Topological resilience in non-normal networked systems

The network of interactions in complex systems strongly influences their resilience and the system capability to resist external perturbations or structural damages and to promptly recover thereafter. The phenomenon manifests itself in different domains, e.g., parasitic species invasion in ecosystems or cascade failures in human-made networks. Understanding the topological features of the networks that affect the resilience phenomenon remains a challenging goal for the design of robust complex systems. We hereby introduce the concept of non-normal networks, namely networks whose adjacency matrices are non-normal, propose a generating model, and show that such a feature can drastically change the global dynamics through an amplification of the system response to exogenous disturbances and eventually impact the system resilience. This early stage transient period can induce the formation of inhomogeneous patterns, even in systems involving a single diffusing agent, providing thus a new kind of dynamical instability complementary to the Turing one. We provide, first, an illustrative application of this result to ecology by proposing a mechanism to mute the Allee effect and, second, we propose a model of virus spreading in a population of commuters moving using a non-normal transport network, the London Tube.

Figure 1

Transient growth and non-normality in a linear generic example. We represent the time evolution of the node density $x_i\left(t\right)$ [colored curves in panels (a) and (b)] for the linear model, $\dot{\mathbf{x}}=-a\mathbf{x}+D\mathbf{L}\mathbf{x}$, on top of a unidirectional (a) ring and (b) path, both made of $M=11$ nodes. The norm $\left|\right|x\left(t\right)\left|\right|$ (black curves) describes the global behavior of the system. The model in panel (a) corresponds to a normal system and thus, independently of the system parameters and the initial conditions close to the equilibrium point, the perturbations are doomed to die out. On the other hand, using a non-normal network (b), and assuming diffusion to be faster than the capture rate into the nodes sinks ($D\gg a$), then the density on the terminal node experiences a strong transient amplification before reaching definitively the equilibrium. This transient regime will influence also the return time [see panel (c)] where the return time ${\tau }_2$ corresponding to the non-normal network in (b) is larger than the return time ${\tau }_1$ associated to the network used in (a). Observe that the stronger the system non-normality, the longer the return time and also the larger gets the transient deviation from the stable equilibrium as measured by $max{x}_i\left(t\right)$ [panel (d)]. For both networked systems we used the same initial conditions and the parameters $a=0.1$ and $D=10$.

Figure 2

Numerical abscissa as a function of the directional parameter $\gamma$ in a non-normal network. Each point has been obtained with 100 replicas using the same set of parameters. For $\gamma =4$, $\gamma =10$, and $\gamma =16$ we show three representative adjacency matrices; the darker the blue the larger the value of the link weight. One can appreciate the presence of a ringlike structure: The weights on the upper diagonal are quite strong as well as the weight connecting the node $N$ to node 1 (bottom left corner in the matrices).

Figure 3

Resilience and the Allee effect. (a) The nonspatial Allee model, $dx/dt=rx(1-x)(x/A-1)$, in normalized variables, $x=n/K$, where $n$ is the population size and $K$ the carrying capacity, $r$ intrinsic growth rate, and $A$ Allee critical parameter (here $A=0.3$). The Allee effect states that for $x\left(0\right)<A$ the species fails to survive. (b) Non-normal versus normal system. For the normal systems, an initial condition $\mathbf{x}\left(0\right)$ converges to the stable state $\widehat{\mathbf{x}}$ of the attraction basin it belongs to, while if the numerical abscissa is large enough, hence the transient phase is strong enough, it can reach another stable solution ${\mathbf{x}}^{*}$. (c) Bifurcation diagram of the networked Allee model. For a normal system (blue), the size of the survival zone, $\sim 1-0.25=0.75$, is very close to that of the nonspatial model, $1-A=0.7$, on the other hand for non-normal systems (red) the survival zone is strongly increased, $\sim 1-0.05=0.95$. (d) Turing patterns mechanism versus the non-normal one. In the former two species are needed, with the inhibitor diffusing faster than the activator (denoted by a thick dashed line); in the non-normal scheme only one inhibitor species is enough to create patterns once it diffuses on a non-normal network playing the role of the activator.

Figure 4

Patterns emergence in non-normal networked systems. We represent the time evolution of the node density $x_i\left(t\right)$ (colored curves on the left panel) for the Allee model (2) on top of a non-normal network made of $M=10$ nodes(middle panel), together with the norm $\left|\right|x\left(t\right)\left|\right|$ (black curve) to appreciate the global behavior of the system. One can observe the onset of a stable spatial patterns instigated by a “terminal node” (red node on the middle panel and red curve on the left panel) of a backbone directed weighted path. On the right panel the patterns amplitude measured by $\left|\right|x\left(t\right)\left|\right|$ for large enough $t$ is reported as a function of $\gamma$ (black curve), a proxy for the non-normality strength (see Appendix pp1). Parameters are $r=0.1$, $A=0.3$, and $D=10$. The core ring has stronger links compared to the long-range ones.

Figure 5

The London Tube map. Top panel: The whole London Tube map where the yellow Circle Line and the magenta Metropolitan Line have been emphasized to show their different geographical emplacement and shape. In the bottom panels we separately present the magenta line (left) and the yellow line (right) and their respective flows of commuters (black lines with arrows).

Figure 6

Hypothetical measles outbreak in the London Tube network. Time evolution of the (normalized) number of infected individuals during the peak hours in the yellow Circle Line taking into account 27 stations (a) and in the magenta Metropolitan Line (b) considering 23 stations with the fluxes illustrated in Fig. 5. In the former case (a) we assumed the average number of passengers to be constant because in each station, the average number of passengers getting on equals that of those getting off. The circular topology and the Allee effect impede the outbreak of the epidemic. On the contrary, for the second case (b), we assume that, on average, as long the trains approach downtown, the number of passengers increases, because most of the individuals have their destination in the city center. In this case, despite the presence of the Allee effect, the topology contributes to the outbreak of the measles epidemic once the train reaches the center. The parameters of the Allee model are $a=1$, $A=0.3$, and $D=10$ for both cases.

Figure 7

Time evolution of the norm of the solution of the linear ODE $\dot{\mathbf{x}}=\mathbf{Ax}$. The red curve corresponds to a non-normal matrix $\omega \left({\mathbf{A}}_2\right)=3.52$ while the blue curve to a normal one $\omega \left({\mathbf{A}}_1\right)=-0.79$; one can clearly appreciate the temporal growth arising in the former case. In the inset we still report the norm of the solution but in logarithmic scale to emphasize the short-time behavior described by the numerical abscissa (the straight black line has slope 3.52) and the long-time one related to the spectral abscissa (the dashed and dot-dashed straight lines have slope $-1$).