## A4

Project A4: Modeling internal fluctuations in the description of sparse networks of dynamical systems

Research Team: T. Pereira (USP), A. Pikovsky (UP), R. Toenjes (UP)

Outline: We want to analyze the origin and the effect of internal, dynamically generated noise due to a small, finite number of neighbors in sparse, random networks of coupled dynamical systems and study how such noise can be incorporated in a low dimensional description as an appropriately defined thermodynamical limit. When the nodes of a network interact weakly with only a few local neighbors, the effect of the network coupling is often described by a deterministic mean value with some additional fluctuations that depend on the number of neighbors. We aim to understand the precise nature of such internal, dynamic fluctuations and their effect on the dynamics as a whole. We have recently found that the transition to synchronization in sparse random networks of phase oscillators is of first order. As similar hysteretic transition is observed in dynamics on power grids, where it has become known as explosive synchronization transition. This is qualitatively and quantitatively very different from the synchronization transition of noisy oscillators coupled to a mean field. How to incorporate the internal fluctuations correctly into a mean field theory remains a challenging open problem which we want to study in this project.

Research in the German group (PhD Supervisor: A. Pikovsky, R. Toenjes): (i) The nature of dynamical fluctuations: In a first part we will study local fluctuations in feed forward networks on layers or directed trees. The nodes in one layer are used as input for the dynamics of nodes in the next layer. We want to observe, the convergence of characteristic quantities of the fluctuations as a function of the layer depth as well as the correlation length near critical transitions. (ii) The role of spatial distance in networks: Non-trivial effects can be expected if the strength of the local fluctuations depends on the order parameters of the system or, if the noise excites global dynamical features that would otherwise be transient. In this second part we will explicitly study if noise induced, spatial effects, such as correlation length and dynamical differentiation in sparse, homogenous networks, need to be considered in a mean field theory for dynamics on sparse networks.

Research in the Brazilian group (PhD Supervisor: T. Pereira): (iii) Analysis of fluctuations in infinite tree networks: Complementary to the analysis in the German group, we will focus on exactly solvable deterministic or stochastic models which can display a phase transition on infinite tree lattices, depending on a system parameter or the network topology (e.g. degree distribution) and verify the predictions in numerical simulations of these processes in large uncorrelated random networks. (iv) Langevin and mean field description: One of the major open problems in the field is to describe the nature of the fluctuations from first principles. In a recent breakthrough, we were able to develop techniques to understand how the fluctuations depend on the network structure parameters. Combining the research of the German group with these results, we will extend mean field theory for sparse random networks to correctly incorporate fluctuations from a finite number of neighbors and to predict dynamic phase transitions quantitatively.