Challenging the assumption of simple scaling in the observables of network growth

Identifying power law scaling in real networks - indicative of preferential attachment - has proved controversial. Critics argue that directly measuring the temporal evolution of a network is better than measuring the degree distribution when looking for preferential attachment. However, many of the established methods do not account for any potential time-dependence in the attachment kernels of growing networks, or assume that the node degree is the key observable determining node evolution. In this paper, we argue that these assumptions may lead to misleading conclusions about the future evolution, and past origin, of these networks. We illustrate the risks of these assumptions by introducing a simple adaptation of the Barabasi-Albert model, the ``k2 model'', where new nodes attach to nodes in the existing network proportionally to the number of nodes within distance two of the target node, the effect of which we demonstrate both numerically and analytically. This k2 model results in time dependent degree distributions and attachment kernels, despite initially appearing to grow as linear preferential attachment. We support these results by showing that similar effects are seen in real networks using the American Physical Society citation network. Our results suggest that significantly more care should go into the analysis of the evolution of growing networks, and that existing results regarding the evolution of growing networks may be false.