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Topology-driven instabilities: the theory of pattern formation on directed networks
Malbor Asllani, Joseph D. Challenger, Francesco Saverio Pavone, Leonardo Sacconi, Duccio Fanelli
TL;DR
The paper examines pattern formation in reaction-diffusion systems on directed graphs, extending analysis beyond settings where patterns emerge on regular lattices. Using linear stability analysis and graph-topology variation, it finds topology-driven instabilities that produce travelling waves or stationary inhomogeneous patterns.
Problem
The paper addresses how directed-network topology contributes to self-organised patterns in reaction-diffusion systems when analogous patterns cannot emerge on regular lattices.
Method
The authors derive deterministic-instability conditions through linear stability analysis, expanding perturbations in the complete eigenvector basis of the Laplacian operator.
Results
Topology-driven instabilities develop on directed graphs, producing travelling waves and stationary inhomogeneous patterns at parameters where patterns cannot emerge on symmetric networks.
Takeaways & Limitations
Directed-network topology can generate a class of reaction-diffusion instabilities unavailable on undirected graphs.
Takeaways & Limitations
The discussion considers the topology of space when it is assumed to be a directed random network.
Abstract
from arXiv · showhide
The theory of pattern formation in reaction-diffusion systems is extended to the case of 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. Results from a linear stability analysis allow the instability region to be analytically traced. Numerical simulations show that the instability can lead to travelling waves, or quasi-stationary patterns, depending on the characteristics of the underlying graph. The results presented here could impact on the diverse range of disciplines where directed networks are found, such as neuroscience, computer networks and traffic systems.
3. European Laboratory for Non-linear Spectroscopy,
The section lists an affiliation in Sesto Fiorentino, Florence, Italy.
- The affiliation is located in Sesto Fiorentino.
- Sesto Fiorentino is identified as being in Florence.
- The listed country is Italy.
4. National Institute of Optics, National Research Council, 50125 Florence, Italy
The paper extends reaction-diffusion pattern-formation theory to directed networks and shows that network topology can itself generate instabilities unavailable on symmetric networks. Linear analysis and simulations connect graph characteristics with travelling waves or stationary inhomogeneous patterns.
- The theory uses linear stability analysis by expanding perturbations in the eigenvector basis of the discrete Laplacian.
- Directed networks have asymmetric connections, as in human mobility, internet routing, traffic, and neural connectivity.
- Directed-network topology can destabilize a homogeneous fixed point even when the same reaction-diffusion system cannot destabilize on a regular lattice or symmetric network.This introduces instabilities that cannot be induced on undirected graphs.
- The findings place the spatial support alongside interaction rules as a determinant of dynamical instability and pattern formation.
- Changing graph topology while fixing model parameters produces travelling waves or stationary inhomogeneous patterns that do not emerge on symmetric spatial supports.
I. RESULTS
Linear stability analysis shows that directed-network Laplacian spectra can destabilize a homogeneous fixed point, producing topology-driven patterns unavailable on symmetric supports. Depending on graph characteristics and dispersion-relation components, the resulting patterns include travelling waves and quasi-stationary inhomogeneous states.
- Linear stability analysis: A homogeneous fixed point becomes unstable when the real part of a dispersion eigenvalue is positive, with the unstable eigenmode guiding the emerging pattern.Because network spatial support is discrete, only finitely many eigenmodes can destabilize, and the mode with largest |Λ(α)| guides the instability.
- Linear stability analysis: Directed-network Laplacian eigenvalues can have nonzero imaginary parts, giving the dispersion relation a topology-dependent imaginary contribution.For symmetric graphs, the Laplacian spectrum is real and the standard instability constraints apply; directed graphs require relaxing the orthonormal-eigenbasis setting.
- Topology-driven instabilities: Directed topology enables instabilities that cannot occur on symmetric supports and can produce travelling waves in two-species reaction-diffusion systems.The paper identifies this as a topology-driven instability associated with the imaginary part of the dispersion relation.
- Pattern types: When the imaginary part of the dispersion eigenvalue is much smaller than its real part, the system evolves toward a stationary inhomogeneous pattern resembling a Turing instability.The paper contrasts this regime with travelling-wave outcomes and attributes the distinction to network characteristics as well as dynamical rules.
- Numerical results: Changing network construction and rewiring alters the instability: Newman-Watts and Watts-Strogatz graphs can place spectra inside the instability region and yield different pattern types.For Newman-Watts networks, increasing p moves spectral points leftward; Watts-Strogatz networks exhibit instability for relatively large p, while changing p can produce a transition from travelling waves to stationary patterns.
4. Alternative formulation of the transport operator
The directed-network formulation uses a Laplacian based on incoming weights and node degree, with balanced networks supporting a homogeneous fixed point and the stated linear-stability analysis. Numerical examples show topology-dependent quasi-stationary patterns and travelling waves.
