Structure and dynamical behavior of non-normal networks

Abstract

We analyze a collection of empirical networks in a wide spectrum of disciplines and show that strong non-normality is ubiquitous in network science. Dynamical processes evolving on non-normal networks exhibit a peculiar behavior, as initial small disturbances may undergo a transient phase and be strongly amplified in linearly stable systems. In addition, eigenvalues may become extremely sensible to noise and have a diminished physical meaning. We identify structural properties of networks that are associated with non-normality and propose simple models to generate networks with a tunable level of non-normality. We also show the potential use of a variety of metrics capturing different aspects of non-normality and propose their potential use in the context of the stability of complex ecosystems.

Fig. 1. Structure of non-normal networks.

(A) DAG (blue links) embedded in a weighted non-normal network, with red links corresponding to entries in the lower triangular part of the adjacency matrix (once nodes are reordered). The thickness of a link is proportional to its weight. (B) Normalized Henrici’s departure from normality versus the structural measure of asymmetry Δ, for the networks of table S1 and for the nSF model.

Fig. 2. nSF network.

(A) Local breaking of reciprocity: If j’s influence over i is larger than the reciprocal one, the pairwise relation is nonreciprocal and presents a broken symmetry. (B) Generating model of nSF network: Once a new node enters the system (dashed green arrow in B₁), it establishes, with higher probability, a link pointing toward the node with larger in-degree, i (solid green link in B₂). With a lower probability, the latter, i.e., the hub, can create a weaker link pointing to the reciprocal direction (thin magenta link in B₃).

Fig. 3. Models of non-normal networks.

Spectra and pseudospectra [computed using the software EigTool (44)] of non-normal models (top): (A) ER with link probability p_ER = 0.1 and weights from a normal distribution $\mathcal{N}(0,1)$, (B) (unweighted) WS with initial number of neighbor nodes k = 2 and rewiring probability p_WS = 1, (C) nSF network with probability of backward links p_{i → j} = 0.001 for j > i and weights from a uniform distribution, $\mathcal{U}[0,1]$. The color bar on the right represents the different levels of ||E|| in log scale, e.g., ϵ = 10^x, where x is the numerical value reported on the bar. Visual inspection reveals that the pseudospectrum of diverse network models is affected differently by ϵ and, in particular, that its size is increased more substantially for the nSF model. In the bottom panels, we quantify how the size increase of the pseudospectrum affects the stability of its network as follows. We consider the difference between the ϵ-pseudoabscissa of A and that of its Hermitian part, δ_ϵ = α_ϵ(A) − α_ϵ(H(A)), hence measuring how the non-Hermitian aspect of the system affects its dominant eigenvalue. This quantity is always positive, as the pseudospectrum of a non-normal matrix is larger than that of a normal one (19), for any given fixed ϵ > 0. (D) ER with weights from a normal distribution $\mathcal{N}(0,\mathrm{\sigma })$ for several values of the variance and different link probabilities, p_ER. (E) (Unweighted) WS for different rewiring probabilities p_WS and different initial number of neighbor nodes k. (F) nSF with varying backward link probability p_{i → j} and different upper bound, b, of the uniform distribution from which the weights are chosen, $\mathcal{U}[0,\mathit{b}]$. In each case, the networks are composed of 100 nodes, and the adjacency matrices have been diagonally shifted with their respective spectral abscissa α(A), to set the real part of the maximum of each spectrum exactly at 0. For the bottom panels, the value of ϵ has been set to 10^−0.5.

Fig. 4. GLV model: x.i=xi(ri−sixi+∑j≠iMijxj).

We consider an ecosystem composed of 25 species. For the sake of simplicity, the intraspecies interactions are all set equal, s_i = 1 ∀ i, and M is the (weighted and signed) adjacency matrix of an nSF for the structured case (main panels) or a random matrix (insets) whose weights are drawn from a normal distribution $\mathcal{N}(0,1/5)$. In the structured cases, the strengths in the upper triangular part of the matrix M are 15 times larger than those in the lower one, thus enhancing non-normality, as can be seen from the pseudospectrum levels [computed using the software EigTool (44)]. In the top panels, we show the master stability function close to the asymptotically stable equilibrium point, based on the use of the pseudospectra. The corresponding dynamical evolution is shown in the bottom panels, where different colors correspond to different levels of ϵ in log scale (as in the top panels); initial conditions have been uniformly randomly chosen. Different cases are considered depending on the signs of the interaction strengths: (A) competition (−/−), (B) prey-predator (−/+), and (C) mutualism (+/+). We observe that, even if the system is asymptotically stable (A₁ and B₁), the ϵ–pseudospectral abscissa is positive for sufficiently large ϵ, thus inducing an unstable system behavior if the perturbation (in the adjacency matrix and/or in the initial conditions) is strong enough (A₂ and B₂). Yet, the unstructured systems still converge to the homogeneous equilibrium [see insets in (A₂) and (B₂)]. Overall, this effect is more pronounced in the structured systems than in the random ones, as the ϵ levels are much larger in the former case, for a fixed value of ϵ.