Statistics of Leaders in Index-Driven Networks

We study the statistical properties of leaders in growing networks with age. A leader of a network is defined as the node with the largest degree and the age of the node is trivially labeled by its index, i.e., the time it joins the network. As networks evolve with the addition of new nodes connecting to old ones with the possibility that is proportional to the index of the target, we investigate both the average number and index of leaders as well as the degree distribution of nodes. The average number of leaders first increases quickly with time and then saturates to a finite value and the average index of leaders increases algebraically with time. Both features result from the degree distribution with an exponential tail. Analytical calculations based on the rate equation are verified by numerical simulations.