scholarly journals Random Graphs' Robustness in Random Environment

2017 ◽  
Vol 46 (3-4) ◽  
pp. 89-98
Author(s):  
Marina Leri ◽  
Yury Pavlov
Keyword(s):  
Random Graphs ◽  
Power Law ◽  
Random Environment ◽  
Degree Distribution ◽  
Vertex Degree ◽  
Power Law Distribution ◽  
Destruction Process ◽  
Random Breakdown ◽  
Uniform Distributions ◽  
Vertex Degrees

We consider configuration graphs the vertex degrees of which are independent and  follow the power-law distribution. Random graphs dynamics takes place in a random  environment with the parameter of vertex degree distribution following  uniform distributions on finite fixed intervals. As the number of vertices tends  to infinity the limit distributions of the maximum vertex degree and the number  of vertices with a given degree were obtained. By computer simulations we study  the robustness of those graphs from the viewpoints of link saving and node survival  in the two cases of the destruction process: the ``targeted attack'' and the  ``random breakdown''. We obtained and compared the results under the conditions that  the vertex degree distribution was averaged with respect to the distribution of the  power-law parameter or that the values of the parameter were drawn from the uniform  distribution separately for each vertex.

A Small-World Model of Scale-Free Networks: Features and Verifications

2011 ◽  
Vol 50-51 ◽  
pp. 166-170 ◽  
Author(s):  
Wen Jun Xiao ◽  
Shi Zhong Jiang ◽  
Guan Rong Chen
Keyword(s):  
Complex Networks ◽  
Power Law ◽  
Degree Sequence ◽  
Small World ◽  
Static Condition ◽  
Vertex Degree ◽  
Power Law Distribution ◽  
Least Common Multiple ◽  
Scale Free ◽  
Vertex Degrees

It is now well known that many large-sized complex networks obey a scale-free power-law vertex-degree distribution. Here, we show that when the vertex degrees of a large-sized network follow a scale-free power-law distribution with exponent  2, the number of degree-1 vertices, if nonzero, is of order N and the average degree is of order lower than log N, where N is the size of the network. Furthermore, we show that the number of degree-1 vertices is divisible by the least common multiple of , , . . ., , and l is less than log N, where l = < is the vertex-degree sequence of the network. The method we developed here relies only on a static condition, which can be easily verified, and we have verified it by a large number of real complex networks.

On the asymptotics of degree structure of configuration graphs with bounded number of edges

2019 ◽  
Vol 29 (4) ◽  
pp. 219-232 ◽  
Author(s):  
Yurii L. Pavlov ◽  
Irina A. Cheplyukova
Keyword(s):  
Random Graphs ◽  
Power Law ◽  
Random Variables ◽  
Random Variable ◽  
Positive Parameter ◽  
Power Law Distribution ◽  
Bounded Number ◽  
Degree Structure ◽  
Vertex Degrees ◽  
Independent Identically Distributed

Abstract We consider configuration graphs with N vertices. The degrees of vertices are independent identically distributed random variables having the power-law distribution with positive parameter $\tau .$We study properties of random graphs such that the sum of vertex degrees does not exceed n and the parameter is a random variable uniformly distributed on the interval $\left[ a,\,\,b \right],0<a<b<\infty .$We find limit distributions of the number ${{\mu }_{r}}$of vertices with degree r for various types of variation of N, n and r.

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 Ultra-small scale-free geometric networks

2006 ◽  
Vol 43 (3) ◽  
pp. 665-677 ◽  
Author(s):  
J. E. Yukich
Keyword(s):  
Long Range ◽  
Power Law ◽  
Degree Distribution ◽  
Small Scale ◽  
Graph Distance ◽  
Power Law Distribution ◽  
Scale Free

We consider a family of long-range percolation models (Gp)p>0on ℤdthat allow dependence between edges and have the following connectivity properties forp∈ (1/d, ∞): (i) the degree distribution of vertices inGphas a power-law distribution; (ii) the graph distance between pointsxandyis bounded by a multiple of logpdlogpd|x-y| with probability 1 -o(1); and (iii) an adversary can delete a relatively small number of nodes fromGp(ℤd∩ [0,n]d), resulting in two large, disconnected subgraphs.

