Clustering and Cliques in P.A random graphs with edge insertion

In this paper, we investigate the global clustering coefficient (a.k.a transitivity) and clique number of graphs generated by a preferential attachment random graph model with an additional feature of allowing edge connections between existing vertices. Specifically, at each time step $t$, either a new vertex is added with probability $f(t)$, or an edge is added between two existing vertices with probability $1-f(t)$. We establish concentration inequalities for the global clustering and clique number of the resulting graphs under the assumption that $f(t)$ is a regularly varying function at infinity with index of regular variation $-\gamma$, where $\gamma \in [0,1)$. We also demonstrate an inverse relation between these two statistics: the clique number is essentially the reciprocal of the global clustering coefficient.Comment: arXiv admin note: text overlap with arXiv:1902.1016

Critical Point and Percolation Probability in a Long Range Site Percolation Model on Zd\Z^d

Consider an independent site percolation model with parameter $p \in (0,1)$ on $\Z^d,\ d\geq 2$ where there are only nearest neighbor bonds and long range bonds of length $k$ parallel to each coordinate axis. We show that the percolation threshold of such model converges to $p_c(\Z^{2d})$ when $k$ goes to infinity, the percolation threshold for ordinary (nearest neighbour) percolation on $\Z^{2d}$. We also generalize this result for models whose long range bonds have several lengths.Comment: 5 pages; Acepted in Stochastic Processes and their Applications 201

Decay Properties of the Connectivity for Mixed Long Range Percolation Models on Zd\Z^d

In this short note we consider mixed short-long range independent bond percolation models on $\Z^{k+d}$. Let $p_{uv}$ be the probability that the edge $(u,v)$ will be open. Allowing a $x,y$-dependent length scale and using a multi-scale analysis due to Aizenman and Newman, we show that the long distance behavior of the connectivity $\tau_{xy}$ is governed by the probability $p_{xy}$. The result holds up to the critical point.Comment: 6 page