Improved degree bounds and full spectrum power laws in preferential attachment networks

Consider a random preferential attachment model G(p) for network evolution that allows both node and edge arrivals. Starting with an arbitrary nonempty graph G0, at each time step, there are two possible events: with probability p > 0 a new node arrives and a new edge is added between the new node and an existing node, and with probability 1 - p a new edge is added between two existing nodes. In both cases, the involved existing nodes are chosen at random according to preferential attachment, i.e., with probability proportional to their degree. G(p) is known to generate power law networks, i.e., the fraction of nodes with degree k is proportional to k^-β. Here β = (4 - p)/(2 - p) is in the range (2, 3]. Denoting the number of nodes of degree k at time t by m_{k t}, we significantly improve some long-standing results. In particular, we show that m_{k t} is concentrated around its mean with a deviation of O(t), which is independent of k. We also tightly bound the expectation E [m_kt] with an additive error of O(1/k), which is independent of t. These new bounds allow us to tightly estimate m_k,t for a considerably larger k values than before. This, in turn, enables us to estimate other important quantities, e.g., the size of the k-rich club, namely, the set of all nodes with a degree at least k. Finally, we introduce a new generalized model, G(pt,rt,qt), which extends G(p) by allowing also time-varying probabilities for node and edge arrivals, as well as the formation of new components. We show that the extended model can produce power law networks with any exponent β in the range (1,∞). Furthermore, the concentration bounds established for m_k,t in G(p) also apply in G(pt,rt,qt).