Turing patterns in Matrix-Weighted Networks

TL;DR

This paper extends Turing's diffusion-driven pattern formation to Matrix-Weighted Networks (MWNs), where each edge carries a matrix weight. By enforcing a coherence condition—the product of edge transformations along any oriented cycle equals the identity—it becomes possible to disentangle node interactions with an orthonormal change of variables and reduce diffusion to a scalar topology via a transformed Laplacian. The authors derive a general dispersion relation linking MWN spectrum and local Jacobians, enabling precise instability criteria, and validate the theory on three models: a Stuart–Landau network with analytic thresholds, an abstract rotation-invariant system, and the Lorenz system, revealing when matrix weights promote or suppress pattern formation. A constructive coherence characterization and algorithm allow building MWNs of any size, broadening the design space for Turing patterns in complex networked systems and paving the way for applications in higher-order network dynamics. Overall, the work reveals a fundamental link between MWN structure, coherence, and pattern formation, and provides a practical framework to analyze and engineer Turing patterns on matrix-weighted and higher-order networks.

Abstract

Diffusion-driven instability is a fundamental mechanism underlying pattern formation in spatially extended systems. In almost all existing works, diffusion across the links of the underlying network is modeled through scalar weights, possibly complemented by cross-diffusion terms that are homogeneous across links. In this work, we investigate the emergence of Turing patterns on Matrix Weighted Networks (MWNs), a recently introduced framework in which each edge is associated with a matrix weight. Focusing on the class of coherent MWNs, we provide a novel characterization of coherence in terms of node-dependent orthonormal matrices, showing that link transformations can be written as relative rotations between nodes. This representation allows us to deal with coherent MWNs of any size and to introduce an orthonormal change of variables capable to reduce diffusion on a coherent MWN to diffusion on a standard weighted network with scalar weights. Building on this, we extend the classical Turing instability analysis to MWNs and derive the conditions under which a homogeneous equilibrium of the local dynamics loses stability due to matrix-weighted diffusion. Our results show how network topology, scalar weights, and inter-node transformations jointly shape pattern formation, and provide a constructive framework to analyze and design Turing patterns on matrix-weighted and higher-order networked systems.

Figures (8)

