TY - JOUR
T1 - The theory of pattern formation on directed networks
AU - Asllani, Malbor
AU - Challenger, Joseph D.
AU - Pavone, Francesco Saverio
AU - Sacconi, Leonardo
AU - Fanelli, Duccio
PY - 2014/7/31
Y1 - 2014/7/31
N2 - Dynamical processes on networks have generated widespread interest in recent years. The theory of pattern formation in reaction-diffusion systems defined on symmetric networks has often been investigated, due to its applications in a wide range of disciplines. Here we extend the theory to the case of directed networks, which are found in a number of different fields, such as neuroscience, computer networks and traffic systems. Owing to the structure of the network Laplacian, the dispersion relation has both real and imaginary parts, at variance with the case for a symmetric, undirected network. 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. Results from a linear stability analysis allow the instability region to be analytically traced. Numerical simulations show travelling waves, or quasi-stationary patterns, depending on the characteristics of the underlying graph.
AB - Dynamical processes on networks have generated widespread interest in recent years. The theory of pattern formation in reaction-diffusion systems defined on symmetric networks has often been investigated, due to its applications in a wide range of disciplines. Here we extend the theory to the case of directed networks, which are found in a number of different fields, such as neuroscience, computer networks and traffic systems. Owing to the structure of the network Laplacian, the dispersion relation has both real and imaginary parts, at variance with the case for a symmetric, undirected network. 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. Results from a linear stability analysis allow the instability region to be analytically traced. Numerical simulations show travelling waves, or quasi-stationary patterns, depending on the characteristics of the underlying graph.
UR - http://www.scopus.com/inward/record.url?scp=84905455107&partnerID=8YFLogxK
U2 - 10.1038/ncomms5517
DO - 10.1038/ncomms5517
M3 - Article
AN - SCOPUS:84905455107
SN - 2041-1723
VL - 5
JO - Nature Communications
JF - Nature Communications
M1 - 4517
ER -