*Project Description*

Synchronization on networks and along with related phenomena such as consensus dynamics and partial synchronization, is ubiquitous in nature. Disorders of the human brain, flocking of birds, political consensus, schooling of fish, distribution of electricity in the power grid, and traffic flow over road networks, among many others, are examples of applications of the study of synchronization. Generally we are interested in the stability of the synchronous state, i.e. we want to know how likely a perturbation away from a synchronous state is to lead to desynchronization. A major challenge which arises in this study is that this stability is related to the network structure, and in many cases different networks lead to different stability landscapes. For states which are very close to the synchronous state (linear stability), the theoretical advances have allowed for a complete understanding of the stability, that is given a network structure one can immediately determine whether or not the system has a linearly stable synchronous state. However for finite sized dynamical perturbations away from the synchronous state the theory breaks down. Recently a method for statistically analyzing finite sized perturbations, known as basin stability has gained popularity. However basin stability is by its nature very computationally expensive, as it requires sampling a large number of states.

Examples of several related projects (any of which the prospective student could choose from) which could lead to a better understanding of synchronization in the context of basin stability:

(1) The development and release of a software package (preferably in python, though this is not a requirement) which given a network structure, dynamics, region of initial conditions and coupling function, allows for the fast computation of basin stability in a distributed context.

(2) Though closely related to (1), study of the basin stability of partial synchronous states is significantly more challenging because unlike complete synchronization where there is a single synchronous state, there are numerous potential partially synchronous state. In this project we would develop and release a software package which can quickly estimate the basin stability around a particular partial synchronous state.

(3) Closely related to (2), it has been recently observed that the basin structure of partially synchronous states is very rich, and apparently in many cases leads to a fractal structure. In this project we would develop software to rapidly examine this structure given various potential planes to view the basin from (as a note the basin is generally very high dimensional, so it is easiest to choose a plane from which to view the basin structure).

(4) While the stability of the partially synchronous state has been analyzed in particular contexts (i.e. external equitable partitions), most network structures do not contain such partitions. In this project we would develop software to rapidly create random network structures and explore (a) whether or not they have external equitable partitions and (b) if there are any nearby network structures which do contain such partitions.

(5) Generally the stability of a given synchronous (or partially synchronous) state is intimately related to the graph (or network) Laplacian. In this project we would develop software for rapid and accurate sampling of the eigenvalues of the graph Laplacian of random networks and study the structure (potentially with the methods which have been recently developed for this study such as those for Bohemian matrices or algebraic starscapes).

(6) Any related project the prospective student might be interested in so long as it relates to both network dynamics and distributed computing!