The theory of pattern formation on directed networks

Abstract

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.

Self‐Organized Stationary Patterns in Networks of Bistable Chemical Reactions

“…[1,2] Similarly, recent advances in microfabrication allow the generation of engineered networks of reaction units, e.g., with labon-chip microdroplets, [3] electrode arrays, [4,5] BZ beads, [6] or microelectrofusion. [7] The interplays between local reaction kinetics (nodes), the physical processes that create coupling (link), and the architecture of the network in such systems can lead to a wealth of self-organized phenomena, including synchronization, [4,6,8] stationary Turing and oscillatory patterns, [9,10,11,12,13] or excitation waves. [14,15,16] Stationary patterns generated via the Turing [17] mechanism have been observed in experiments for both continuous [18] and networked [19] systems.…”

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

“…D x and D y are the diffusion coefficients. To make contact with the analysis carried out in [14], we shall deal with perfectly balanced networks, namely, graphs characterized by an identical number of ingoing and outgoing links. We will then assume that Eqs.…”

“…To solve the above linear system, one needs to introduce the eigenvalues (α) and eigenvectors φ (α) of the Laplacian operator [7,14]. These are solutions of the eigenvalue problem…”

“…Surprisingly, patterns can rise on a directed support, even if they are formally impeded on a regular, continuum or discrete, spatial medium. The mathematics of this process has been recently investigated in [14], where the conditions for the instability are obtained in the framework of a standard linear analysis calculation. The patterns which manifest as a byproduct of the aforementioned instability reflect, however, the nonlinearities which are accommodated for in the model and that are, by definition, omitted in the linear analysis theory.…”

Pattern Formation over Multigraphs

“…Recent efforts focusing on pattern formation over directed graphs [27], [28] and multigraphs [28], [29], [30], [31] share a common characteristic: the matrices describing coupling among nodes are the Laplacian operators associated with the network structures, yielding P ( k ) with zero row-sum. While this assumption is appropriate in the context of diffusion-driven instabilities, it does not permit the study of pattern formation propelled by mechanisms without diffusible molecules, e.g., in the case of lateral inhibition P ( k ) is row-stochastic (studied in [25], [26] when P ( k ) are identical and symmetric).…”