Universal transient behavior in large dynamical systems on networks

Tarnowski, Wojciech; Neri, Izaak; Vivo, Pierpaolo

doi:10.1103/physrevresearch.2.023333

Cited by 32 publications

(37 citation statements)

References 89 publications

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“…Lastly, we remark that the subleading eigenvalue, and its associated right (left) eigenvector, provide not only information about the asymptotic stability of large dynamical systems, but also about their response to random perturbations as shown in Ref. [46]. Taken together, we conclude that the spectral theory presented in this paper can be used in various contexts.…”

Section: Discussionsupporting

confidence: 64%

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Linear stability analysis of large dynamical systems on random directed graphs

Neri

Metz

2020

Phys. Rev. Research

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We present a linear stability analysis of stationary states (or fixed points) in large dynamical systems defined on random, directed graphs with a prescribed distribution of indegrees and outdegrees. We obtain two remarkable results for such dynamical systems. First, infinitely large systems on directed graphs can be stable even when the degree distribution has unbounded support; this result is surprising since their counterparts on nondirected graphs are unstable when system size is large enough. Second, we show that the phase transition between the stable and unstable phase is universal in the sense that it depends only on a few parameters, such as, the mean degree and a degree correlation coefficient. In addition, in the unstable regime, we characterize the nature of the destabilizing mode, which also exhibits universal features. These results follow from an exact theory for the leading eigenvalue of infinitely large graphs that are locally treelike and oriented, as well as, the right and left eigenvectors associated with the leading eigenvalue. We corroborate analytical results for infinitely large graphs with numerical experiments on random graphs of finite size. We discuss how the presented theory can be extended to graphs with diagonal disorder and to graphs that contain nondirected links. Finally, we discuss the influence of cycles and how they can destabilize large dynamical systems when they induce strong feedback loops.

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Section: Discussionsupporting

confidence: 64%

“…[39][40][41], and recently also spectral properties of random, directed graphs have been studied, see Refs. [42][43][44][45][46], but the properties of the leading eigenvalue of the adjacency matrices of random, directed graphs have not been studied so far.…”

Section: Introductionmentioning

confidence: 99%

Linear stability analysis of large dynamical systems on random directed graphs

Neri

Metz

2020

Phys. Rev. Research

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“…Here it is necessary to mention that the interest in statistical properties of the overlap matrix O kl and related objects extends much beyond the issues of eigenvalue stability under perturbation and is driven by numerous applications in theoretical and mathematical physics. In particular, non-orthogonality governs transient dynamics in complex systems [30,32,40] (see also [16,34]), analysis of spectral outliers in non-selfadjoint matrices [36], and, last but not least, the description of the Dyson Brownian motion for non-normal matrices [5,6,31]. Another steady source of interest in the statistics of eigenvector overlaps is due to its role in chaotic wave scattering.…”

