Frequency Response and Transfer Functions of Large Self-Similar Networks

Large-scale dynamical systems, no matter whether possessing interconnected appearances, are frequently modeled as networks. For instance, graphs, multi-agent systems, and materials' intricate behaviors are often treated as networked dynamical systems. However, only a few studies have approached the problem in the frequency domain, mostly due to the complexity of evaluating their frequency response. That gap is filled by this paper, which proposes algorithms computing a general class of self-similar networks' frequency response and transfer functions, no matter they are finite or infinite, damaged or undamaged. In addition, this paper shows that for infinite self-similar networks, even when they are damaged, fractional-order and irrational dynamics naturally come into sight. Most importantly, this paper illustrates that for a network under different operating conditions, its frequency response would form a set of neighboring plants, which sets the basis of applying robust control methods to dynamic networks.