scholarly journals On the Spectra of General Random Graphs

10.37236/702 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Fan Chung ◽  
Mary Radcliffe
Keyword(s):  
Independent Random Variable ◽  
Random Graph ◽  
Random Graphs ◽  
Power Law ◽  
Error Bounds ◽  
Adjacency Matrix ◽  
Random Variable ◽  
Degree Sequences ◽  
Chernoff Inequality

We consider random graphs such that each edge is determined by an independent random variable, where the probability of each edge is not assumed to be equal. We use a Chernoff inequality for matrices to show that the eigenvalues of the adjacency matrix and the normalized Laplacian of such a random graph can be approximated by those of the weighted expectation graph, with error bounds dependent upon the minimum and maximum expected degrees. In particular, we use these results to bound the spectra of random graphs with given expected degree sequences, including random power law graphs. Moreover, we prove a similar result giving concentration of the spectrum of a matrix martingale on its expectation.

scholarly journals Large Cliques in a Power-Law Random Graph

2010 ◽  
Vol 47 (4) ◽  
pp. 1124-1135 ◽  
Author(s):  
Svante Janson ◽  
Tomasz Łuczak ◽  
Ilkka Norros
Keyword(s):  
Random Graph ◽  
Power Law ◽  
High Probability ◽  
Simple Algorithm ◽  
Heavy Tail ◽  
Degree Sequences ◽  
Constant Size ◽  
Tail Distribution ◽  
Law Degree ◽  
Large Clique

In this paper we study the size of the largest clique ω(G(n, α)) in a random graph G(n, α) on n vertices which has power-law degree distribution with exponent α. We show that, for ‘flat’ degree sequences with α > 2, with high probability, the largest clique in G(n, α) is of a constant size, while, for the heavy tail distribution, when 0 < α < 2, ω(G(n, α)) grows as a power of n. Moreover, we show that a natural simple algorithm with high probability finds in G(n, α) a large clique of size (1 − o(1))ω(G(n, α)) in polynomial time.

scholarly journals Pairwise disjoint maximal cliques in random graphs and sequential motion planning on random right angled Artin groups

2019 ◽  
Vol 11 (02) ◽  
pp. 371-386
Author(s):  
Jesús González ◽  
Bárbara Gutiérrez ◽  
Hugo Mas
Keyword(s):  
Motion Planning ◽  
Random Graph ◽  
Random Graphs ◽  
Pairwise Disjoint ◽  
Random Variable ◽  
Clique Number ◽  
Planning Problem ◽  
Artin Group ◽  
Logarithmic Function ◽  
Right Angled Artin Groups

The clique number of a random graph in the Erdös–Rényi model [Formula: see text] yields a random variable which takes values asymptotically almost surely (as [Formula: see text]) within one of an explicit logarithmic function [Formula: see text]. We show that random graphs have, asymptotically almost surely, arbitrarily many pairwise disjoint cliques with [Formula: see text] vertices. Such a result is motivated by, and applied to, the multi-tasking version of Farber’s topological model to study the motion planning problem in robotics. Indeed, we study the behavior of all the higher topological complexities of Eilenberg–MacLane spaces of type [Formula: see text], where [Formula: see text] is a random right angled Artin group.

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>

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 Greed is Good for Deterministic Scale-Free Networks

Algorithmica ◽  
2020 ◽  
Vol 82 (11) ◽  
pp. 3338-3389
Author(s):  
Ankit Chauhan ◽  
Tobias Friedrich ◽  
Ralf Rothenberger
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Power Law ◽  
Real World ◽  
Degree Distribution ◽  
High Probability ◽  
Maximum Independent Set ◽  
Graph Models ◽  
Random Graph Models ◽  
Graph Classes

Abstract Large real-world networks typically follow a power-law degree distribution. To study such networks, numerous random graph models have been proposed. However, real-world networks are not drawn at random. Therefore, Brach et al. (27th symposium on discrete algorithms (SODA), pp 1306–1325, 2016) introduced two natural deterministic conditions: (1) a power-law upper bound on the degree distribution (PLB-U) and (2) power-law neighborhoods, that is, the degree distribution of neighbors of each vertex is also upper bounded by a power law (PLB-N). They showed that many real-world networks satisfy both properties and exploit them to design faster algorithms for a number of classical graph problems. We complement their work by showing that some well-studied random graph models exhibit both of the mentioned PLB properties. PLB-U and PLB-N hold with high probability for Chung–Lu Random Graphs and Geometric Inhomogeneous Random Graphs and almost surely for Hyperbolic Random Graphs. As a consequence, all results of Brach et al. also hold with high probability or almost surely for those random graph classes. In the second part we study three classical $$\textsf {NP}$$ NP -hard optimization problems on PLB networks. It is known that on general graphs with maximum degree $$\Delta$$ Δ , a greedy algorithm, which chooses nodes in the order of their degree, only achieves a $$\Omega (\ln \Delta )$$ Ω ( ln Δ ) -approximation for Minimum Vertex Cover and Minimum Dominating Set, and a $$\Omega (\Delta )$$ Ω ( Δ ) -approximation for Maximum Independent Set. We prove that the PLB-U property with $$\beta >2$$ β > 2 suffices for the greedy approach to achieve a constant-factor approximation for all three problems. We also show that these problems are -hard even if PLB-U, PLB-N, and an additional power-law lower bound on the degree distribution hold. Hence, a PTAS cannot be expected unless = . Furthermore, we prove that all three problems are in if the PLB-U property holds.

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 Large Cliques in a Power-Law Random Graph

2010 ◽  
Vol 47 (04) ◽  
pp. 1124-1135 ◽  
Author(s):  
Svante Janson ◽  
Tomasz Łuczak ◽  
Ilkka Norros
Keyword(s):  
Random Graph ◽  
Power Law ◽  
High Probability ◽  
Simple Algorithm ◽  
Heavy Tail ◽  
Degree Sequences ◽  
Constant Size ◽  
Tail Distribution ◽  
Law Degree ◽  
Large Clique

In this paper we study the size of the largest clique ω(G(n, α)) in a random graph G(n, α) on n vertices which has power-law degree distribution with exponent α. We show that, for ‘flat’ degree sequences with α &gt; 2, with high probability, the largest clique in G(n, α) is of a constant size, while, for the heavy tail distribution, when 0 &lt; α &lt; 2, ω(G(n, α)) grows as a power of n. Moreover, we show that a natural simple algorithm with high probability finds in G(n, α) a large clique of size (1 − o(1))ω(G(n, α)) in polynomial time.

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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