Network coherence in fractal graphs

We study distributed consensus algorithms in fractal networks where agents are subject to external disturbances. We characterize the coherence of these networks in terms of an H₂ norm of the system that captures how closely agents track the consensus value. We show that, in first-order systems, the coherence measure is closely related to the global mean first passage time of a simple random walk. We can therefore draw directly from the literature on random walks in fractal graphs to derive asymptotic expressions for the coherence in terms of the network size and dimension. We then show how techniques employed in the random walks setting can be extended to analyze the coherence of second-order consensus algorithms in fractal graphs with tree-like structures, and we present asymptotic results for these second-order systems.