Impact of directionality on the emergence of Turing patterns on m-directed higher-order structures

TL;DR

The paper develops a theory of Turing pattern formation on $m$-directed $d$-hypergraphs, introducing two higher-order Laplace matrices $\hat{\mathbf{L}}^{(d,m)}$ (symmetric) and $\check{\mathbf{L}}^{(d,m)}$ (asymmetric) that yield an effective Laplacian $\mathbf{M}^{(d,m)} = \alpha^{(d,m)}\hat{\mathbf{L}}^{(d,m)} + \check{\mathbf{L}}^{(d,m)}$ for dispersion analysis. Through a generalized natural coupling and a linear stability framework, the authors project the dynamics onto the (potentially complex) spectrum of $\mathbf{M}$ to derive a dispersion relation and an instability criterion in the complex plane, highlighting how directionality promotes diffusion-driven instability. Using the Brusselator as a two-species RD model, they show that increasing head size $m$ moves eigenvalues toward the imaginary axis, expanding the parameter regions where Turing patterns occur on hypergraphs and random directed hypergraphs. They also develop a method to interpolate directed topologies to undirected ones via a parameter $p$, enabling comparisons and showing that patterns can persist or disappear depending on topology and parameter choices. The results extend Turing pattern theory to directed higher-order structures, with potential implications for brain networks, chemical systems, and other complex media where higher-order directed interactions are relevant.

Abstract

We hereby develop the theory of Turing instability for reaction-diffusion systems defined on m-directed hypergraphs, the latter being generalization of hypergraphs where nodes forming hyperedges can be shared into two disjoint sets, the head nodes and the tail nodes. This framework encodes thus for a privileged direction for the reaction to occur: the joint action of tail nodes is a driver for the reaction involving head nodes. It thus results a natural generalization of directed networks. Based on a linear stability analysis we have shown the existence of two Laplace matrices, allowing to analytically prove that Turing patterns, stationary or wave-like, emerges for a much broader set of parameters in the m-directed setting. In particular directionality promotes Turing instability, otherwise absent in the symmetric case. Analytical results are compared to simulations performed by using the Brusselator model defined on a m-directed d-hyperring as well as on a m-directed random hypergraph.

Figures (11)

