We investigate the degree sequence of the geometric preferential attachment model of Flaxman, Vera (2006), (2007) in the case where the self-loop parameter α is set to 0. We show that, given certain conditions on the attractiveness function F , the degree sequence converges to the same sequence as found for standard preferential attachment in Bollobás et al. (2001). We also apply our method to the extended model introduced in van der Esker (2008) which allows for an initial attractiveness term, proving similar results.

We investigate the degree sequence of the geometric preferential attachment model of Flaxman, Frieze and Vera (2006), (2007) in the case where the self-loop parameter α is set to 0. We show that, given certain conditions on the attractiveness function F, the degree sequence converges to the same sequence as found for standard preferential attachment in Bollobás et al. (2001). We also apply our method to the extended model introduced in van der Esker (2008) which allows for an initial attractiveness term, proving similar results.

ABSTRACT:We investigate the degree sequences of scale-free random graphs. We obtain a formula for the limiting proportion of vertices with degree d, confirming non-rigorous arguments of Dorogovtsev, Mendes, and Samukhin (Phys Rev Lett 85 (2000), 4633). We also consider a generalization of the model with more randomization, proving similar results. Finally, we use our results on the degree sequence to show that for certain values of parameters localized eigenfunctions of the adjacency matrix can be found.

We consider the connected component of the partial duplication model for a random graph, a model which was introduced by Bhan, Galas and Dewey as a model for gene expression networks. The most rigorous results are due to Hermann and Pfaffelhuber, who show a phase transition between a subcritical case where in the limit almost all vertices are isolated and a supercritical case where the proportion of the vertices which are connected is bounded away from zero.We study the connected component in the subcritical case, and show that, when the duplication parameter p < e −1 , the degree distribution of the connected component has a limit, which we can describe in terms of the stationary distribution of a certain Markov chain and which follows an approximately power law tail, with the power law index predicted by Ispolatov, Krapivsky and Yuryev. Our methods involve analysing the quasi-stationary distribution of a certain continuous time Markov chain associated with the evolution of the graph.

We investigate the degree sequences of geometric preferential attachment graphs in general compact metric spaces. We show that, under certain conditions on the attractiveness function, the behaviour of the degree sequence is similar to that of the preferential attachment with multiplicative fitness models investigated by Borgs et al. When the metric space is finite, the degree distribution at each point of the space converges to a degree distribution which is an asymptotic power law whose index depends on the chosen point. For infinite metric spaces, we can show that for vertices in a Borel subset of S of positive measure the degree distribution converges to a distribution whose tail is close to that of a power law whose index again depends on the set.