scholarly journals Epidemics on Random Graphs with Tunable Clustering

2008 ◽  
Vol 45 (03) ◽  
pp. 743-756 ◽  
Author(s):  
Tom Britton ◽  
Maria Deijfen ◽  
Andreas N. Lagerås ◽  
Mathias Lindholm
Keyword(s):  
Random Graphs ◽  
Branching Process ◽  
Intersection Graph ◽  
Epidemic Threshold ◽  
Random Intersection Graph ◽  
Large Outbreak ◽  
Process Approximation

In this paper a branching process approximation for the spread of a Reed-Frost epidemic on a network with tunable clustering is derived. The approximation gives rise to expressions for the epidemic threshold and the probability of a large outbreak in the epidemic. We investigate how these quantities vary with the clustering in the graph and find that, as the clustering increases, the epidemic threshold decreases. The network is modeled by a random intersection graph, in which individuals are independently members of a number of groups and two individuals are linked to each other if and only if there is at least one group that they are both members of.

scholarly journals Epidemics on Random Graphs with Tunable Clustering

2008 ◽  
Vol 45 (3) ◽  
pp. 743-756 ◽  
Author(s):  
Tom Britton ◽  
Maria Deijfen ◽  
Andreas N. Lagerås ◽  
Mathias Lindholm
Keyword(s):  
Random Graphs ◽  
Branching Process ◽  
Intersection Graph ◽  
Epidemic Threshold ◽  
Random Intersection Graph ◽  
Large Outbreak ◽  
Process Approximation

In this paper a branching process approximation for the spread of a Reed-Frost epidemic on a network with tunable clustering is derived. The approximation gives rise to expressions for the epidemic threshold and the probability of a large outbreak in the epidemic. We investigate how these quantities vary with the clustering in the graph and find that, as the clustering increases, the epidemic threshold decreases. The network is modeled by a random intersection graph, in which individuals are independently members of a number of groups and two individuals are linked to each other if and only if there is at least one group that they are both members of.

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 Size of the Largest Component in a Critical Graph

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Bin Wang ◽  
Lang Zou
Keyword(s):  
Branching Process ◽  
Intersection Graph ◽  
Martingale Method ◽  
Critical Graph ◽  
Random Intersection Graph ◽  
Scaling Window

In this paper, by the branching process and the martingale method, we prove that the size of the largest component in the critical random intersection graph Gn,n5/3,p is asymptotically of order n2/3 and the width of scaling window is n−1/3.

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.

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 The Size of the Giant Joint Component in a Binomial Random Double Graph

10.37236/8846 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Mark Jerrum ◽  
Tamás Makai
Keyword(s):  
Phase Transition ◽  
Critical Point ◽  
Random Graphs ◽  
Spanning Tree ◽  
Branching Process ◽  
Graph Model ◽  
First Order ◽  
Blue Edge ◽  
Critical Pairs

We study the joint components in a random 'double graph' that is obtained by superposing red and blue binomial random graphs on $n$~vertices.  A joint component is a maximal set of vertices that supports both a red and a blue spanning tree.  We show that there are critical pairs of red and blue edge densities at which a giant joint component appears.  In contrast to the standard binomial graph model, the phase transition is first order:  the size of the largest joint component jumps from $O(1)$ vertices to $\Theta(n)$ at the critical point.  We connect this phenomenon to the properties of a certain bicoloured branching process. 

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 asymptotic final size distribution of multitype chain-binomial epidemic processes

1999 ◽  
Vol 31 (01) ◽  
pp. 220-234 ◽  
Author(s):  
Mikael Andersson
Keyword(s):  
Branching Process ◽  
Counting Process ◽  
Contact Structure ◽  
Finite Population ◽  
Convergence Result ◽  
System Of Equations ◽  
Final Size ◽  
Linear System Of Equations ◽  
Non Linear System ◽  
Process Approximation

A multitype chain-binomial epidemic process is defined for a closed finite population by sampling a simple multidimensional counting process at certain points. The final size of the epidemic is then characterized, given the counting process, as the smallest root of a non-linear system of equations. By letting the population grow, this characterization is used, in combination with a branching process approximation and a weak convergence result for the counting process, to derive the asymptotic distribution of the final size. This is done for processes with an irreducible contact structure both when the initial infection increases at the same rate as the population and when it stays fixed.

scholarly journals An epidemic model with exposure-dependent severities

2005 ◽  
Vol 42 (04) ◽  
pp. 932-949 ◽  
Author(s):  
Frank Ball ◽  
Tom Britton
Keyword(s):  
Branching Process ◽  
Large Population ◽  
Final Outcome ◽  
Model Parameters ◽  
Final Size ◽  
Strong Law ◽  
Sir Epidemic ◽  
Major Outbreak ◽  
Infectious State ◽  
Process Approximation

We consider a stochastic model for the spread of a susceptible–infective–removed (SIR) epidemic among a closed, finite population, in which there are two types of severity of infectious individuals, namely mild and severe. The type of severity depends on the amount of infectious exposure an individual receives, in that infectives are always initially mild but may become severe if additionally exposed. Large-population properties of the model are derived. In particular, a coupling argument is used to provide a rigorous branching process approximation to the early stages of an epidemic, and an embedding argument is used to derive a strong law and an associated central limit theorem for the final outcome of an epidemic in the event of a major outbreak. The basic reproduction number, which determines whether or not a major outbreak can occur given few initial infectives, depends only on parameters of the mild infectious state, whereas the final outcome in the event of a major outbreak depends also on parameters of the severe state. Moreover, the limiting final size proportions need not even be continuous in the model parameters.

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