• Figure 1: Emergence of Turing patterns in a MWN with stochastic block model topology of coupled Stuart-Landau systems. Panels $(a),(b)$ - The dispersion relation is reported as a function of the network Laplacian eigenvalues $\Lambda^{(\alpha)}$ (red dots), the blue curve is represented to emphasize the linear dependence: $(a)$ stable regime $\lambda(\Lambda^{(\alpha)})<0$ for all $\alpha$; $(b)$ unstable regime, there exist $\Lambda^{(\alpha)}$ associated to a positive dispersion relation, returning thus Turing pattern formation. Panels $(c),(d)$ - Temporal evolution of $\xi_j(t)$ across nodes: $(c)$ convergence to the homogeneous equilibrium of the oscillators in the stable regime; $(d)$ Turing patterns emerge in the unstable regime. Panels $(e),(f)$ - Network visualizations with node colors indicating dynamical states, values of $\xi_j(t)$ after a sufficiently long time: $(e)$ nodes present the same color, meaning that oscillators assume the same value; $(f)$ nodes present different colors indicating that in the unstable regime, Turing patterns emerge, i.e., $\xi_j$ vary from node to node. The model parameters are $\sigma = -1-0.5i$, $\beta = 1+i$, $\mu=2 + 5.5i$ for the top panels $(a)$, $(c)$, and $(e)$, while $\mu=-0.12+5.5i$ for the bottom panels $(b)$, $(d)$, and $(f)$. The underlying topology is given by a stochastic bloc model of Erdős–Rényi network composed by $n = 50$ nodes and with $p_{in} = 0.6$ and $p_{out}=0.02$, and $K=4$ blocks.
• Figure 2: Emergence of Turing patterns in a MWN with random Erdős–Rényi topology of coupled Stuart-Landau systems. Panels $(a),(b)$ - Dispersion relations is shown as a function of the network Laplacian eigenvalues $\Lambda^{(\alpha)}$ (red dots), the blue curve is displayed to emphasize the linear dependence: $(a)$ stable regime with a negative dispersion relation for all eigenvalues; $(b)$ unstable regime with dispersion relation assuming positive values for some eigenvalues, Turing pattern can thus develop. Panels $(c),(d)$ - Temporal evolution of $\xi_j(t)$ across nodes: $(c)$ convergence to the homogeneous solution in the stable regime; $(d)$ Turing patterns emerge in the unstable regime. Panels $(e),(f)$ - Network visualizations with node colors indicating dynamical states, i.e., value of $\xi_j(t)$ after a sufficiently long time: $(e)$ nodes present the same color, meaning that oscillators assumed the same value independently from the node index; $(f)$ nodes present different colors indicating that in the unstable regime, Turing patterns emerge, i.e., nodes differentiate among themselves. The model parameters are $\sigma = -1-0.5i$, $\beta = 1+i$, $\mu=2 + 5.5i$ for the top panels $(a)$, $(c)$, and $(e)$, while $\mu=-0.12+5.5i$ for the bottom panels $(b)$, $(d)$, and $(f)$. The underlying topology is given by a Erdős–Rényi network composed by $n = 50$ nodes and $p=0.15.$
• Figure 3: Emergence of Turing patterns for the abstract model \ref{['eq:modelk']} defined on top of a MWN whose underlying topology is given by a Barabási-Albert network composed by $n=50$ nodes. Panels $(a),(b)$ - Dispersion relations as a function of the Laplace eigenvalues $\Lambda^{(\alpha)}$ (red dots), the blue curve has been drawn to help the reader and can be obtained by replacing the eigenvalues with a continuous variable. In panel $(a)$, the dispersion relation remains negative for any value of $\Lambda^{(\alpha)}$, which prevents the emergence of Turing patterns, as can be observed in panel $(c)$, where we report the time evolution of $\Re(\zeta_j(t))$, and in panel $(e)$ where we show the network whose nodes are colored according to the asymptotic stationary values of $\Re(\zeta_j(t))$. In panel $(b)$, we report the dispersion relations and we can observe that it assumes positive values (red dots) for some $\Lambda^{(\alpha)}\gtrsim 3$, Turing patterns can thus emerge as visible in panel $(d)$, where we report $\Re(\zeta_j(t))$ versus time and panels $(f)$, where the network is displayed with the nodes again colored according to the asymptotic stationary values of $\Re(\zeta_j(t))$. The model parameters are $k=4$, $\sigma = 1+i$, $\beta = 1-2i$, $\varepsilon=1.4+2i$ for the top panels $(a)$, $(c)$, and $(d)$, while $\varepsilon=-1.4+2i$ for the bottom panels $(b)$, $(d)$, and $(f)$.
• Figure 4: On the use of the rotated variables and the original ones. We consider the model \ref{['eq:modelk']} with $k=4$ defined on top of a MWN whose underlying topology is given by a Barabási-Albert network composed of $n=50$ nodes. The parameters have been fixed so as to have a negative dispersion relation, $\sigma = 1+i$, $\beta = 1-2i$, and $\varepsilon=1.4+2i$. Hence, patterns cannot emerge as clearly shown in the top panels: (a) the time evolution of $\Re(\zeta_j(t))$, and (b) the network with nodes colored according to the asymptotic values of $\Re(\zeta_j(t))$. In the bottom panels, we show the same results but by using the original variables, (c) the time evolution of $\Re(z_j(t))$, and (d) the network with nodes colored according to the asymptotic values of $\Re(z_j(t))$. In both cases, a heterogeneous solution appears.
• Figure 5: Emergence of Turing patterns in a MWN with random Erdős–Rényi topology of coupled Lorenz systems. Panels $(a),(b)$ - The dispersion relations is displayed as a function of network Laplacian eigenvalue $\Lambda^{(\alpha)}$ (reds dots), the blue curve have been obtained by replacing the latter with a continuous variable and it is shown to help the reader to appreciate the nonlinear behavior: $(a)$ the dispersion relation remains negative for any value of $\Lambda^{(\alpha)}$, which prevents the emergence of Turing patterns; $(b)$ negative dispersion relationship up to a critical threshold of $\Lambda^{(\alpha)}$, beyond which it becomes positive enabling the emergence of Turing patterns. Panels $(c),(d)$ - Temporal evolution of $\xi_j(t)$ across nodes: $(c)$ all variables $\xi_j(t)$ converge to the equilibrium point $\xi^*$; $(d)$ emergence of Turing patterns. Panels $(e),(f)$ - Network visualizations with node colors indicating dynamical states, i.e., $\xi_j(t)$ after a sufficiently long period of time: $(e)$ nodes present same color meaning that oscillators reached the same value regardless of the node index; $(f)$ nodes present different colors indicating the onset of Turing patterns. The model parameters used to obtain the presented results are $\sigma = 13$, $\rho = 28$, $\beta=8$. The underlying topology is given by a Erdős–Rényi network composed by $n = 50$ nodes and $p=0.15.$, and the coupling is obtained with $\varepsilon=4$ and $E_{11}=1$ (top panels) while $E_{31}=1$ (bottom panels).

Theorems & Definitions (5)

• Proposition 1: Characterization of coherent MWNs
• proof
• Remark 2: Routh--Hurwitz criterion for cubic polynomials
• Lemma 3: Coherence on a cycle basis
• proof