Duccio Fanelli - Università di Firenze #
Deterministic and Stochastic  Pattern formation for reaction-diffusion  models on networks. #
The process of pattern formation for reaction-diffusion systems  
defined on complex networks is discussed. According to the  
deterministic picture, partial differential equations are assumed to  
govern the evolution of  the concentrations of the interacting species  
that populate the nodes of the network. A small perturbation of a  
homogeneous fixed point can spontaneously amplify  as follow a  
symmetry breaking instability and eventually yield to asymptotically  
stable non homogeneous patterns, the celebrated Turing patterns.  
Traveling waves can also manifest as a byproduct of the instability.  
Beyond the deterministic scenario, single individual effects, stemming  
from the intimate discreteness of the analyzed medium,  prove crucial  
by significantly modifying the mean-field predictions. The stochastic  
component of the microscopic dynamics can in particular induce the  
emergence of regular macroscopic patterns, in time and space, outside  
the region of deterministic instability.  To gain insight into the  
role of fluctuations and eventually work out the conditions for the  
emergence of stochastic patterns, one can operate under the linear  
noise approximations scheme adapted to network based applications, as  
I shall discuss. Furthermore, I will also consider reaction-diffusion  
models defined on a directed network. Due to the structure of the  
network Laplacian of the scrutinised system, the dispersion relation  
has both real and imaginary parts, at variance with the conventional  
case for a symmetric network. It is found that the homogeneous fixed  point can become unstable due to the topology of the network,  
resulting in a new class of instabilities which cannot be induced on  undirected graphs.