Fragility Limits Performance in Complex Networks

While numerous studies have suggested that large natural, biological, social, and technological networks are fragile, convincing theories are still lacking to explain why natural evolution and human design have failed to optimize networks and avoid fragility. In this paper we provide analytical and numerical evidence that a tradeoff exists in networks with linear dynamics, according to which general measures of robustness and performance are in fact competitive features that cannot be simultaneously optimized. Our findings show that large networks can either be robust to variations of their weights and parameters, or efficient in responding to external stimuli, processing noise, or transmitting information across long distances. As illustrated in our numerical studies, this performance tradeoff seems agnostic to the specific application domain, and in fact it applies to simplified models of ecological, neuronal, and traffic networks.

where t f > 0 denotes the control horizon, and A T denotes the transpose of A.
If the network (1) is controllable, then there exist control inputs to drive the network state from any initial state x(0) = x 0 to any final state x(t f ) = x f . It is known that this happens if and only if G t f is invertible. In particular, the input with minimum energy to drive the network state from x 0 to x f is given by whose energy is .
When t f = ∞ and the network is stable, and the energy to drive the state from x 0 to x f equals The controllability Gramian G can be computed in different ways. For instance, G is the unique solution of the Lyapunov equation 1, Thm 3.3.1 Moreover 2 , where Γ is a curve in the complex plane that encloses all the eigenvalues of A. By choosing Γ as the semi-circle with infinite radius enclosing the stable half plane, we obtain where (ωiI − A) −H denotes the inverse of the complex conjugate of ωiI − A. While equations (4), (7) and (6), are valid only when A is stable, the expression (2) is valid also for unstable networks. The controllability Gramian can be used to quantify the responsiveness of a network to external stimuli. In particular, several scalar metrics can be defined to measure the "size" of the controllability Gramian, and therefore quantify the control energy needed to reach particular states. In this paper, we useσ(G) = 1 n n i=1 σ i (G) = tr(G)/n to quantify the responsiveness of a network, which is also an indirect measure of the average energy needed to control the network. Different metrics are also interesting. For instance, σ min (G) = min{σ 1 (G), . . . , σ n (G)} = 1/ G −1 , whose inverse 1/σ min (G) quantifies the largest control energy over all possible target states. Clearly, σ min (G) ≤ n/tr(G −1 ) ≤ (det(G)) 1/n ≤σ(G) ≤ σ max (G), where tr(G), det(G), and σ max (G) denote the trace, determinant, and largest eigenvalue of the Gramian G. Notice that tr(AB) ≤ tr(A 2 )tr(B 2 ) for any positive semidefinite matrices A and B. Then, the following inequality holds: This implies that tr(G −1 )/n grows whenσ(G) = tr(G)/n becomes small.

The stability radius of a network and its fragility
When the network (1) is stable, the following definition of stability radius quantifies its distance to instability ( · denotes the Euclidean norm): r(A) := min{ ∆ : ∆ ∈ C n×n , A + ∆ is unstable}.
The index r(A), namely, the stability radius of A, quantifies the degree of stability of (1), as it quantifies the minimum size of a perturbation of the weights that renders the network unstable. Conversely, 1/r(A) can be used to measure the degree of fragility of (1) with respect to changes of its weights. It can be shown that 3, Prop 4.1

Network responsiveness and fragility
In this section we characterize analytical relationships between the responsiveness and fragility degrees of a network. Recall that 4, 5 σ min (X)tr(Y ) ≤ tr(XY ) ≤ σ max (X)tr(Y ). Then, from (4) we obtain We will now derive a family of upper bounds forσ(G) = tr(G)/n, which will be parametrized by the scalar α ∈ [0, 1]. For the results in the main text, only the case of α = 1/2 will be used. However, the family of upper bounds derived here remain of general and independent interest, as they provide different insights intoσ(G) for different values of α. Let ω satisfy Observe that A T − A is skew symmetric. Then, i(A T − A) is a Hermitian matrix, and it features only real eigenvalues that are symmetric with respect to the origin. Namely, if µ is an eigenvalue of i(A T − A), so is −µ. This implies that the maximum and the minimum eigenvalues of i( where Notice that Similarly, Then, Consequently, we have where σ min (A) is the minimum singular value of A. Notice now that r(A) ≤ σ min (A). Thus, Finally, for each value of α we obtain , Substituting (9) into equation (8) yields .
If A is symmetric, then it is more convenient to choose α = 1, which yields

The role of the non-normality degree of A
In this section we assume that the matrix A is diagonalizable, and characterize the role of the non-normality degree of A with respect to its fragility and responsiveness. Observe that Observe that s(A) represents the distance of the eigenvalues of A from the instability region.
On the other hand, κ(V ) is instead related the sensitivity of the eigenvalues of A to possible perturbations 13 . Thus, both the distance of the eigenvalues of A from the imaginary axis as well as their sensitivity to perturbations contribute to the fragility degree of a network.
3 Numerical studies

