ON THE DENSE PREFERENTIAL ATTACHMENT GRAPH MODELS AND THEIR GRAPHON INDUCED COUNTERPART

Letting $\mathcal{M}$ denote the space of finite measures on $\mathbb{N}$, and $\mu_\lambda\in\mathcal{M}$ denote the Poisson distribution with parameter $\lambda$, the function $W:[0,1]^2\to\mathcal{M}$ given by $W(x,y)=\mu_{c\log x\log y}$ is called the PAG graphon with density $c$. It is known that this is the limit, in the multigraph homomorphism sense, of the dense Preferential Attachment Graph (PAG) model with edge density $c$. This graphon can then in turn be used to generate the so-called W-random graphs in a natural way, and similar constructions also work in the slightly more general context of the so-called $\mathrm{PAG}_{\kappa}$ models.\\ The aim of this paper is to compare these dense $\mathrm{PAG}_{\kappa}$ models with the W-random graph models obtained from the corresponding graphons. Motivated by the multigraph limit theory, we investigate the expected jumble norm distance of the two models in terms of the number of vertices $n$. We present a coupling for which the expectation can be bounded from above by $O(\log^{3/2} n\cdot n^{-1/2})$, and provide a universal lower bound that is coupling-independent, but without the logarithmic term

Preferential attachment processes approaching the Rado multigraph

We consider a preferential attachment process in which a multigraph is built one node at a time. The number of edges added at stage t, emanating from the new node, is given by some prescribed function f(t), generalising a model considered by Kleinberg and Kleinberg in 2005 where f was presumed constant. We show that if f(t) is asymptotically bounded above and below by linear functions in t, then with probability 1 the infinite limit of the process will be isomorphic to the Rado multigraph. This structure is the natural multigraph analogue of the Rado graph, which we introduce here