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A preferential attachment model with random initial degrees Maria Deijfen
, 2007
"... In this paper, a random graph process {G(t)}t≥1 is studied and its degree sequence is analyzed. Let {Wt}t≥1 be an i.i.d. sequence. The graph process is defined so that, at each integer time t, a new vertex with Wt edges attached to it, is added to the graph. The new edges added at time t are then pr ..."
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In this paper, a random graph process {G(t)}t≥1 is studied and its degree sequence is analyzed. Let {Wt}t≥1 be an i.i.d. sequence. The graph process is defined so that, at each integer time t, a new vertex with Wt edges attached to it, is added to the graph. The new edges added at time t are then preferentially connected to older vertices, i.e., conditionally on G(t−1), the probability that a given edge of vertex t is connected to vertex i is proportional to di(t − 1) + δ, where di(t − 1) is the degree of vertex i at time t − 1, independently of the other edges. The main result is that the asymptotical degree sequence for this process is a power law with exponent τ = min{τW, τP}, where τW is the powerlaw exponent of the initial degrees {Wt}t≥1 and τP the exponent predicted by pure preferential attachment. This result extends previous work by Cooper and Frieze. 1
A stochastic model for competing growth
 on R d . Markov Processes and Related
, 2003
"... A stochastic model, describing the growth of two competing infections on R d, is introduced. The growth is driven by outbursts in the infected region, an outburst in the type 1 (2) infected region transmitting the type 1 (2) infection to the previously uninfected parts of a ball with stochastic radi ..."
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Cited by 8 (3 self)
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also extend a shape theorem of Deijfen for the corresponding model with just one type of infection.
A STOCHASTIC MODEL FOR COMPETING GROWTH ON Rd
, 2003
"... A stochastic model, describing the growth of two competing infections on Rd, is introduced. The growth is driven by outbursts in the infected region, an outburst in the type 1 (2) infected region transmitting the type 1 (2) infection to the previously uninfected parts of a ball with stochastic radiu ..."
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Cited by 1 (0 self)
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also extend a shape theorem of Deijfen for the corresponding model with just one type of infection.
Bipartite Stable Poisson graphs on R
, 2012
"... Let red and blue points be distributed on R according to two independent Poisson processes R and B and let each red (blue) point independently be equipped with a random number of halfedges according to a probability distribution ν (µ). We consider translationinvariant bipartite random graphs with ..."
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. The twocolor model is hence qualitatively different from the onecolor model, where Deijfen, Holroyd and Peres have given strong evidence that there is an infinite component. We also present simulation results for other degree distributions.
Epidemics on random graphs with tunable clustering
, 2007
"... In this paper, a branching process approximation for the spread of a ReedFrost 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. It is investigated how these quant ..."
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Cited by 32 (3 self)
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In this paper, a branching process approximation for the spread of a ReedFrost 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. It is investigated how these quantities varies with the clustering in the graph and it turns out for instance that, as the clustering increases, the epidemic threshold decreases. The network is modelled 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 they share at least one group.
Random intersection graphs with tunable degree distribution and clustering
, 2008
"... 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 paper a model is developed in which each vertex is given a random weight, and vertices with ..."
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Cited by 29 (2 self)
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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 paper 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 so 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.
Coexistence in a twotype continuum growth model
, 2004
"... We consider a stochastic model, describing the growth of two competing infections on R d. The growth takes place by way of spherical outbursts in the infected region, an outburst in the type 1 (2) infected region causing all previously uninfected points within a stochastic distance from the outburst ..."
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Cited by 3 (1 self)
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We consider a stochastic model, describing the growth of two competing infections on R d. The growth takes place by way of spherical outbursts in the infected region, an outburst in the type 1 (2) infected region causing all previously uninfected points within a stochastic distance from the outburst location to be type 1 (2) infected. The main result is that, if the infection types have the same intensity, then there is a strictly positive probability that both infection types grow unboundedly. Keywords: Richardson’s model, competing growth, coexistence.
Results 1  10
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37