- Alternative formulation of the transport operator: The transport operator is defined as Δ_ij = W_ji − k_iδ_ij for diffusive spreading on the network.This operator models the diffusive spreading of material between network nodes.
- Alternative formulation of the transport operator: The analysis assumes a balanced network, where incoming and outgoing connection counts are equal, so the homogeneous fixed point solves the spatially extended system.Under this condition, the homogeneous fixed point (φ*, ψ*) is retained by the spatially extended dynamics.
- Alternative formulation of the transport operator: For unbalanced networks, the same analysis requires linearising around a non-homogeneous state; that generalisation is deferred.The stated scope therefore focuses on balanced networks.
- Numerical results: On a Watts-Strogatz network, simulations display quasi-Turing patterns while the dispersion relation has a real and an imaginary component.The network examples use N = 100 and p = 0.2, with parameters set as in Figure 1.
II. DISCUSSION
The discussion argues that directed-network topology can generate instabilities unavailable on symmetric networks. The resulting patterns depend on the spatial support, with rewiring producing transitions between travelling waves and stationary patterns.
- Discussion: The paper extends reaction-diffusion pattern-formation analysis from symmetric networks to directed, non-symmetric networks.Directed links can represent movement between nodes without equivalent reverse movement.
- Discussion: A topology-driven instability can arise on directed graphs even when the same system is stable on a regular lattice or continuous spatial support.Symmetric networks cannot turn an otherwise stable homogeneous fixed point unstable in this way.
- Discussion: The spatial support determines whether the instability produces travelling waves or asymptotically stationary inhomogeneous patterns.In Watts–Strogatz networks, tuning the rewiring probability p yields transitions between these regimes.
- Discussion: A compact mathematical criterion accounts for the underlying network’s spectral properties and was validated against selected case studies.The criterion provides a rigorous formulation of the instability condition.
- Discussion: The discussion concludes that network topology plays an equally crucial role with interaction rules in determining the system’s asymptotic fate.This conclusion motivates broader applications to domains where networks are essential.
III. METHODS
The analysis linearizes perturbations around a homogeneous fixed point and expands them in network-Laplacian eigenvectors. It applies these tools to reaction-diffusion dynamics and directed small-world graph constructions.
- Small perturbations are introduced around the fixed point and the dynamics are linearized to search for instabilities.
- Expanding perturbations in network-Laplacian eigenvectors yields an eigenvalue problem for each network mode.
- The mode matrix is Jα = J + DΛ(α), and its largest-real-part eigenvalue defines the dispersion relation.
- Reaction-diffusion model: The Brusselator has a unique homogeneous fixed point φ∗ = 1, ψ∗ = b/c, stable for b < 1 + c, with b > 1 imposed.
- Directed network construction: Directed WS networks are generated by rewiring one-sided lattice edges without imposing reverse symmetric edges or duplicate links.
- Directed network construction: Newman-Watts networks preserve original directed-lattice links and add approximately ΩKp long-range directed links, with a balanced variant equalizing incoming and outgoing counts.
Appendix A: Topology induced patterns in FitzHugh-Nagumo model
The FitzHugh-Nagumo analysis examines whether directed network topology can destabilize a fixed point that remains stable under symmetric spatial coupling. The appendix evaluates instability regions and associated wave patterns on Newman-Watts networks.
- Directed graphs can support pattern formation even when classical instability conditions on symmetric supports are not satisfied.
- The FitzHugh-Nagumo model represents membrane potential and recovery dynamics with parameters a = 0.5, b = 0.04, c = 26, Du = 0.2, and Dv = 15.
- For these parameters, the homogeneous fixed point is stable to inhomogeneous perturbations on a symmetric spatial support.
- For a Newman-Watts network with N = 100 and p = 0.27, the dispersion relation has both real and imaginary parts, while the instability region is redrawn for this model.
- The linear instability theory indicates that no instability develops on a symmetric support for the selected parameters.
- Simulations display instability and an emerging wave for a balanced Newman-Watts network with a purely diffusive transport operator.
Appendix B: Evolution of wave patterns in small-world networks
The appendix presents time evolution of wave patterns for Brusselator and FitzHugh-Nagumo dynamics on Newman-Watts small-world networks. The visualizations use node ordering inherited from the original lattice and compare network configurations.
- The Brusselator simulation starts from the homogeneous fixed point φ∗ = 1 on a 100-node Newman-Watts network with p = 0.27.
- A travelling wave is shown for the Brusselator dynamics on the same 100-node Newman-Watts network generated with p = 0.27.
- The appendix also presents FitzHugh-Nagumo wave-pattern results for the balanced Newman-Watts configuration and its Laplacian convention.
- The visualizations order nodes according to the original lattice, while the vertical axis represents the magnitude of species φ.