Core size and densification in preferential attachment networks

Consider a preferential attachment model for network evolution that allows both node and edge arrival events: at time t, with probability pt a new node arrives and a new edge is added between the new node and an existing node, and with probability 1 - pt a new edge is added between two existing nodes. In both cases existing nodes are chosen at random according to preferential attachment, i.e., with probability proportional to their degree. For ? ? (0, 1), the ?-founders of the network at time t is the minimal set of the first nodes to enter the network (i.e., founders) guaranteeing that the sum of degrees of nodes in the set is at least a ? fraction of the number of edges in the graph at time t. We show that for the common model where pt is constant, i.e., when pt = p for every t and the network is sparse with linear number of edges, the size of the ?-founders set is concentrated around ?2/pnt, and thus is linear in nt, the number of nodes at time t. In contrast, we show that for pt = min{1, (equation found)} and when the network is dense with super-linear number of edges, the size of the ?-founders set is sub-linear in nt and concentrated around (image found)((nt)?), where ? = ?1/a Springer-Verlag Berlin Heidelberg 2015.