Core Size and Densification in Preferential Attachment Networks

Abstract

Consider a preferential attachment model for network evolution that allows both node and edge arrival events: at time t, with probability $p_t$ a new node arrives and a new edge is added between the new node and an existing node, and with probability $1-p_t$ 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 $\delta \in (0,1)$, the $\delta$ -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 $\delta$ fraction of the number of edges in the graph at time t. We show that for the common model where $p_t$ is constant, i.e., when $p_t=p$ for every t and the network is sparse with linear number of edges, the size of the $\delta$-founders set is concentrated around $\delta ^{2/p} n_t$, and thus is linear in $n_t$, the number of nodes at time t. In contrast, we show that for $p_t=\min \{1,\frac{2a}{\ln t}\}$ and when the network is dense with super-linear number of edges, the size of the $\delta$-founders set is sub-linear in $n_t$ and concentrated around $\tilde{\Theta }((n_t)^{\eta })$, where $\eta =\delta ^{1/a}$.

Supported in part by the Israel Science Foundation (grant 1549/13).