Bence Rozner | Searcharxiv

arXiv2019

Barabási-Albert random graph with multiple type edges with perturbation

In this paper we introduce the perturbed version of the Barabási-Albert random graph with multiple type edges and prove the existence of the (generalized) asymptotic degree distribution. Similarly to the non-perturbed case, the asymptotic degree distribution depends on the almost sure limit of the proportion of edges of different types. However, if there is perturbation then the resulting degree distribution will be deterministic, which is a major difference compared to the non-perturbed case.

math.PR↗

arXiv2019

Asymptotic degree distribution in preferential attachment graph models with multiple type edges

We deal with a general preferential attachment graph model with multiple type edges. The types are chosen randomly, in a way that depends on the evolution of the graph. In the $N$-type case, we define the (generalized) degree of a given vertex as $\boldsymbol{d}=(d_{1},d_{2},\dots,d_{N})$, where $d_{k}\in\mathbb{Z}_{0}^{+}$ is the number of type $k$ edges connected to it. We prove the existence of an a.s.\ asymptotic degree distribution for a general family of preferential attachment random graph models with multi-type edges. More precisely, we show that the proportion of vertices with (generalized) degree $\boldsymbol{d}$ tends to some random variable as the number of steps goes to infinity. We also provide recurrence equations for the asymptotic degree distribution. Finally, we generalize the scale-free property of random graphs to the multi-type case.

math.PR↗