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.

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 Degree Distributions in General Random Intersection Graphs

10.37236/295 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Yilun Shang
Keyword(s):  
Graph Model ◽  
Intersection Graph ◽  
The Other ◽  
Intersection Graphs ◽  
Degree Of A Vertex ◽  
Random Weights ◽  
Random Intersection Graph

We study $G(n,m,F,H)$, a variant of the standard random intersection graph model in which random weights are assigned to both vertex types in the bipartite structure. Under certain assumptions on the distributions of these weights, the degree of a vertex is shown to depend on the weight of that particular vertex and on the distribution of the weights of the other vertex type.

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 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.

scholarly journals Isoperimetric Numbers of Randomly Perturbed Intersection Graphs

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 452
Author(s):  
Yilun Shang
Keyword(s):  
Social Networks ◽  
Social Interactions ◽  
Bipartite Graph ◽  
Scientific Collaboration ◽  
Graph Model ◽  
Intersection Graph ◽  
Collaboration Network ◽  
Intersection Graphs ◽  
Mild Conditions ◽  
Isoperimetric Number

Social networks describe social interactions between people, which are often modeled by intersection graphs. In this paper, we propose an intersection graph model that is induced by adding a sparse random bipartite graph to a given bipartite graph. Under some mild conditions, we show that the vertex–isoperimetric number and the edge–isoperimetric number of the randomly perturbed intersection graph on n vertices are Ω ( 1 / ln n ) asymptomatically almost surely. Numerical simulations for small graphs extracted from two real-world social networks, namely, the board interlocking network and the scientific collaboration network, were performed. It was revealed that the effect of increasing isoperimetric numbers (i.e., expansion properties) on randomly perturbed intersection graphs is presumably independent of the order of the network.

scholarly journals On hamiltonicity of uniform random intersection graphs

2010 ◽  
Vol 51 ◽  
Author(s):  
Mindaugas Bloznelis ◽  
Irmantas Radavičius
Keyword(s):  
Hamilton Cycle ◽  
Intersection Graph ◽  
Sufficient Condition ◽  
Intersection Graphs ◽  
Random Intersection Graph

We give a sufficient condition for the hamiltonicity of the uniform random intersection graph G{n,m,d}. It is a graph on n vertices, where each vertex is assigned d keys drawn independently at random from a given set of m keys, and where any two vertices establish an edge whenever they share at least one common key. We show that with probability tending to 1 the graph Gn,m,d has a Hamilton cycle provided that n = 2-1m(ln m + ln ln m + ω(m)) with some ω(m) → +∞ as m → ∞.

scholarly journals A Note on the Component Structure in Random Intersection Graphs with Tunable Clustering

10.37236/885 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Andreas N. Lagerås ◽  
Mathias Lindholm
Keyword(s):  
Phase Transition ◽  
Linear Order ◽  
Average Degree ◽  
Logarithmic Order ◽  
Large Component ◽  
Intersection Graphs ◽  
Component Structure

We study the component structure in random intersection graphs with tunable clustering, and show that the average degree works as a threshold for a phase transition for the size of the largest component. That is, if the expected degree is less than one, the size of the largest component is a.a.s. of logarithmic order, but if the average degree is greater than one, a.a.s. a single large component of linear order emerges, and the size of the second largest component is at most of logarithmic order.

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 The Largest Component in an Inhomogeneous Random Intersection Graph with Clustering

10.37236/382 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Mindaugas Bloznelis
Keyword(s):  
Probability Measure ◽  
Branching Process ◽  
Intersection Graph ◽  
Connected Component ◽  
Intersection Graphs ◽  
First Order ◽  
Random Intersection Graph ◽  
Vertex Set

Given integers $n$ and $m=\lfloor\beta n \rfloor$ and a probability measure $Q$ on $\{0, 1,\dots, m\}$, consider the random intersection graph on the vertex set $[n]=\{1,2,\dots, n\}$ where $i,j\in [n]$ are declared adjacent whenever $S(i)\cap S(j)\neq\emptyset$. Here $S(1),\dots, S(n)$ denote the iid random subsets of $[m]$ with the distribution $\bf{P}(S(i)=A)={{m}\choose{|A|}}^{-1}Q(|A|)$, $A\subset [m]$. For sparse random intersection graphs, we establish a first-order asymptotic as $n\to \infty$ for the order of the largest connected component $N_1=n(1-Q(0))\rho+o_P(n)$. Here $\rho$ is the average of nonextinction probabilities of a related multitype Poisson branching process.

The configuration model

2018 ◽  
Author(s):  
Mark Newman
Keyword(s):  
Degree Sequence ◽  
Graph Model ◽  
Network Models ◽  
Clustering Coefficient ◽  
Giant Component ◽  
Configuration Model ◽  
Bipartite Networks ◽  
Small Component ◽  
Random Graph Model ◽  
Average Size

A discussion of the most fundamental of network models, the configuration model, which is a random graph model of a network with a specified degree sequence. Following a definition of the model a number of basic properties are derived, including the probability of an edge, the expected number of multiedges, the excess degree distribution, the friendship paradox, and the clustering coefficient. This is followed by derivations of some more advanced properties including the condition for the existence of a giant component, the size of the giant component, the average size of a small component, and the expected diameter. Generating function methods for network models are also introduced and used to perform some more advanced calculations, such as the calculation of the distribution of the number of second neighbors of a node and the complete distribution of sizes of small components. The chapter ends with a brief discussion of extensions of the configuration model to directed networks, bipartite networks, networks with degree correlations, networks with high clustering, and networks with community structure, among other possibilities.

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