Pattern reconstruction through generalized eigenvectors on defective networks

TL;DR

This work proves the validity of Turing idea also in the case of a network with a defective Laplace matrix and shows that it can reconstruct the asymptotic pattern with a relatively small discrepancy.

Abstract

Self-organization in natural and engineered systems causes the emergence of ordered spatio-temporal motifs. In presence of diffusive species, Turing theory has been widely used to understand the formation of such patterns on continuous domains obtained from a diffusion-driven instability mechanism. The theory was later extended to networked systems, where the reaction processes occur locally (in the nodes), while diffusion takes place through the networks links. The condition for the instability onset relies on the spectral property of the Laplace matrix, i.e., the diffusive operator, and in particular on the existence of an eigenbasis. In this work we make one step forward and we prove the validity of Turing idea also in the case of a network with defective Laplace matrix. Moreover, by using both eigenvectors and generalized eigenvectors we show that we can reconstruct the asymptotic pattern with a relatively small discrepancy. Because a large majority of empirical networks are non-normal and often defective, our results pave the way for a thorough understanding of self-organization in real-world systems.

Figures (17)

• Figure 1: Region of the complex plane associated to Turing instability (top panel) and dispersion relation (bottom panel) computed for the Brusselator model with parameters $b=3.92$, $c=3$, $D_u=0.2$ and $D_v=0.8$. Top panel: The black dots denote the eigenvalues of the Laplace matrix, $\Lambda^{(1)}=0$ (multiplicity $2$), $\Lambda^{(2)}=-1$ (multiplicity $5$), $\Lambda^{(3)}=-2$ (multiplicity $1$), $\Lambda^{(4)}=-4$ (multiplicity $2$), the green region is associated to a positive dispersion relation, while the white one to the negative case. Bottom panel: the largest real part of the spectrum of the matrix $\mathbf{J}_0+\zeta \mathbf{D}$ is shown in blue as a function of $\zeta$, the dispersion relation evaluated on the Laplace spectrum is reported by using red triangles.
• Figure 2: Random non-normal defective network composed by $n=10$ nodes, built by using a directed Erdős-Rényi algorithm where the probability to create a bidirectional link is $0.2$ and the probability to transform it into a directed one is $0.6$. Nodes have been colored according to the value of species $u$ at time $\hat{t}=200$ (see colorbar).
• Figure 3: Evolution of the concentration of the specie $u_i$ over time for the Brusselator model with parameters $b=3.92$, $c=3$, $D_u=0.2$ and $D_v=0.8$. The underlying network is the one shown in Fig. \ref{['fig:3 unstable eigenvalue - network']}. The nodes concentrations have been initialized to the homogeneous equilibrium, $u_*=1$, upon which a random, node dependent, perturbation of order $0.01$ has been added. The resulting orbits have been obtained by using a Runge-Kutta 4th method with time step $0.01$. Each trajectory has been represented by using the same color of the corresponding node in Fig. \ref{['fig:3 unstable eigenvalue - network']}, namely the value at time $\hat{t}=200$.
• Figure 4: Pattern vs. reconstructed pattern for the Brusselator model with parameters $b=3.92$, $c=3$, $D_u=0.2$ and $D_v=0.8$. Upper panel: the reconstruction is obtained with only the eigenvectors (EV). Lower panel: the eigenvectors and generalized eigenvectors (GEV) are used for the reconstruction. In the first case, the reconstruction error is $\varepsilon=0.088$ while in the second one we have $\varepsilon=0.015$.
• Figure 5: An example of directed defective network obtained with the previous algorithm by using the multiplicity : $m_0=1$, $m_1=2$, $m_2=3$ and $m_3=1$.