Controllability of fractal networks: An analytical approach

¹ School of Systems Science, Beijing Normal University - Beijing, 10085, PRC
² School of Mathematics and Statistics, Minnan Normal University - Fujian, 363000, PRC

^(a) zyuan@mnnu.edu.cn

Received: 19 December 2013
Accepted: 21 February 2014

Abstract

Fractal networks are ubiquitous in nature, ranging from river networks to vascular networks. The ultimate goal of exploring these fractal networked systems lies in controlling the dynamical processes that take place on them. We offer analytical results to exactly understand our ability to control the dynamics of regular fractal networks in terms of identifying the minimum number of driver nodes that are required to achieve full control. According to the exact controllability theory, the controllability of an undirected network is completely determined by the eigenvalue spectrum of the coupling matrix that captures the network structure. The self-similarity in the fractal networks allows us to solve exactly the eigenvalue spectrum from the growth unit and the steps of the iterations, enabling an analytical quantification of the controllability of the fractal networks via the eigenvalue spectrum. We validate our exact analytical results in three typical regular fractal networks. Our results have implications for the control of many real networked systems that have fractal characteristics.