scholarly journals Contagions in random networks with overlapping communities

2015 ◽  
Vol 47 (4) ◽  
pp. 973-988 ◽  
Author(s):  
Emilie Coupechoux ◽  
Marc Lelarge
Keyword(s):  
Phase Transition ◽  
Random Graph ◽  
Epidemic Model ◽  
Branching Process ◽  
Branching Processes ◽  
Graph Model ◽  
Single Individual ◽  
Overlapping Communities ◽  
Random Graph Model ◽  
The Mean

We consider a threshold epidemic model on a clustered random graph model obtained from local transformations in an alternating branching process that approximates a bipartite graph. In other words, our epidemic model is such that an individual becomes infected as soon as the proportion of his/her infected neighbors exceeds the threshold q of the epidemic. In our random graph model, each individual can belong to several communities. The distributions for the community sizes and the number of communities an individual belongs to are arbitrary. We consider the case where the epidemic starts from a single individual, and we prove a phase transition (when the parameter q of the model varies) for the appearance of a cascade, i.e. when the epidemic can be propagated to an infinite part of the population. More precisely, we show that our epidemic is entirely described by a multi-type (and alternating) branching process, and then we apply Sevastyanov's theorem about the phase transition of multi-type Galton-Watson branching processes. In addition, we compute the entries of the mean progeny matrix corresponding to the epidemic. The phase transition for the contagion is given in terms of the largest eigenvalue of this matrix.

scholarly journals Contagions in random networks with overlapping communities

2015 ◽  
Vol 47 (04) ◽  
pp. 973-988 ◽  
Author(s):  
Emilie Coupechoux ◽  
Marc Lelarge
Keyword(s):  
Phase Transition ◽  
Random Graph ◽  
Epidemic Model ◽  
Branching Process ◽  
Branching Processes ◽  
Graph Model ◽  
Single Individual ◽  
Overlapping Communities ◽  
Random Graph Model ◽  
The Mean

We consider a threshold epidemic model on a clustered random graph model obtained from local transformations in an alternating branching process that approximates a bipartite graph. In other words, our epidemic model is such that an individual becomes infected as soon as the proportion of his/her infected neighbors exceeds the threshold q of the epidemic. In our random graph model, each individual can belong to several communities. The distributions for the community sizes and the number of communities an individual belongs to are arbitrary. We consider the case where the epidemic starts from a single individual, and we prove a phase transition (when the parameter q of the model varies) for the appearance of a cascade, i.e. when the epidemic can be propagated to an infinite part of the population. More precisely, we show that our epidemic is entirely described by a multi-type (and alternating) branching process, and then we apply Sevastyanov's theorem about the phase transition of multi-type Galton-Watson branching processes. In addition, we compute the entries of the mean progeny matrix corresponding to the epidemic. The phase transition for the contagion is given in terms of the largest eigenvalue of this matrix.

scholarly journals Rigorous Result for the CHKNS Random Graph Model

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Rick Durrett
Keyword(s):  
Phase Transition ◽  
Random Graph ◽  
Cluster Size ◽  
Graph Model ◽  
Physical Review ◽  
Critical Value ◽  
Polynomial Decay ◽  
Rigorous Result ◽  
Random Graph Model ◽  
International Audience

International audience We study the phase transition in a random graph in which vertices and edges are added at constant rates. Two recent papers in Physical Review E by Callaway, Hopcroft, Kleinberg, Newman, and Strogatz, and Dorogovstev, Mendes, and Samukhin have computed the critical value of this model, shown that the fraction of vertices in finite clusters is infinitely differentiable at the critical value, and that in the subcritical phase the cluster size distribution has a polynomial decay rate with a continuously varying power. Here we sketch rigorous proofs for the first and third results and a new estimates about connectivity probabilities at the critical value.

scholarly journals On the phase transition curve in a directed exponential random graph model

2018 ◽  
Vol 50 (01) ◽  
pp. 272-301 ◽  
Author(s):  
David Aristoff ◽  
Lingjiong Zhu
Keyword(s):  
Phase Transition ◽  
Free Energy ◽  
Random Graph ◽  
Graph Model ◽  
Transition Curve ◽  
Exponential Random Graph Models ◽  
Normalization Constant ◽  
Unusual Behavior ◽  
Random Graph Model ◽  
Exponential Random Graph

Abstract We consider a family of directed exponential random graph models parametrized by edges and outward stars. Much of the important statistical content of such models is given by the normalization constant of the models, and, in particular, an appropriately scaled limit of the normalization, which is called the free energy. We derive precise asymptotics for the normalization constant for finite graphs. We use this to derive a formula for the free energy. The limit is analytic everywhere except along a curve corresponding to a first-order phase transition. We examine unusual behavior of the model along the phase transition curve.

scholarly journals Estimation in branching processes with restricted observations

2006 ◽  
Vol 38 (4) ◽  
pp. 1098-1115 ◽  
Author(s):  
Ronald Meester ◽  
Pieter Trapman
Keyword(s):  
Discrete Time ◽  
Epidemic Model ◽  
Branching Process ◽  
Branching Processes ◽  
Infected Individual ◽  
Classical Swine Fever ◽  
Lattice Parameters ◽  
Time Interval ◽  
The Mean ◽  
Watson Process

We consider an epidemic model where the spread of the epidemic can be described by a discrete-time Galton-Watson branching process. Between times n and n + 1, any infected individual is detected with unknown probability π and the numbers of these detected individuals are the only observations we have. Detected individuals produce a reduced number of offspring in the time interval of detection, and no offspring at all thereafter. If only the generation sizes of a Galton-Watson process are observed, it is known that one can only estimate the first two moments of the offspring distribution consistently on the explosion set of the process (and, apart from some lattice parameters, no parameters that are not determined by those moments). Somewhat surprisingly, in our context, where we observe a binomially distributed subset of each generation, we are able to estimate three functions of the parameters consistently. In concrete situations, this often enables us to estimate π consistently, as well as the mean number of offspring. We apply the estimators to data for a real epidemic of classical swine fever.

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.

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. 

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