Ecological networks
An ecological dynamical network is described by the following set of differential equations 6 : where n denotes the number of species, x i (t) is the density of the species i, and c i and M ij are network parameters that regulate the interaction rates among the species. The network (10) can be written in vector form as where diag(x) is the diagonal matrix defined by the species vector x, c is the vector of c i , and M is the matrix of the coefficients M ij . Let x * ∈ R n + an equilibrium point of (11). Then, either x * = 0, which corresponds to the case where all species are extinct, or x * solves the equations c = −M x * . The stability of an equilibrium point x * can be assessed through the linearized system where A = diag(x * )M is the Jacobian matrix of (11) at the point x * .
An ecological network is called mutualistic if the species can be divided into two classes, where the species of each class benefit from the species in the other class. In a mutualistic network, the matrix M can be partitioned as where the matrices M P P and M AA have non-positive entries, while the matrices M P A and M AP have non-negative entries. In Figure 1 in the main text we consider a three-dimensional network of two species of plants x 1 and x 2 and one species of animals x 3 . Figures 1(a) and 1(b) highlight the difference between the dynamics of a stable and an unstable equilibrium: in both cases the three states are at equilibrium until time t = 10, when they are slightly perturbed by a vector ε, with ε = 0.1. Figures 1(c) and 1(d), instead, highlight the difference between a robust and a fragile system. The state is at equilibrium until time t = 10, when a slight variation of the parameters changes M into M + ∆, with ∆ = 0.01. The parameters used to obtain Figure 1 are below: Parameters of Figure 1

Neuronal networks
Following 8 a network of neurons can be modelled by the differential equation where x(t) is the vector of spiking rates of the neurons, e(t) is the column vector with the external inputs, τ is the time constant of the neurons, and the matrix M describes the strength of connections among neurons. Because each neuron can be either excitatory or inhibitory, then the matrix M obeys Dale's law, namely, its columns are either non-negative or non-positive. This implies that x(t) and M can be partitioned as follows where x E (t) and x I (t) contain the states of the excitatory and inhibitory neurons, respectively, and the matrices M EE , M EI , M IE and M II are non-negative. We follow the algorithm in 9 to construct a sequence of matrices M that obey Dale's law and tend to minimize the value of s such that ∞ 0 e (M −sI) T t e (M −sI)t dt = 1 ( -smoothed spectral abscissa 10 ). We refer interested reader to 9 for a detailed description of this algorithm. To generate Figure 5 (b), we consider a network of dimension n = 100 and n E = n I = 50. Let M (k) be the coupling matrix at the k-th iteration of the algorithm in 9 , and let A (k) = (M (k) − I)/τ . We then compute the controllability Gramian G (k) with B = I. Figure 5 (b) in the main text shows the relationship between the stability radius r(A (k) ) and the average singularσ(G (k) ).

Traffic Networks
Following 11 , a traffic network where vehicles drive as an aligned platoon is described by the equations where p i and v i are the position and the velocity of the i-th vehicle i ∈ {1, . . . , n − 1}, respectively, tanh is the hyperbolic tangent function, and u i is an external input. We assume that the n-th vehicle plays the role of leader, whose velocity is constant and equal to α, and whose position enters as external input to the system (16). When all vehicles also move with velocity α, the system (16) read as whose solution is In order to analyze the dynamics of system (16) in the neighborhood of the particular trajectory (18), we linearize the nonlinear system (16) around the trajectory (18). Let us define δ i (t) = p i (t)−p i (t), and consider x = δ 1 , d dt δ 1 , . . . , δ n−1 , d dt δ n−1 T and u = [u 1 , · · · , u n−1 , δ n ] T as the state and input vectors of the linearized system. Then, where the matrices A ∈ R 2(n−1)×2(n−1) and B ∈ R 2(n−1)×n are defined as

Networks obtained from discretization of the wave equation
Consider the first-order wave equation 12 with z ∈ (−1, 1) and boundary values w(1, t) = 0 for all t ≥ 0. We discretize (19) using a regular grid and a centered difference scheme for the spatial coordinate. This yields, and Eq. (19) becomes , and where we have used w(t, 1) = w(t, −1 + N ∆z) = 0 and w(t, −1) = w(t, −1 + ∆z) at all times. We then discretize the temporal coordinate using the third-order Adams-Bashforth formula and obtain v(k + 1) ≈ v(k) + ∆t 12 where v(k) = w(k∆t) ∈ R N −1 . Finally, letting we obtain x(k + 1) = Ax(k), where δ = ∆t ∆x and Finally, we add a control input and use the following equations to evaluate the network controllability Gramian, and its eigenvalues as a function of the parameters N and δ. Figure 6(a) in the main text shows the fragility versus responsiveness tradeoff for the discrete-time network (22) for the value of δ that ranges from 0.1 to 0.7. Figure 6(b) shows the condition number of the network matrix, as a function of δ. It can be seen, the smaller δ, the larger the non-normality and fragility degrees of the network.