Higher-order Connection Laplacians for Directed Simplicial Complexes

Abstract

Higher-order networks encode the many-body interactions existing in complex systems, such as the brain, protein complexes, and social interactions. Simplicial complexes are higher-order networks that allow a comprehensive investigation of the interplay between topology and dynamics. However, simplicial complexes have the limitation that they only capture undirected higher-order interactions while in real-world scenarios, often there is a need to introduce the direction of simplices, extending the popular notion of direction of edges. On graphs and networks the Magnetic Laplacian, a special case of Connection Laplacian, is becoming a popular operator to treat edge directionality. Here we tackle the challenge of treating directional simplicial complexes by formulating Higher-order Connection Laplacians taking into account the configurations induced by the simplices' directions. Specifically, we define all the Connection Laplacians of directed simplicial complexes of dimension two and we discuss the induced higher-order diffusion dynamics by considering instructive synthetic examples of simplicial complexes. The proposed higher-order diffusion processes can be adopted in real scenarios when we want to consider higher-order diffusion displaying non-trivial frustration effects due to conflicting directionalities of the incident simplices.

Figures (13)

• Figure 1: Higher-order Connection Laplacians rotate vectors along nodes, edges, or triangles. (a) 0-up Connection Laplacian rotates vectors defined on vertices along edges. (b) 1-down Connection Laplacian rotates vectors defined on edges along nodes. (c) 1-up Connection Laplacian rotates vectors defined on edges along triangles. (d) 2-down Connection Laplacian rotates vectors defined on triangles along edges.
• Figure 2: The complete spectrum of $\hbox{$\mathcal{L}$}_{1}^{c, up}$ (top-left), $\hbox{$\mathcal{L}$}_{1}^{c, down}$ (top-right), and $\hbox{$\mathcal{L}$}_{1}^{c}$ (bottom-left), and commutator $[\hbox{$\mathcal{L}$}_{1}^{c, up}, \hbox{$\mathcal{L}$}_{1}^{c, down}]$ (bottom-right) for Case 1 directed simplicial triangles is plotted as a function of $\delta$.
• Figure 3: Higher-order diffusion driven by the $1$-Connection Laplacians $\hbox{$\mathcal{L}$}_1^{up}$ (Up), $\hbox{$\mathcal{L}$}_1^{down}$ (Down) and $\hbox{$\mathcal{L}$}_1$ (Up+Down) on directed simplicial triangle in Case 1 are plotted for single initial conditions and for $\delta=\pi/3$ (upper row) and $\delta=2\pi/3$ (lower row). In the upper-left plot, the final state is an eigenvector of $\hbox{$\mathcal{L}$}_{1}^{c, up}$ corresponding to the 0 eigenvalue, while in the lower-middle plot, the equilibrium vector is an eigenvector of $\hbox{$\mathcal{L}$}_{1}^{c, down}$ corresponding to the 0 eigenvalue. In the remaining plots, the final states converge to the slowest eigenmodes associated with the smallest positive eigenvalue.
• Figure 4: The complete spectra of $\hbox{$\mathcal{L}$}_{1}^{c, up}$ (top-left), $\hbox{$\mathcal{L}$}_{1}^{c, down}$ (top-right), and $\hbox{$\mathcal{L}$}_{1}^{c}$ (bottom-left), and commutator $[\hbox{$\mathcal{L}$}_{1}^{c, up}, \hbox{$\mathcal{L}$}_{1}^{c, down}]$ (bottom-right) for Case 2 directed simplicial triangles is plotted as a function of $\delta$.
• Figure 5: Higher-order diffusion driven by the $1$-Connection Laplacians $\hbox{$\mathcal{L}$}_1^{up}$ (Up), $\hbox{$\mathcal{L}$}_1^{down}$ (Down) and $\hbox{$\mathcal{L}$}_1$ (Up+Down) on directed simplicial triangle in Case 2 are plotted for single initial conditions and for $\delta=\pi/3$ (upper row) and $\delta=2\pi/3$ (lower row). In the upper-left plot, vectors converge towards an eigenvector of $\hbox{$\mathcal{L}$}_{1}^{c, \text{up}}$ with an eigenvalue of zero. In the lower plot, where $\delta = 2\pi/3$, both equilibrium vectors for $\hbox{$\mathcal{L}$}_{1}^{c, \text{up}}$ (on the left) and $\hbox{$\mathcal{L}$}_{1}^{c, \text{down}}$ (in the middle) are eigenvectors associated with a zero eigenvalue. Across the remaining plots, the final states converge to the slowest eigenmodes linked to the smallest positive eigenvalue due to the absence of a zero eigenvalue.