 286 (5439):509–512) (for the special case of out-degree 1) and the random recursive tree model ( Theory of Probability and Mathematical Statistics , (51):1–28). We give on-the-fly generation algorithms for both models. With probability 1-1/poly( n ), each and every query is answered in polylog( n ) time, and the increase in space and the number of random bits consumed by any single query are both polylog( n ), where n denotes the number of vertices in the graph. Our work thus proposes a new approach for the access to huge graphs sampled from a given distribution, and our results show that, although the BA random graph model is defined by a sequential process, efficient random access to the graph’s nodes is possible. In addition to the conceptual contribution, efficient on-the-fly generation of random graphs can serve as a tool for the efficient simulation of sublinear algorithms over large BA-graphs, and the efficient estimation of their on such graphs.

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>

TOPOLOGICAL ANALYSIS OF SPECIFIC SPATIAL COMPLEX NETWORKS

2009 ◽  
Vol 12 (01) ◽  
pp. 45-71 ◽  
Author(s):  
JUN WANG ◽  
GREGORY PROVAN
Keyword(s):  
Random Graph ◽  
Optimization Model ◽  
Preferential Attachment ◽  
Topological Analysis ◽  
Brain Networks ◽  
Graph Model ◽  
Spatial Networks ◽  
Electronic Circuits ◽  
Random Graph Model ◽  
Attachment Model

Based on analyses of specific spatial networks, we compare the accuracy of three models in capturing topologies of two types of spatial networks: electronic circuits and brain networks. The models analyzed are an optimization model trading off multiple-objective constraints, an extended preferential attachment model with spatial constraints, and the generalized random graph model. First, we find that the optimization model and the spatial preferential attachment model can generate similar topological structures under appropriate parameters. Second, our experiments surprisingly show that the simple generalized random graph model outperforms the two proposed models. Third, we find that a series of spatial networks under global optimization of wire length, including the electronic circuits, brain networks, neuronal networks and transportation networks, have high s-metric values close to those of the corresponding generalized random graph models. These s-metric observations explain why the generalized random graph model can match the electronic circuits and the brain networks well from a probabilistic viewpoint, and distinguish their structures from self-organized spatial networks, such as the Internet.

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 Limit theorems for the weights and the degrees in anN-interactions random graph model

2016 ◽  
Vol 14 (1) ◽  
pp. 414-424 ◽  
Author(s):  
István Fazekas ◽  
Bettina Porvázsnyik
Keyword(s):  
Asymptotic Behaviour ◽  
Random Graph ◽  
Limit Theorems ◽  
Preferential Attachment ◽  
Graph Model ◽  
Maximal Degree ◽  
Random Graph Model ◽  
Maximal Weight ◽  
Martingale Methods

Abstract A random graph evolution based on interactions of N vertices is studied. During the evolution both the preferential attachment rule and the uniform choice of vertices are allowed. The weight of an M-clique means the number of its interactions. The asymptotic behaviour of the weight of a fixed M-clique is studied. Asymptotic theorems for the weight and the degree of a fixed vertex are also presented. Moreover, the limits of the maximal weight and the maximal degree are described. The proofs are based on martingale methods.

scholarly journals Network growth with preferential attachment and without “rich get richer” mechanism

2015 ◽  
Vol 27 (02) ◽  
pp. 1650020
Author(s):  
A. Lachgar ◽  
A. Achahbar
Keyword(s):  
Power Law ◽  
Degree Distribution ◽  
Preferential Attachment ◽  
Target Node ◽  
Network Growth ◽  
Perfect Agreement ◽  
Equation Solution ◽  
Attachment Model ◽  
Law Degree ◽  
Growing Network

We propose a simple preferential attachment model of growing network using the complementary probability of Barabási–Albert (BA) model, i.e. [Formula: see text]. In this network, new nodes are preferentially attached to not well connected nodes. Numerical simulations, in perfect agreement with the master equation solution, give an exponential degree distribution. This suggests that the power law degree distribution is a consequence of preferential attachment probability together with “rich get richer” phenomena. We also calculate the average degree of a target node at time t[Formula: see text] and its fluctuations, to have a better view of the microscopic evolution of the network, and we also compare the results with BA model.

scholarly journals Joint Vertex Degrees in the Inhomogeneous Random Graph Model ℊ(n,{pij })

2012 ◽  
Vol 44 (01) ◽  
pp. 139-165
Author(s):  
Kaisheng Lin ◽  
Gesine Reinert
Keyword(s):  
Random Graph ◽  
Power Law ◽  
Graph Model ◽  
Multivariate Normal ◽  
Random Graph Model ◽  
Inhomogeneous Random Graphs ◽  
Vertex Degrees ◽  
Law Degree ◽  
Process Approximation ◽  
Inhomogeneous Random Graph

In a random graph, counts for the number of vertices with given degrees will typically be dependent. We show via a multivariate normal and a Poisson process approximation that, for graphs which have independent edges, with a possibly inhomogeneous distribution, only when the degrees are large can we reasonably approximate the joint counts as independent. The proofs are based on Stein's method and the Stein-Chen method with a new size-biased coupling for such inhomogeneous random graphs, and, hence, bounds on the distributional distance are obtained. Finally, we illustrate that apparent (pseudo-)power-law-type behaviour can arise in such inhomogeneous networks despite not actually following a power-law degree distribution.

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