On connectivity,conductance and bootstrap percolation for a random k‐out,age‐biased graph

A uniform attachment graph (with parameter k), denoted G_n,k in the paper, is a random graph on the vertex set n], where each vertex v makes k selections from v ? 1] uniformly and independently, and these selections determine the edge set. We study several aspects of this graph. Our motivation comes from two similarly constructed, well‐studied random graphs: k‐out graphs and preferential attachment graphs. In this paper, we find the asymptotic distribution of its minimum degree and connectivity, and study the expansion properties of G_n,k to show that the conductance of G_n,k is of order . We also study the bootstrap percolation on G_n,k, where r infected neighbors infect a vertex, and show that if the probability of initial infection of a vertex is negligible compared to then with high probability (whp) the disease will not spread to the whole vertex set, and if this probability exceeds by a sub‐logarithmical factor then the disease whp will spread to the whole vertex set.