Investigating the synchronizability of complex networks
Abstract
In 1972 Robert May posed the fundamental question: "Will a Large Complex System be Stable?" [1] May focused on the stability of fixed points of complex systems such as large ecosystems with random connectivity and concluded that stability would inevitably be compromised as the number of the states of the system is increased. Here we are interested in a separate but related question "Will a Large Complex System Synchronize?" Synchronization is a fundamental phenomenon that arises in many technological and natural systems and can be seen as the hallmark of dynamical order in such systems. According to the application, synchronization may either be a desirable property (like in the case of emergent coordinated behavior in the animal world [2]) or an undesired property (like in the case of excessive synchronization in the brain which is associated with epilepsy [3].) However, it is not clear whether systems of increasing complexity will eventually lose their ability to synchronize. Here we are mostly concerned with the structure of such complex systems, i.e., the underlying network of connectivity between the individual components that form the system and whether this structure supports or not synchronization. This proposal investigates how the particular structure of a network and its complexity affect the amount of order displayed by the network dynamics. Structural indices such as the synchronizability and the reactivity are at the core of this relation. A well known measure for the ability of a network to achieve asymptotic stability is the socalled synchronizability [4], which describes the range of the coupling strength over which the synchronous solution is stable. A separate, but perhaps more important problem, is the robustness of the synchronous solution to finite perturbations, i.e., transient stability. However, there is no known measure that quantifies the ability of a network to synchronize against finite perturbations. Such a measure will be introduced as part of the proposed research. In particular, it is known that systems characterized by a non-normal Jacobian are prone to transient instabilities, which may steer their long term dynamics away from a fixed point, even when this is asymptotically stable [5]. For fixed points, transient stability can be measured by the reactivity, which is defined as the initial rate of growth of a perturbation about the fixed point [6]. Synthetic and real graphs will be compared in terms of their synchronizability and their reactivity.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Aug 11, 2023
- Source ID
- N000142312755
Entities
People
- Francesco Sorrentino
Organizations
- Office of Naval Research
- United States Navy
- University of New Mexico