scholarly journals Degree Distribution of an Inhomogeneous Random Intersection Graph

10.37236/2786 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Mindaugas Bloznelis ◽  
Julius Damarackas
Keyword(s):  
Degree Distribution ◽  
Intersection Graph ◽  
Random Intersection Graph

We show the asymptotic degree distribution of the typical vertex of a sparse inhomogeneous random intersection graph.

scholarly journals RANDOM INTERSECTION GRAPHS WITH TUNABLE DEGREE DISTRIBUTION AND CLUSTERING

2009 ◽  
Vol 23 (4) ◽  
pp. 661-674 ◽  
Author(s):  
Maria Deijfen ◽  
Willemien Kets
Keyword(s):  
Power Law ◽  
Weight Distribution ◽  
Degree Distribution ◽  
Asymptotic Expression ◽  
Intersection Graph ◽  
Intersection Graphs ◽  
Random Intersection Graph ◽  
Random Weight

A random intersection graph is constructed by assigning independently to each vertex a subset of a given set and drawing an edge between two vertices if and only if their respective subsets intersect. In this article a model is developed in which each vertex is given a random weight and vertices with larger weights are more likely to be assigned large subsets. The distribution of the degree of a given vertex is characterized and is shown to depend on the weight of the vertex. In particular, if the weight distribution is a power law, the degree distribution will be as well. Furthermore, an asymptotic expression for the clustering in the graph is derived. By tuning the parameters of the model, it is possible to generate a graph with arbitrary clustering, expected degree, and—in the power-law case—tail exponent.

The Vertex Degree Distribution of Passive Random Intersection Graph Models

2008 ◽  
Vol 17 (4) ◽  
pp. 549-558 ◽  
Author(s):  
JERZY JAWORSKI ◽  
DUDLEY STARK
Keyword(s):  
Degree Distribution ◽  
Graph Model ◽  
Fixed Number ◽  
Vertex Degree ◽  
Intersection Graph ◽  
Graph Models ◽  
Random Size ◽  
Random Intersection Graph

In a random passive intersection graph model the edges of the graph are decided by taking the union of a fixed number of cliques of random size. We give conditions for a random passive intersection graph model to have a limiting vertex degree distribution, in particular to have a Poisson limiting vertex degree distribution. We give related conditions which, in addition to implying a limiting vertex degree distribution, imply convergence of expectation.

scholarly journals Large Cliques in Sparse Random Intersection Graphs

10.37236/6233 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Mindaugas Bloznelis ◽  
Valentas Kurauskas
Keyword(s):  
Degree Distribution ◽  
Maximum Clique ◽  
Polynomial Time Algorithm ◽  
Clique Number ◽  
Time Algorithm ◽  
Intersection Graph ◽  
Optimal Order ◽  
Intersection Graphs ◽  
Random Intersection Graph ◽  
Finite Set

An intersection graph defines an adjacency relation between subsets $S_1,\dots, S_n$ of a finite set $W=\{w_1,\dots, w_m\}$: the subsets $S_i$ and $S_j$ are adjacent if they intersect. Assuming that the subsets are drawn independently at random according to the probability distribution $\mathbb{P}(S_i=A)=P(|A|){\binom{m}{|A|}}^{-1}$, $A\subseteq W$, where $P$ is a probability on $\{0, 1, \dots, m\}$, we obtain the random intersection graph $G=G(n,m,P)$. We establish  the asymptotic order of the clique number $\omega(G)$ of  a sparse random intersection graph as $n,m\to+\infty$. For $m = \Theta(n)$ we show that the maximum clique is of size $(1-\alpha/2)^{-\alpha/2}n^{1-\alpha/2}(\ln n)^{-\alpha/2}(1+o_P(1))$ in the case where the asymptotic degree distribution of $G$ is a power-law with exponent $\alpha \in (1,2)$. It is of size $\frac {\ln n} {\ln \ln n}(1+o_P(1))$ if the degree distribution has bounded variance, i.e., $\alpha>2$. We construct a simple polynomial-time algorithm which finds a clique of the optimal order $\omega(G) (1-o_P(1))$.

scholarly journals On the Isolated Vertices and Connectivity in Random Intersection Graphs

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Yilun Shang
Keyword(s):  
Random Graphs ◽  
Intersection Graph ◽  
Intersection Graphs ◽  
Random Intersection Graph ◽  
Probability Of Connectivity ◽  
Asymptotic Probability

We study isolated vertices and connectivity in the random intersection graph . A Poisson convergence for the number of isolated vertices is determined at the threshold for absence of isolated vertices, which is equivalent to the threshold for connectivity. When and , we give the asymptotic probability of connectivity at the threshold for connectivity. Analogous results are well known in Erdős-Rényi random graphs.

scholarly journals Component Evolution in Random Intersection Graphs

10.37236/935 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Michael Behrisch
Keyword(s):  
Real World ◽  
Linear Order ◽  
Graph Model ◽  
Vertex Degree ◽  
Intersection Graph ◽  
Giant Component ◽  
Logarithmic Order ◽  
Intersection Graphs ◽  
Similarity Network ◽  
Random Intersection Graph

We study the evolution of the order of the largest component in the random intersection graph model which reflects some clustering properties of real–world networks. We show that for appropriate choice of the parameters random intersection graphs differ from $G_{n,p}$ in that neither the so-called giant component, appearing when the expected vertex degree gets larger than one, has linear order nor is the second largest of logarithmic order. We also describe a test of our result on a protein similarity network.

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