• Figure 1: Some examples of $m$-directed $d$-hyperedges and their associated Laplace matrices $\hat{\;\mathbf{L}}^{(d,m)}$ and $\check{\;\mathbf{L}}^{(d,m)}$. In the top panels we schematically show $2$-hyperedges ($1$-directed in panel a) and $2$-directed in panel b), while the bottom panels refer to $3$-hyperedges ($1$-directed in panel c), $2$-directed in panel d) and $3$-directed in panel e)). The hyperedge heads are denoted by a darker blue color and the arrow helps to visualize the directionality. The adjacency tensors are given by $A_{(1)(23)}^{(2,1)}=A_{(1)(32)}^{(2,1)}=1$ (a), $A_{(12)(3)}^{(2,2)}=A_{(21)(3)}^{(2,2)}=1$ (b), $A_{(1)(234)}^{(3,1)}=A_{(1)(243)}^{(3,1)}=A_{(1)(324)}^{(3,1)}=A_{(1)(342)}^{(3,1)}=A_{(1)(423)}^{(3,1)}=A_{(1)(432)}^{(3,1)}=1$ (c), $A_{(12)(34)}^{(3,2)}=A_{(21)(34)}^{(3,2)}=A_{(12)(43)}^{(3,2)}=A_{(21)(43)}^{(3,2)}=1$ (d) and $A_{(123)(4)}^{(3,3)}=A_{(132)(4)}^{(3,3)} =A_{(213)(4)}^{(3,3)}=A_{(231)(4)}^{(3,3)}=A_{(312)(4)}^{(3,3)}=A_{(321)(4)}^{(3,3)}=1$ (e), all the remaining entries vanish. Beside each hyperedge we show the associated matrices $\hat{\;\mathbf{L}}^{(d,m)}$ and $\check{\;\mathbf{L}}^{(d,m)}$.
• Figure 2: Undirected (top panels) and $m$-directed $2$-hyperring, $m=1$ (middle panels) and $m=2$ (bottom panels). On the left columns we show, for visualization purpose, a representative hyperring composed by $5$ hyperedges each one containing $3$ nodes, $m$ of which form the head and $q=3-m$ determine the tail, for a total of $10$ nodes, $m=1$ (panel $a_2$)) while $m=2$ (panel $a_3$). The results of remaining panels have been obtained by using a $2$-hyperring composed by $Q=15$ hyperedges and thus $N=30$ nodes. The hyperedge heads are emphasized in darker blue and the arrows help the reader to visualize the directionality. In panel $a_1)$ there is no directionality. Panels $b_1)$, $b_2)$ and $b_3)$ present the instability region in the complex plane (green area) and the complex spectrum of the effective Laplace matrix $\mathbf{M}^{(d,m)}$ (black dots). Panels $c_1)$, $c_2)$ and $c_3)$ report the dispersion relation $\lambda_s$ as a function of $-\Re\Lambda^{(s)}$. The time evolution of $u_i(t)$ is presented in panels $d_1)$, $d_2)$ an $d_3)$. The parameter of the Brusselator model are given by $b=5$, $c=7$ and the coupling functions are given by $h_1^{(2,1)}(u_1,u_2)=\frac{D_u}{2}u_1^2u_2^2$, $h_2^{(2,1)}(v_1,v_2)=\frac{D_v}{2}v_1^2v_2^2$, $h_1^{(2,2)}(u_1,u_2)=D_u u_1^2u_2^2$ and $h_2^{(2,2)}(v_1,v_2)=D_v v_1^2v_2^2$, with $D_u=1$, $D_v=9$, moreover $\sigma_2=1$. In the undirected case, we used ${h}_1^{(2)}(u_{1},u_{2})=\frac{D_u}{2}u_1^2u_2^2$, ${h}_2^{(2)}(v_{1},v_{2})=\frac{D_v}{2}v_1^2v_2^2$.
• Figure 3: Undirected (top panels) and $m$-directed $3$-hyperring, $m=1$ (second panels from the top), $m=2$ (third panels from the top) and $m=3$ (bottom panels). On the left columns we show, for visualization purpose, a representative hyperring, composed by $5$ hyperedges each one containing $4$ nodes, $m$ of which form the head and $q=4-m$ determine the tail, for a total of $15$ nodes, $m=1$ (panel $a_2)$, $m=2$ (panel $a_3)$) and $m=3$ (panel $a_4)$). The results in the remaining panels have been obtained by using a $3$-hyperring composed by $Q=15$ hyperedges and thus $N=45$ nodes. The hyperedge heads are emphasized in darker blue and the arrows help the reader to determine the directionality. In panel $a_1)$ we show the undirected hyperring. Panels $b_1)$, $b_2)$, $b_3)$ and $b_4)$ present the instability region in the complex plane (green area) and the complex spectrum of the effective Laplace matrix $\mathbf{M}^{(d,m)}$ (black dots). Panels $c_1)$, $c_2)$, $c_3)$ and $c_4)$ report the dispersion relation $\lambda_s$ as a function of $-\Re\Lambda^{(s)}$. The time evolution of $u_i(t)$ is presented in panels $d_1)$, $d_2)$, $d_3)$ and $d_4)$. The parameter of the Brusselator model are given by $b=5.3$, $c=7$, $D_u=1$, $D_v=9$ and the coupling functions are given by $h_1^{(3,1)}(u_1,u_2,u_3)=\frac{D_u}{2}u_1^2u_2u_3$, $h_2^{(3,1)}(v_1,v_2,v_3)=\frac{D_v}{2}v_1^2v_2v_3$, $h_1^{(3,2)}(u_1,u_2,u_3)=D_u u_1^2u_2u_3$, $h_2^{(3,2)}(v_1,v_2,v_3)=D_v v_1^2v_2v_3$, $h_1^{(3,3)}(u_1,u_2,u_3)=D_u u_1u_2u_3^2$ and $h_2^{(3,3)}(v_1,v_2,v_3)=D_v v_1v_2v_3^2$ moreover $\sigma_2=1$. In the undirected case, we used ${h}_1^{(3)}(u_{1},u_{2},u_{3})=D_u u_{1}u_{2}u_{3}^{2}$, ${h}_2^{(3)}(v_{1},v_{2},v_{3})=D_v v_{1}v_{2}v_{3}^{2}$ as coupling functions.
• Figure 4: Directionality and emergence of patterns. We report the maximum of the dispersion relation as a function of the parameters $b$ and $c$ (left panel for $D_{u}=1$ and $D_{v}=9$) and the same quantity as a function of $D_u$ and $D_v$ (right panel for $b=5.3$ and $c=7$) by assuming the underlying hypergraph to be a $3$-hyperring composed by $Q=15$ hyperedges and thus $N=45$ nodes. The white region corresponds to a negative dispersion relation and thus to absence of patterns. The colored areas are associated to a positive maximum and thus to the presence of Turing patterns. More precisely, parameters $(b,c)$ or $(D_u,D_v)$ in the yellow region return patterns for $1$, $2$ and $3$-directed hyperring, the orange region denotes presence of patterns for $2$ and $3$-directed hyperring, while the green one only for $3$-directed hyperring. The region bounded by the black curve is associated to patterns for the undirected hyperring. The coupling functions are the same used in Fig. \ref{['fig:d3m123Motif']}.
• Figure 5: $3$-directed versus undirected $3$-hyperedges. We show how to obtain the undirected $3$-hyperedge by "juxtaposition" of four $3$-directed hyperedges weighted with positive coefficients $q_1=q_2=q_3=q_4=1/4$. For the remaining values of the latter, we obtain a directed $3$-hyperedge. The hyperedge heads are emphasized in darker blue and the arrows help the reader to determine the directionality.

Theorems & Definitions (2)

• Remark 1: Some examples
• Remark 2