Age, Innovations and Time Operator of Networks

•Extension of the Time Operator theory with network evolution.•Innovations and Age of Markov Networks.•A new distribution emerging from the innovations of the Barabási–Albert network evolution model.•Age of Markov and Barabási–Albert Networks expressed in terms of Tsallis Entropy.

We extend the Time Operator and Age to Network Evolution models. Internal Age formulas and the distribution of innovations are computed for Erdős–Rényi Random Networks, for Markov Networks and Barabási–Albert preferential Attachment Networks. The innovation probabilities are found to be proportional to the quadratic entropy (which coincides with the Tsallis entropy for entropic index q=2q=2) in all Markov networks, as well as in the linear growth mechanism. The distribution of innovations in the Barabási–Albert model is a new probability distribution of the logarithmic type.