Universal transient behavior in large dynamical systems on networks

TL;DR

Analytical results reveal that unidirectional systems exhibit a high degree of universality in the sense that the average squared norm only depends on a single parameter encoding the average interaction strength between the individual constituents.

Abstract

We analyze how the transient dynamics of large dynamical systems in the vicinity of a stationary point, modeled by a set of randomly coupled linear differential equations, depends on the network topology. We characterize the transient response of a system through the evolution in time of the squared norm of the state vector, which is averaged over different realizations of the initial perturbation. We develop a mathematical formalism that computes this quantity for graphs that are locally tree-like. We show that for unidirectional networks the theory simplifies and general analytical results can be derived. For example, we derive analytical expressions for the average squared norm for random directed graphs with a prescribed degree distribution. These analytical results reveal that unidirectional systems exhibit a high degree of universality in the sense that the average squared norm only depends on a single parameter encoding the average interaction strength between the individual constituents. In addition, we derive analytical expressions for the average squared norm for unidirectional systems with fixed diagonal disorder and with bimodal diagonal disorder. We illustrate these results with numerical experiments on large random graphs and on real-world networks.

Figures (7)

• Figure 1: $S_N(t) =\overline{\langle |\bm y(t)|^2\rangle}$ for weighted oriented graphs with Poissonian connectivity with mean degree $c=2$, with fixed diagonal (Top Row) and bimodal diagonal disorder (Bottom Row). The theoretical result for $S(t)$ is provided in black solid line (see Eqs. \ref{['besseldiag']} and \ref{['StBimodal']}, respectively). Symbols denote numerical solution of the differential equation \ref{['dynamical1']} for $N=5000$, averaged over $25$ initial conditions and $5$ realizations of the underlying graph. Red circles stand for Gaussian bond disorder, blue triangles for uniform bond disorder, and green squares for Laplace-distributed disorder. The parameters for different panels are described below. We show schematically in the insets the location of the continuous part of the spectrum and the outlier (if present), according to Eqs. \ref{['boundary']} and \ref{['outlier']}. Top row: Fixed diagonal at $-\mu=-5$. (a) $\overline{J}=2$ and $\overline{J^2}=5$, (b) $\overline{J}=3$ and $\overline{J^2}=10$, and (c) $\overline{J}=4$ and $\overline{J^2}=32$. In panels (b) and (c), the red dashed curves represent $\tilde{S}(t)$ with fitted values of parameters $a=7.7\cdot 10^{-4}$, $b=1.04$ and $a=4.2\cdot 10^{-3}$, $b=4.03$, respectively. Eq. \ref{['boundary']} simplifies in this case as $r^2=c\overline{J^2}=|\lambda_{\rm b}-\mu|^2$ and $\lambda_{\mathrm{isol}}=c\overline{J}-\mu$. Bottom row: Diagonal entries taken at random between $-\mu_1=-5$ and $-\mu_2=-14$ with equal probability ($q=1/2$). (a) $\overline{J}=2$ and $\overline{J^2}=5$, (b) $\overline{J}=4$ and $\overline{J^2}=17$, and (c) $\overline{J}=4$ and $\overline{J^2}=32$. In panels (b) and (c), the red dashed curves represent $\tilde{S}(t)$ with fitted values of parameters $a=1.1\cdot 10^{-3}$, $b=0.6$ and $a=1.8\cdot 10^{-4}$, $b=4.0$, respectively.
• Figure 2: $S_N(t)=\overline{\langle |\bm y(t)|^2\rangle}$ for the simulated dynamics of Eq. \ref{['dynamical1']} on two examples of real-world graphs (circles for a food web with $N=180$ and $c=10.7$, and stars for a signaling network with $N=2288$ and $c=13.3$) are compared with the theoretical prediction $S(t)$ (blue and red lines), given by Eq. \ref{['besseldiag']}, as well as with, $e^{2t(r-\mu)}$ and $e^{-2\mu t}$ (black and magenta lines). We have weighted the networks with couplings $J_{ij}$ that are i.i.d. random variables drawn from the distribution $p_J(x) = (1/2) \delta(x-1) +(1/2) \delta(x+1)$ and we have used a single realization of the matrix $J$ for both the food web and the signaling network. The diagonal elements are fixed to a constant $-\mu = -r+0.27$, such that the system is asymptotically unstable. Insets (a) and (b) show the spectra of the adjacency matrices $A$ for the food web and the signaling network considered with random couplings. The red circle has radius $r = \sqrt{c}$ and is the predicted boundary of the spectrum according to \ref{['boundary']}. We have estimated $S_N(t)=\overline{\langle |\bm y(t)|^2\rangle}$ by simulating the dynamics on the generated networks for $25$ realizations of the initial condition $\bm y(0)$.
• Figure 3: Markers denote numerical results for $S_N(t) =\overline{\langle |\bm y(t)|^2\rangle}$ for model \ref{['dynamical1']} with matrix \ref{['eq:x']} as a function of $t$ and for different values of $N$. The $X_{ij}$ are i.i.d. random variables taken from a Gaussian distribution with zero mean and variance $1/N$ and we have set $\mu=-1$. Dashed black line is the curve $S(t)=e^{-2\mu t}I_0(2\rho t)$ (see Eq. \ref{['Sfullyconnected']}). Markers are averages over $2000$ matrix samples with $25$ realizations of the initial conditions for each sample.
• Figure 4: Numerical results for the crossover time $t^\star$ as a function of $N$ for the four considered models: (i) Ginibre ensemble, (ii) Gaussian Orthogonal Ensemble, (iii) Ginibre ensemble with an outlier, (iv) sparse random graph with Poissonian connectivity, mean degree $c=2$ and Gaussian bond disorder. Solid lines denote fitted functions to the empirical data. In cases (i) and (ii) we have set $\mu$ such that the system is at the edge of stability (in the $N\to\infty$ limit, the average of the leading eigenvalue $\overline{\lambda_1}=0$), and therefore $S(t)$ decays asymptotically as $t^{-1/2}$. The parameters in (iii) and (iv) are set such that the system is transiently stable but asymptotically unstable, as in the case (b) of the top panel of Fig. \ref{['Fig1']}. In cases (i)-(iii), markers are averages over $2000$ matrix samples with $25$ realizations of the initial conditions each. In case (iv), we have used $20$ matrix samples with $25$ realizations each.
• Figure 5: Dependence of $t^\star$ on the system size for the normal Ginibre ensemble, see Appendix \ref{['normalgin']}. Points are the numerical solution of Eq. \ref{['trans']}. The blue solid line is obtained by fitting the model $t^\star\approx \alpha N^{\beta}$ with 2 free parameters.