Dimension reduction for large-scale networked systems

A methodology is proposed to approximate large-scale networked systems by a lower dimensional networked system. We first group the nodes into m communities which will form the m vertices of the reduced network. We then associate an appropriate scalar dynamics to each community; in that way, the dimension of the new model is equal to m. The main idea is to approximate each node trajectory by the trajectory of its community. The edges are derived by considering some linear combinations of the link strengths between the elements of each community. Finally, the initial conditions are selected to guarantee the asymptotic consistency of the reduced model with the original system. Thus, we prove the asymptotic convergence of any state of the original system to its corresponding community state according to some distance. It has to be emphasized that our approach is flexible as the user is free to select the reduced system dimension m.