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We introduce a model of a preferential attachment based random graph which extends the family of models in which condensation phenomena can occur. Each vertex has an associated uniform random variable which we call its location. Our model evolves in discrete time by selecting $r$ vertices from the graph with replacement, with probabilities proportional to their degrees plus a constant $alpha$. A new vertex joins the network and attaches to one of these vertices according to a given probability associated to the ranking of their locations. We give conditions for the occurrence of condensation, showing the existence of phase transitions in $alpha$ below which condensation occurs. The condensation in our model differs from that in preferential attachment models with fitness in that the condensation can occur at a random location, that it can be due to a persistent hub, and that there can be more than one point of condensation.
We consider the degree distributions of preferential attachment random graph models with choice similar to those considered in recent work by Malyshkin and Paquette and Krapivsky and Redner. In these models a new vertex chooses $r$ vertices according
We introduce a new model of preferential attachment with fitness, and establish a time reversed duality between the model and a system of branching-coalescing particles. Using this duality, we give a clear and concise explanation for the condensation
In this paper, a random graph process ${G(t)}_{tgeq 1}$ is studied and its degree sequence is analyzed. Let $(W_t)_{tgeq 1}$ be an i.i.d. sequence. The graph process is defined so that, at each integer time $t$, a new vertex, with $W_t$ edges attache
We consider an evolving preferential attachment random graph model where at discrete times a new node is attached to an old node, selected with probability proportional to a superlinear function of its degree. For such schemes, it is known that the g
Preferential attachment models form a popular class of growing networks, where incoming vertices are preferably connected to vertices with high degree. We consider a variant of this process, where vertices are equipped with a random initial fitness r