Multiple-scale theory of topology-driven patterns on directed networks

Abstract

Dynamical processes on networks are currently being considered in different domains of cross-disciplinary interest. Reaction-diffusion systems hosted on directed graphs are in particular relevant for their widespread applications, from computer networks to traffic systems. Due to the peculiar spectrum of the discrete Laplacian operator, homogeneous fixed points can turn unstable, on a directed support, because of the topology of the network, a phenomenon which cannot be induced on undirected graphs. A linear analysis can be performed to single out the conditions that underly the instability. The complete characterization of the patterns, which are eventually attained beyond the linear regime of exponential growth, calls instead for a full nonlinear treatment. By performing a multiple time scale perturbative calculation, we here derive an effective equation for the nonlinear evolution of the amplitude of the most unstable mode, close to the threshold of criticality. This is a Stuart-Landau equation the complex coefficients of which appear to depend on the topological features of the embedding directed graph. The theory proves adequate versus simulations, as confirmed by operating with a paradigmatic reaction-diffusion model.

Ginzburg-Landau approximation for self-sustained oscillators weakly coupled on complex directed graphs

“…In doing so, we will generalize the work of Nakao [20] to the interesting setting where asymmetry in the couplings needs to be accommodated for and, at the same time, reformulate the classical work of Kuramoto [18], on a discrete spatial backing. To anticipate our findings, and at variance with the analysis reported in [22], we will finally obtain a CGLE as a minimal description for the dynamics of the self-sustained oscillators coupled on a complex and asymmetric graph. The obtained CGLE enables one to analytically probe the stability of the synchronous uniform state, as displayed by the reaction-diffusion system.…”

“…As remarked in [21], directionality matters and can seed the emergence of non trivial collective dynamics which cannot manifest when the scrutinized system is made to evolve on a symmetric discrete support. Motivated by this finding, we considered in [22] the dynamics of a reaction-diffusion system defined on a directed graph which displays a stable fixed point and obtained an effective description for the evolution mode triggered unstable, just above the threshold of criticality. The analysis exploits a multiple time-scale analysis and eventually yields a Stuart-Landau for the amplitude of the unstable mode, whose complex coefficients reflect the topology of the network, the factual drive to the instability.…”

“…In this paper we aim at following the similar strategy of [22] to derive an approximate equation for the evolution of a reaction diffusion system on a directed (and balanced) graph, in the vicinity of a supercritical Hopf bifurcation. As a matter of fact we will assume weak the strength of the coupling that links adjacent nodes.…”

“…Panel (a): eigenvalues of the Laplacian for a balanced Newman-Watts[26] network generated with p = 0.27 and N = 100 nodes. The algorithm to built the network is detailed in[22]. The shaded area marks the instability region for the Brusselator model, as obtained under the CGL normal form representation.…”

Networks of planar Hamiltonian systems

“…For now we simply remark that homogeneous states in networks of planar Hamiltonian systems remain homogeneous following perturbation provided that D x ≈ D y , but may evolve towards a stable heterogeneous state when diffusion coefficients satisfy (15). Several authors have recently studied the effect of discrete network topology on pattern morphology [9,26,27].…”