Section: Gin2mentioning

confidence: 99%

Condition Numbers for Real Eigenvalues in the Real Elliptic Gaussian Ensemble

Fyodorov

Tarnowski

2020

Ann. Henri Poincaré

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We study the distribution of the eigenvalue condition numbers $$\kappa _i=\sqrt{ ({\mathbf{l}}_i^* {\mathbf{l}}_i)({\mathbf{r}}_i^* {\mathbf{r}}_i)}$$ κ i = ( l i ∗ l i ) ( r i ∗ r i ) associated with real eigenvalues $$\lambda _i$$ λ i of partially asymmetric $$N\times N$$ N × N random matrices from the real Elliptic Gaussian ensemble. The large values of $$\kappa _i$$ κ i signal the non-orthogonality of the (bi-orthogonal) set of left $${\mathbf{l}}_i$$ l i and right $${\mathbf{r}}_i$$ r i eigenvectors and enhanced sensitivity of the associated eigenvalues against perturbations of the matrix entries. We derive the general finite N expression for the joint density function (JDF) $${{\mathcal {P}}}_N(z,t)$$ P N ( z , t ) of $$t=\kappa _i^2-1$$ t = κ i 2 - 1 and $$\lambda _i$$ λ i taking value z, and investigate its several scaling regimes in the limit $$N\rightarrow \infty $$ N → ∞ . When the degree of asymmetry is fixed as $$N\rightarrow \infty $$ N → ∞ , the number of real eigenvalues is $$\mathcal {O}(\sqrt{N})$$ O ( N ) , and in the bulk of the real spectrum $$t_i=\mathcal {O}(N)$$ t i = O ( N ) , while on approaching the spectral edges the non-orthogonality is weaker: $$t_i=\mathcal {O}(\sqrt{N})$$ t i = O ( N ) . In both cases the corresponding JDFs, after appropriate rescaling, coincide with those found in the earlier studied case of fully asymmetric (Ginibre) matrices. A different regime of weak asymmetry arises when a finite fraction of N eigenvalues remain real as $$N\rightarrow \infty $$ N → ∞ . In such a regime eigenvectors are weakly non-orthogonal, $$t=\mathcal {O}(1)$$ t = O ( 1 ) , and we derive the associated JDF, finding that the characteristic tail $${{\mathcal {P}}}(z,t)\sim t^{-2}$$ P ( z , t ) ∼ t - 2 survives for arbitrary weak asymmetry. As such, it is the most robust feature of the condition number density for real eigenvalues of asymmetric matrices.

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“…Even when a fixed point is linearly stable in a nonlinear system described by ordinary differential equations, if the corresponding Jacobian matrix is non-normal, a small but finite perturbation can transiently grow beyond the validity of the linear approximation and enter into the nonlinear regime, preventing the perturbation from decaying to zero. The discovery of this phenomenon has led to the thorough study of the spectral properties of non-normal matrices in the context of transient dynamics [2]; it has also inspired recent works on implications of non-normality for network and spatiotemporal dynamics [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Given the common perception that linear dynamics are fully understood, the possibility of such transient growth offers interesting alternative interpretations for behavior usually attributed to nonlinearity, such as ignition dynamics in combustion and temporary activation of biochemical signals.…”

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confidence: 99%

Non-normality and non-monotonic dynamics in complex reaction networks

Nicolaou

Nishikawa

Nicholson

et al. 2020

Phys. Rev. Research

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Complex chemical reaction networks, which underlie many industrial and biological processes, often exhibit non-monotonic changes in chemical species concentrations. Such non-monotonic dynamics are in principle possible even in a linear model if the matrix defining the model is non-normal, as characterized by a necessarily non-orthogonal set of eigenvectors. However, the extent to which non-normality is responsible for non-monotonic behavior remains an open question. Here, using a master equation to model the reaction dynamics, we derive a general condition for observing non-monotonic dynamics of individual species, establishing that non-normality promotes non-monotonicity but is not a requirement for it. In contrast, we show that non-normality is a requirement for non-monotonic dynamics to be observed in the Rényi entropy. Using hydrogen combustion as an example application, we demonstrate that non-monotonic dynamics under experimental conditions are supported by a linear chain of connected components, at variance with the dominance of a single giant component observed in typical random reaction networks. The exact linearity of the master equation enables development of a rigorous theory and simulations for dynamical networks of unprecedented sizes (approaching 10 5 dynamical variables, even for a network of only 20 reactions and involving less than 100 atoms). Our conclusions are expected to hold for other combustion processes, and the general theory we develop is applicable to all chemical reaction networks, including biological ones.

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Universal transient behavior in large dynamical systems on networks

Cited by 32 publications

References 89 publications

Linear stability analysis of large dynamical systems on random directed graphs

Linear stability analysis of large dynamical systems on random directed graphs

Condition Numbers for Real Eigenvalues in the Real Elliptic Gaussian Ensemble

Non-normality and non-monotonic dynamics in complex reaction networks

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