scholarly journals Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution

2017 ◽  
Vol 173 (3-4) ◽  
pp. 806-844 ◽  
Author(s):  
Pim van der Hoorn ◽  
Gabor Lippner ◽  
Dmitri Krioukov
Keyword(s):  
Maximum Entropy ◽  
Random Graphs ◽  
Power Law ◽  
Degree Distribution ◽  
Law Degree

scholarly journals Ultra-small scale-free geometric networks

2006 ◽  
Vol 43 (03) ◽  
pp. 665-677
Author(s):  
J. E. Yukich
Keyword(s):  
Long Range ◽  
Power Law ◽  
Degree Distribution ◽  
Small Scale ◽  
Graph Distance ◽  
Power Law Distribution ◽  
Scale Free

We consider a family of long-range percolation models (G p ) p&gt;0 on ℤ d that allow dependence between edges and have the following connectivity properties for p ∈ (1/d, ∞): (i) the degree distribution of vertices in G p has a power-law distribution; (ii) the graph distance between points x and y is bounded by a multiple of log pd log pd | x - y | with probability 1 - o(1); and (iii) an adversary can delete a relatively small number of nodes from G p (ℤ d ∩ [0, n] d ), resulting in two large, disconnected subgraphs.

scholarly journals Moduli stabilisation and the statistics of SUSY breaking in the landscape

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Igor Broeckel ◽  
Michele Cicoli ◽  
Anshuman Maharana ◽  
Kajal Singh ◽  
Kuver Sinha
Keyword(s):  
Supersymmetry Breaking ◽  
Power Law ◽  
Complex Structure ◽  
Susy Breaking ◽  
Power Law Distribution ◽  
High Scale ◽  
The Past ◽  
Uniform Distributions ◽  
Logarithmic Distribution ◽  
Breaking Scale

Abstract The statistics of the supersymmetry breaking scale in the string landscape has been extensively studied in the past finding either a power-law behaviour induced by uniform distributions of F-terms or a logarithmic distribution motivated by dynamical supersymmetry breaking. These studies focused mainly on type IIB flux compactifications but did not systematically incorporate the Kähler moduli. In this paper we point out that the inclusion of the Kähler moduli is crucial to understand the distribution of the supersymmetry breaking scale in the landscape since in general one obtains unstable vacua when the F-terms of the dilaton and the complex structure moduli are larger than the F- terms of the Kähler moduli. After taking Kähler moduli stabilisation into account, we find that the distribution of the gravitino mass and the soft terms is power-law only in KKLT and perturbatively stabilised vacua which therefore favour high scale supersymmetry. On the other hand, LVS vacua feature a logarithmic distribution of soft terms and thus a preference for lower scales of supersymmetry breaking. Whether the landscape of type IIB flux vacua predicts a logarithmic or power-law distribution of the supersymmetry breaking scale thus depends on the relative preponderance of LVS and KKLT vacua.

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 Using Power-Law Degree Distribution to Accelerate PageRank

2012 ◽  
Vol 1 (2) ◽  
pp. 63-70
Author(s):  
Zhaoyan Jin ◽  
Quanyuan Wu
Keyword(s):  
World Wide Web ◽  
Power Law ◽  
Real World ◽  
Degree Distribution ◽  
World Wide ◽  
Power Law Distribution ◽  
The World ◽  
Real World Applications ◽  
Real World Datasets ◽  
Law Degree

The PageRank vector of a network is very important, for it can reflect the importance of a Web page in the World Wide Web, or of a people in a social network. However, with the growth of the World Wide Web and social networks, it needs more and more time to compute the PageRank vector of a network. In many real-world applications, the degree and PageRank distributions of these complex networks conform to the Power-Law distribution. This paper utilizes the degree distribution of a network to initialize its PageRank vector, and presents a Power-Law degree distribution accelerating algorithm of PageRank computation. Experiments on four real-world datasets show that the proposed algorithm converges more quickly than the original PageRank algorithm.DOI: 10.18495/comengapp.12.063070

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